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• 301.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Generalized axiomatic scale-space theory2013Ingår i: Advances in Imaging and Electron Physics, Vol 178 / [ed] P. Hawkes, Elsevier , 2013, Vol. 178, s. 1-96Kapitel i bok, del av antologi (Refereegranskat)

A fundamental problem in vision is what types of image operations should be used at the first stages of visual processing. I discuss a principled approach to this problem by describing a generalized axiomatic scale-space theory that makes it possible to derive the notions of linear scale-space, affine Gaussian scale-space, and linear spatio-temporal scale-space using similar sets of assumptions (scale-space axioms).

Based on a requirement that new image structures should not be created with increasing scale formalized into a condition of non-enhancement of local extrema, a complete classification is given of the linear (Gaussian) scale-space concepts that satisfy these conditions on isotropic spatial, non-isotropic spatial, and spatio-temporal domains, which results in a general taxonomy of Gaussian scale-spaces for continuous image data. The resulting theory allows filter shapes to be tuned from specific context information and provides a theoretical foundation for the recently exploited mechanisms of affine shape adaptation and Galilean velocity adaptation, with highly useful applications in computer vision. It is also shown how time-causal and time-recursive spatio-temporal scale-space concepts can be derived from similar or closely related assumptions.

The receptive fields arising from the spatial, spatio-chromatic, and spatio-temporal derivatives resulting from these scale-space concepts can be used as a general basis for expressing image operations for a large class of computer vision or image analysis methods. The receptive field profiles generated by necessity from these theories also have close similarities to receptive fields measured by cell recordings in biological vision, specifically regarding space-time separable cells in the retina and the lateral geniculate nucleus (LGN), as well as both space-time separable and non-separable cells in the striate cortex (V1) of higher mammals.

• 302.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Generalized Gaussian Scale-Space Axiomatics Comprising Linear Scale-Space, Affine Scale-Space and Spatio-Temporal Scale-Space2011Ingår i: Journal of Mathematical Imaging and Vision, ISSN 0924-9907, E-ISSN 1573-7683, Vol. 40, nr 1, s. 36-81Artikel i tidskrift (Refereegranskat)

This paper describes a generalized axiomatic scale-space theory that makes it possible to derive the notions of linear scale-space, affine Gaussian scale-space and linear spatio-temporal scale-space using a similar set of assumptions (scale-space axioms). The notion of non-enhancement of local extrema is generalized from previous application over discrete and rotationally symmetric kernels to continuous and more general non-isotropic kernels over both spatial and spatio-temporal image domains. It is shown how a complete classification can be given of the linear (Gaussian) scale-space concepts that satisfy these conditions on isotropic spatial, non-isotropic spatial and spatio-temporal domains, which results in a general taxonomy of Gaussian scale-spaces for continuous image data. The resulting theory allows filter shapes to be tuned from specific context information and provides a theoretical foundation for the recently exploited mechanisms of shape adaptation and velocity adaptation, with highly useful applications in computer vision. It is also shown how time-causal spatio-temporal scale-spaces can be derived from similar assumptions. The mathematical structure of these scale-spaces is analyzed in detail concerning transformation properties over space and time, the temporal cascade structure they satisfy over time as well as properties of the resulting multi-scale spatio-temporal derivative operators. It is also shown how temporal derivatives with respect to transformed time can be defined, leading to the formulation of a novel analogue of scale normalized derivatives for time-causal scale-spaces. The kernels generated from these two types of theories have interesting relations to biological vision. We show how filter kernels generated from the Gaussian spatio-temporal scale-space as well as the time-causal spatio-temporal scale-space relate to spatio-temporal receptive field profiles registered from mammalian vision. Specifically, we show that there are close analogies to space-time separable cells in the LGN as well as to both space-time separable and non-separable cells in the striate cortex. We do also present a set of plausible models for complex cells using extended quasi-quadrature measures expressed in terms of scale normalized spatio-temporal derivatives. The theories presented as well as their relations to biological vision show that it is possible to describe a general set of Gaussian and/or time-causal scale-spaces using a unified framework, which generalizes and complements previously presented scale-space formulations in this area.

• 303.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Image matching using generalized scale-space interest points2015Ingår i: Journal of Mathematical Imaging and Vision, ISSN 0924-9907, E-ISSN 1573-7683, Vol. 52, nr 1, s. 3-36Artikel i tidskrift (Refereegranskat)

The performance of matching and object recognition methods based on interest points depends on both the properties of the underlying interest points and the choice of associated image descriptors. This paper demonstrates advantages of using generalized scale-space interest point detectors in this context for selecting a sparse set of points for computing image descriptors for image-based matching.

For detecting interest points at any given scale, we make use of the Laplacian, the determinant of the Hessian and four new unsigned or signed Hessian feature strength measures, which are defined by generalizing the definitions of the Harris and Shi-and-Tomasi operators from the second moment matrix to the Hessian matrix. Then, feature selection over different scales is performed either by scale selection from local extrema over scale of scale-normalized derivates or by linking features over scale into feature trajectories and computing a significance measure from an integrated measure of normalized feature strength over scale.

A theoretical analysis is presented of the robustness of the differential entities underlying these interest points under image deformations, in terms of invariance properties under affine image deformations or approximations thereof. Disregarding the effect of the rotationally symmetric scale-space smoothing operation, the determinant of the Hessian is a truly affine covariant differential entity and two of the new Hessian feature strength measures have a major contribution from the affine covariant determinant of the Hessian, implying that local extrema of these differential entities will bemore robust under affine image deformations than local extrema of the Laplacian operator or the two other new Hessian feature strength measures.

It is shown how these generalized scale-space interest points allow for a higher ratio of correct matches and a lower ratio of false matches compared to previously known interest point detectors within the same class. The best results are obtained using interest points computed with scale linking and with the new Hessian feature strength measures and the determinant of the Hessian being the differential entities that lead to the best matching performance under perspective image transformations with significant foreshortening, and better than the more commonly used Laplacian operator, its difference-of-Gaussians approximation or the Harris-Laplace operator.

We propose that these generalized scale-space interest points, when accompanied by associated local scale-invariant image descriptors, should allow for better performance of interest point based methods for image-based matching, object recognition and related visual tasks.

• 304.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Image matching using generalized scale-space interest points2013Ingår i: Scale Space and Variational Methods in Computer Vision: 4th International Conference, SSVM 2013, Schloss Seggau, Leibnitz, Austria, , June 2-6, 2013, Proceedings / [ed] A. Kuijper et al, Springer Berlin/Heidelberg, 2013, Vol. 7893, s. 355-367Konferensbidrag (Refereegranskat)

The performance of matching and object recognition methods based on interest points depends on both the properties of the underlying interest points and the associated image descriptors. This paper demonstrates the advantages of using generalized scale-space interest point detectors when computing image descriptors for image-based matching. These generalized scale-space interest points are based on linking of image features over scale and scale selection by weighted averaging along feature trajectories over scale and allow for a higher ratio of correct matches and a lower ratio of false matches compared to previously known interest point detectors within the same class. Specifically, it is shown how a significant increase in matching performance can be obtained in relation to the underlying interest point detectors in the SIFT and the SURF operators. We propose that these generalized scale-space interest points when accompanied by associated scale-invariant image descriptors should allow for better performance of interest point based methods for image-based matching, object recognition and related vision tasks.

• 305.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Invariance of visual operations at the level of receptive fields2013Konferensbidrag (Refereegranskat)

Receptive field profiles measured by cell recordings have shown that mammalian vision has developed receptive fields tuned to different sizes and orientations in the image domain as well as to different image velocities in space-time [1, 2]. This article presents a theory by which families of idealized receptive field profiles can be derived mathematically from a small set of basic assumptions that correspond to structural properties of the environment [3, 4]. The article also presents a theory for how basic invariance properties to variations in scale, viewing direction and relative motion can be obtained from the output of such receptive fields, using complementary selection mechanisms that operate over the output of families of receptive fields tuned to different parameters [4]. Thereby, the theory shows how basic invariance properties of a visual system can be obtained already at the level of receptive fields, and we can explain the different shapes of receptive field profiles found in biological vision from a requirement that the visual system should be invariant to the natural types of image transformations that occur in its environment.

Model.

The brain is able to maintain a stable perception although the visual stimuli vary substantially on the retina due to geometric transformations and lighting variations in the environment. These transformations comprise (i) local scaling transformations caused by objects of different size and at different distances to the observer, (ii) locally linearized image deformations caused by variations in the viewing direction in relation to the object, (iii) locally linearized relative motions between the object and the observer and (iv) local multiplicative intensity transformations caused by illumination variations. Let us assume that receptive fields should be constructed by linear operations that are shift-invariant over space and/or space-time, with an additional requirement that receptive fields must not create new image structures at coarser scales that do not correspond to simplifications of corresponding structures at finer scales.

Results.

Given the above structural conditions, we derive idealized families of spatial and spatio-temporal receptive fields that satisfy these structural requirements by necessity, based on Gaussian kernels, Gaussian derivatives or closely related operators [3, 4].  We show that there are very close similarities between the receptive fields predicted from this theory and receptive fields found by cell recordings in biological vision, including (i) spatial on-center-off-surround and off-center-on-surround receptive fields in the fovea and the LGN, (ii) simple cells with spatial directional preference in V1, (iii) space-time separable spatio-temporal receptive fields in the LGN and V1 and (iv) non-separable space-time tilted receptive fields in V1 [3, 4]. Indeed, from kernels predicted by this theory it is possible to generate receptive fields similar to all the basic types of monocular receptive fields reported by DeAngelis et al [2] in their survey of classical receptive fields.

By complementing such receptive field measurements with selection mechanisms over the parameters in the receptive field families, we show how true invariance of receptive field responses can be obtained under scaling transformations, affine transformations and Galilean transformations [4]. Thereby, the framework provides a mathematically well-founded and biologically plausible model for how basic invariance properties can be achieved already at the level of receptive fields. In this way, the presented theory supports invariant recognition of objects and events under variations in viewpoint, retinal size, object motion and illumination.

References.

1. Hubel DH, Wiesel TN: Brain and Visual Perception, Oxford University Press, 2005.

2. DeAngelis GC, Anzai A: A modern view of the classical receptive field: Linear and non-linear spatio-temporal processing by V1 neurons. The Visual Neurosciences, MIT Press, vol 1, 705-719, 2004.

3. Lindeberg T: Generalized Gaussian scale-space axiomatics comprising linear scale-space, affine scale-space and spatio-temporal scale-space. J Math Imaging Vis, 2011, 40(1):36-81.

4. Lindeberg T: Invariance of visual operations at the level of receptive fields. PLOS One, in press, doi:10.1371/journal.pone.0066990, preprint available from arXiv:1210.0754.

• 306.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Invariance of visual operations at the level of receptive fields2013Ingår i: PLoS ONE, ISSN 1932-6203, E-ISSN 1932-6203, Vol. 8, nr 7, s. e66990-1-e66990-33Artikel i tidskrift (Refereegranskat)

The brain is able to maintain a stable perception although the visual stimuli vary substantially on the retina due to geometric transformations and lighting variations in the environment. This paper presents a theory for achieving basic invariance properties already at the level of receptive fields. Specifically, the presented framework comprises (i) local scaling transformations caused by objects of different size and at different distances to the observer, (ii) locally linearized image deformations caused by variations in the viewing direction in relation to the object, (iii) locally linearized relative motions between the object and the observer and (iv) local multiplicative intensity transformations caused by illumination variations. The receptive field model can be derived by necessity from symmetry properties of the environment and leads to predictions about receptive field profiles in good agreement with receptive field profiles measured by cell recordings in mammalian vision. Indeed, the receptive field profiles in the retina, LGN and V1 are close to ideal to what is motivated by the idealized requirements. By complementing receptive field measurements with selection mechanisms over the parameters in the receptive field families, it is shown how true invariance of receptive field responses can be obtained under scaling transformations, affine transformations and Galilean transformations. Thereby, the framework provides a mathematically well-founded and biologically plausible model for how basic invariance properties can be achieved already at the level of receptive fields and support invariant recognition of objects and events under variations in viewpoint, retinal size, object motion and illumination. The theory can explain the different shapes of receptive field profiles found in biological vision, which are tuned to different sizes and orientations in the image domain as well as to different image velocities in space-time, from a requirement that the visual system should be invariant to the natural types of image transformations that occur in its environment.

• 307.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Linear spatio-temporal scale-space1997Ingår i: Scale-Space Theory in Computer Vision: Proceedings of First International Conference, Scale-Space'97 Utrecht, The Netherlands, July 2–4, 1997, Springer Berlin/Heidelberg, 1997, Vol. 1252, s. 113-127Konferensbidrag (Refereegranskat)

This article shows how a linear scale-space formulation previously expressed for spatial domains extends to spatio-temporal data. Starting from the main assumptions that: (i) the scale-space should be generated by convolution with a semi-group of filter kernels and that (ii) local extrema must not be enhanced when the scale parameter increases, a complete taxonomy is given of the linear scale-space concepts that satisfy these conditions on spatial, temporal and spatio-temporal domains, including the cases with continuous as well as discrete data.

Key aspects captured by this theory include that: (i) time-causal scale-space kernels must not extend into the future, (ii) filter shapes can be tuned from specific context information, permitting mechanisms such local shifting, shape adaptation and velocity adaptation, all expressed in terms of local diffusion operations.

Receptive field profiles generated by the proposed theory show high qualitative similarities to receptive field profiles recorded from biological vision.

• 308.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
On Automatic Selection of Temporal Scales in Time-Casual Scale-Space1997Ingår i: Proceedings of the Algebraic Frames for the Perception-Action Cycle: AFPAC'97 (Kiel, Germany), 1997, Vol. 1315, s. 94-113Konferensbidrag (Refereegranskat)

This paper outlines a general framework for automatic selection in multi-scale representations of temporal and spatio-temporal data, A general principle for automatic scale selection based on local maxima of normalized differential entities is adapted to the temporal domain, and it is shown how the notion of normalized derivatives can be defined for three main types of (continuous and discrete) temporal scale-space representations. Closed-form analysis is carried out for basic model patterns, and shows how the suggested theory applies to motion detection and motion estimation.

• 309.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
On the axiomatic foundations of linear scale-space: Combining semi-group structure with causality vs. scale invariance1996Ingår i: Gaussian Scale-Space Theory: Proceedings of PhD School on Scale-Space (Copenhagen, Denmark) May 1996 / [ed] J. Sporring, M. Nielsen, L. Florack and P. Johansen, Kluwer Academic Publishers, 1996Kapitel i bok, del av antologi (Refereegranskat)

The notion of multi-scale representation is essential to many aspects of early visual processing. This article deals with the axiomatic formulation of the special type of multi-scale representation known as scale-space representation. Specifically, this work is concerned with the problem of how different choices of basic assumptions (scale-space axioms) restrict the class of permissible smoothing operations.

A scale-space formulation previously expressed for discrete signals is adapted to the continuous domain. The basic assumptions are that the scale-space family should be generated by convolution with a one-parameter family of rotationally symmetric smoothing kernels that satisfy a semi-group structure and obey a causality condition expressed as a non-enhancement requirement of local extrema. Under these assumptions, it is shown that the smoothing kernel is uniquely determined to be a Gaussian.

Relations between this scale scale-space formulation and recent formulations based on scale invariance are explained in detail. Connections are also pointed out to approaches based on non-uniform smoothing.

• 310.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
On the behaviour in scale-space of local extrema and blobs1991Ingår i: Theory and Applications of Image Analysis: Selected Papers from the 7th Scandinavian Conference on Image Analysis (Aalborg, Denmark, 1991) / [ed] P. Johansen and S. Olsen, World Scientific, 1991, s. 38-47Kapitel i bok, del av antologi (Refereegranskat)

We apply elementary techniques from real analysis and singularity theory to derive analytical results for the behaviour in scale-space of critical points and related entities. The main results of the treatment comprise:

• a description of the general nature of trajectories of critical points in scale-space.
• an estimation of the drift velocity of critical points and edges.
• an analysis of the qualitative behaviour of critical points in bifurcation situations.
• a classification of what types of blob bifurcations are possible.
• 311.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
On the behaviour in scale-space of local extrema and blobs1991Ingår i: Scandinavian Conference on Image Analysis: SCIA'91 (Aalborg, Denmark), 1991, s. 8-17Konferensbidrag (Refereegranskat)

We apply elementary techniques from real analysis and singularity theory to derive analytical results for the behaviour in scale-space of critical points and related entities. The main results of the treatment comprise:

• a description of the general nature of trajectories of critical points in scale-space.
• an estimation of the drift velocity of critical points and edges.
• an analysis of the qualitative behaviour of critical points in bifurcation situations.
• a classification of what types of blob bifurcations are possible.
• 312.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
On the Construction of a Scale-Space for Discrete Images1988Rapport (Övrigt vetenskapligt)

In this paper we address the formulation of a scale-space theory for discrete images. We denote a one-dimensional kernel a scale-space kernel if it reduces the number of local extrema and discuss which discrete kernels are possible scale-space kernels. Unimodality and positivity properties are shown to hold for such kernels as well as their Fourier transforms. An explicit expression characterizing all discrete scale-space kernels is given.

We propose that there is only one reasonable way to define a scale-space family of images L(x; t) for a one-dimensional discrete signal f(x) namely by convolution with the family of discrete kernels T(n; t) = e^(-t) I_nt(t) where I_n is the modified Bessel function of order n.

With this representation, comprising a continuous scale parameter, we are no longer restricted to specific predetermined levels of scale. Further, T(n; t) appears naturally in the solution of a discretized version of the heat equation, both in one and two dimensions.

The family T(n; t) (t >= 0) is the only one-parameter family of discrete symmetric shift-invariant kernels satisfying both necessary scale-space requirements and the semigroup property T(n; s) * T(n; t) = T(n; s+t). Similar arguments applied in the continuous case uniquely lead to the family of Gaussian kernels.

The commonly adapted technique with a sampled Gaussian produces undesirable effects. It is shown that scale-space violations might occur in the family of functions generated by convolution with the sampled Gaussian kernel. The result exemplifies that properties derived in the continuous case might be violated after discretization.

A discussion about the numerical implementation is performed and an algorithm generating the filter coefficients is supplied.

• 313.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Principles for Automatic Scale Selection1999Ingår i: Handbook on Computer Vision and Applications: volume II / [ed] Berndt Jähne, Academic Press, 1999, s. 239-274Kapitel i bok, del av antologi (Övrigt vetenskapligt)

An inherent property of objects in the world is that they only exist as meaningful entities over certain ranges of scale. If one aims at describing the structure of unknown real-world signals, then a multi-scale representation of data is of crucial importance. Whereas conventional scale-space theory provides a well-founded framework for dealing with image structures at different scales, this theory does not directly address the problem of how to selectappropriate scales for further analysis. This article outlines a systematic methodology of how mechanisms for automatic scale selection can be formulated in the problem domains of feature detection and image matching (flow estimation), respectively.

For feature detectors expressed in terms of Gaussian derivatives, hypotheses about interesting scale levels can be generated from scales at which normalized measures of feature strength assume local maxima with respect to scale. It is shown how the notion of $\gamma$-normalized derivatives arises by necessity given the requirement that the scale selection mechanism should commute with rescalings of the image pattern. Specifically, it is worked out in detail how feature detection algorithms with automatic scale selection can be formulated for the problems of edge detection, blob detection, junction detection, ridge detection and frequency estimation. A general property of this scheme is that the selected scale levels reflect the size of the image structures.

When estimating image deformations, such as in image matching and optic flow computations, scale levels with associated deformation estimates can be selected from the scales at which normalized measures of uncertainty assume local minima with respect to scales. It is shown how an integrated scale selection and flow estimation algorithm has the qualitative properties of leading to the selection of coarser scales for larger size image structures and increasing noise level, whereas it leads to the selection of finer scales in the neighbourhood of flow field discontinuities.

• 314.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Scale invariant feature transform2012Övrigt (Refereegranskat)

Scale Invariant Feature Transform (SIFT) is an image descriptor for image-based matching developed by David Lowe (1999,2004). This descriptor as well as related image descriptors are used for a large number of purposes in computer vision related to point matching between different views of a 3-D scene and view-based object recognition. The SIFT descriptor is invariant to translations, rotations and scaling transformations in the image domain and robust to moderate perspective transformations and illumination variations. Experimentally, the SIFT descriptor has been proven to be very useful in practice for robust image matching and object recognition under real-world conditions.

In its original formulation, the SIFT descriptor comprised a method for detecting interest points from a grey-level image at which statistics of local gradient directions of image intensities were accumulated to give a summarizing description of the local image structures in a local neighbourhood around each interest point, with the intention that this descriptor should be used for matching corresponding interest points between different images. Later, the SIFT descriptor has also been applied at dense grids (dense SIFT) which have been shown to lead to better performance for tasks such as object categorization and texture classification. The SIFT descriptor has also been extended from grey-level to colour images and from 2-D spatial images to 2+1-D spatio-temporal video.

• 315.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Scale selection2014Ingår i: Computer Vision: A Reference Guide / [ed] Katsushi Ikeuchi, Springer US , 2014, s. 701-713Kapitel i bok, del av antologi (Refereegranskat)

The notion of scale selection refers to methods for estimating characteristic scales in image data and for automatically determining locally appropriate scales in a scale-space representation, so as to adapt subsequent processing to the local image structure and compute scale invariant image features and image descriptors.

An essential aspect of the approach is that it allows for a bottom-up determination of inherent scales of features and objects without first recognizing them or delimiting alternatively segmenting them from their surrounding.

Scale selection methods have also been developed from other viewpoints of performing noise suppression and exploring top-down information.

• 316.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Scale Selection Properties of Generalized Scale-Space Interest Point Detectors2013Ingår i: Journal of Mathematical Imaging and Vision, ISSN 0924-9907, E-ISSN 1573-7683, Vol. 46, nr 2, s. 177-210Artikel i tidskrift (Refereegranskat)

Scale-invariant interest points have found several highly successful applications in computer vision, in particular for image-based matching and recognition. This paper presents a theoretical analysis of the scale selection properties of a generalized framework for detecting interest points from scale-space features presented in Lindeberg (Int. J. Comput. Vis. 2010, under revision) and comprising: an enriched set of differential interest operators at a fixed scale including the Laplacian operator, the determinant of the Hessian, the new Hessian feature strength measures I and II and the rescaled level curve curvature operator, as well as an enriched set of scale selection mechanisms including scale selection based on local extrema over scale, complementary post-smoothing after the computation of non-linear differential invariants and scale selection based on weighted averaging of scale values along feature trajectories over scale. A theoretical analysis of the sensitivity to affine image deformations is presented, and it is shown that the scale estimates obtained from the determinant of the Hessian operator are affine covariant for an anisotropic Gaussian blob model. Among the other purely second-order operators, the Hessian feature strength measure I has the lowest sensitivity to non-uniform scaling transformations, followed by the Laplacian operator and the Hessian feature strength measure II. The predictions from this theoretical analysis agree with experimental results of the repeatability properties of the different interest point detectors under affine and perspective transformations of real image data. A number of less complete results are derived for the level curve curvature operator.

• 317.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Scale-Space2009Ingår i: Wiley Encyclopedia of Computer Science and Engineering / [ed] Benjamin Wah, Hoboken, New Jersey: John Wiley & Sons, 2009, s. 2495-2504Kapitel i bok, del av antologi (Refereegranskat)

Scale-space theory is a framework for multiscale image representation, which has been developed by the computer vision community with complementary motivations from physics and biologic vision. The idea is to handle the multiscale nature of real-world objects, which implies that object may be perceived in different ways depending on the scale of observation. If one aims to develop automatic algorithms for interpreting images of unknown scenes, no way exists to know a priori what scales are relevant. Hence, the only reasonable approach is to consider representations at all scales simultaneously. From axiomatic derivations is has been shown that given the requirement that coarse-scale representations should correspond to true simplifications of fine scale structures, convolution with Gaussian kernels and Gaussian derivatives is singled out as a canonical class of image operators forthe earliest stages of visual processing. These image operators can be used as basis to solve a large variety of visual tasks, including feature detection, feature classification, stereo matching, motion descriptors, shape cues, and image-based recognition. By complementing scale-space representation with a module for automatic scale selection based on the maximization of normalized derivatives over scales, early visual modules can be made scale invariant. In this way, visual modules canadapt automatically to the unknown scale variations that may occur because of objects and substructures of varying physical size as well as objects with varying distances to the camera. An interesting similarity to biologic vision is that the scale-space operators resemble closely receptive field profiles registered in neurophysiologic studies of the mammalian retina and visual cortex.

• 318.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Scale-Space Behaviour and Invariance Properties of Differential Singularities1992Ingår i: Shape inPicture: Mathematical Description of Shape in Grey-Level Images: Proc. of Workshop in Driebergen, Netherlands, Sep. 7--11, 1992, Springer, 1992, s. 591-600Konferensbidrag (Refereegranskat)

This article describes how a certain way of expressing low-level feature detectors, in terms of singularities of differential expressions defined at multiple scales in scale-space, simplifies the analysis of the effect of smoothing. It is shown how such features can be related across scales, and generally valid expressions for drift velocities are derived with examples concerning edges, junctions, Laplacean zero-crossings, and blobs. A number of invariance properties are pointed out, and a particular representation defined from such singularities, the scale-space primal sketch, is treated in more detail.

• 319.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Scale-space for discrete images1989Ingår i: Scandinavian Conference on Image Analysis: SCIA'89 (Oulo, Finland), 1989, s. 1098-1107Konferensbidrag (Refereegranskat)

This article addresses the formulation of a scale-space theory for one-dimensional discrete images. Two main subjects are treated:

1. Which linear transformations remove structure in the sense that the number of local extrema (or zero-crossings) in the output image does not exceed the number of local extrema (or zero-crossings) in the original image?
2. How should one create a multi-resolution family of representations with the property that an image at a coarser level of scale never contains more structure than an image at a finer level of scale?

We propose that there is only one reasonable way to define a scale-space for discrete images comprising a continuous scale parameter, namely by (discrete) convolution with the family of kernels T(n; t) = e^{-t} I_n(t),, where $I_n$ are the modified Bessel functions of integer order. Similar arguments applied in the continuous case uniquely lead to the Gaussian kernel.

Some obvious discretizations of the continuous scale-space theory are discussed in view of the results presented. An important result is that scale-space violations might occur in the family of representations generated by discrete convolution with the sampled Gaussian kernel.

• 320.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Scale-space for discrete signals1990Ingår i: IEEE Transaction on Pattern Analysis and Machine Intelligence, ISSN 0162-8828, E-ISSN 1939-3539, Vol. 12, nr 3, s. 234-254Artikel i tidskrift (Refereegranskat)

This article addresses the formulation of a scale-space theory for discrete signals. In one dimension it is possible to characterize the smoothing transformations completely and an exhaustive treatment is given, answering the following two main questions:

• Which linear transformations remove structure in the sense that the number of local extrema (or zero-crossings) in the output signal does not exceed the number of local extrema (or zero-crossings) in the original signal?
• How should one create a multi-resolution family of representations with the property that a signal at a coarser level of scale never contains more structure than a signal at a finer level of scale?

It is proposed that there is only one reasonable way to define a scale-space for 1D discrete signals comprising a continuous scale parameter, namely by (discrete) convolution with the family of kernels T(n; t) = e^{-t} I_n(t), where I_n are the modified Bessel functions of integer order. Similar arguments applied in the continuous case uniquely lead to the Gaussian kernel.

Some obvious discretizations of the continuous scale-space theory are discussed in view of the results presented. It is shown that the kernel T(n; t) arises naturally in the solution of a discretized version of the diffusion equation. The commonly adapted technique with a sampled Gaussian can lead to undesirable effects since scale-space violations might occur in the corresponding representation. The result exemplifies the fact that properties derived in the continuous case might be violated after discretization.

A two-dimensional theory, showing how the scale-space should be constructed for images, is given based on the requirement that local extrema must not be enhanced, when the scale parameter is increased continuously. In the separable case the resulting scale-space representation can be calculated by separated convolution with the kernel T(n; t).

The presented discrete theory has computational advantages compared to a scale-space implementation based on the sampled Gaussian, for instance concerning the Laplacian of the Gaussian. The main reason is that the discrete nature of the implementation has been taken into account already in the theoretical formulation of the scale-space representation.

• 321.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Scale-Space for N-dimensional discrete signals1992Ingår i: Shape inPicture: Mathematical Description of Shape in Grey-Level Images: Proc. of Workshop in Driebergen, Netherlands, Sep. 7--11, 1992, Springer, 1992, s. 571-590Konferensbidrag (Refereegranskat)

This article shows how a (linear) scale-space representation can be defined for discrete signals of arbitrary dimension. The treatment is based upon the assumptions that (i) the scale-space representation should be defined by convolving the original signal with a one-parameter family of symmetric smoothing kernels possessing a semi-group property, and (ii) local extrema must not be enhanced when the scale parameter is increased continuously.

It is shown that given these requirements the scale-space representation must satisfy the differential equation \partial_t L = A L for some linear and shift invariant operator A satisfying locality, positivity, zero sum, and symmetry conditions. Examples in one, two, and three dimensions illustrate that this corresponds to natural semi-discretizations of the continuous (second-order) diffusion equation using different discrete approximations of the Laplacean operator. In a special case the multi-dimensional representation is given by convolution with the one-dimensional discrete analogue of the Gaussian kernel along each dimension.

• 322.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Scale-space theory2001Ingår i: Encyclopaedia of Mathematics / [ed] Michiel Hazewinkel, Springer , 2001Kapitel i bok, del av antologi (Refereegranskat)

# Scale-space theory

A theory of multi-scale representation of sensory data developed by the image processing and computer vision communities. The purpose is to represent signals at multiple scales in such a way that fine scale structures are successively suppressed, and a scale parameter  is associated with each level in the multi-scale representation.

For a given signal , a linear scale-space representation is a family of derived signals , defined by  and

for some family  of convolution kernels [a1], [a2] (cf. also Integral equation of convolution type). An essential requirement on the scale-space family  is that the representation at a coarse scale constitutes a simplification of the representations at finer scales. Several different ways of formalizing this requirement about non-creation of new structures with increasing scales show that the Gaussian kernel

constitutes a canonical choice for generating a scale-space representation [a3], [a4], [a5], [a6]. Equivalently, the scale-space family satisfies the diffusion equation

The motivation for generating a scale-space representation of a given data set originates from the basic fact that real-world objects are composed of different structures at different scales and may appear in different ways depending on the scale of observation. For example, the concept of a "tree" is appropriate at the scale of meters, while concepts such as leaves and molecules are more appropriate at finer scales. For a machine vision system analyzing an unknown scene, there is no way to know what scales are appropriate for describing the data. Thus, the only reasonable approach is to consider descriptions at all scales simultaneously [a1], [a2].

From the scale-space representation, at any level of scale one can define scale-space derivatives by

where  and  constitute multi-index notation for the derivative operator . Such Gaussian derivative operators provide a compact way to characterize the local image structure around a certain image point at any scale. Specifically, the output from scale-space derivatives can be combined into multi-scale differential invariants, to serve as feature detectors (see Edge detection and Corner detection for two examples).

More generally, a scale-space representation with its Gaussian derivative operators can serve as a basis for expressing a large number of early visual operations, including feature detection, stereo matching, computation of motion descriptors and the computation of cues to surface shape [a3], [a4]. Neuro-physiological studies have shown that there are receptive field profiles in the mammalian retina and visual cortex, which can be well modeled by the scale-space framework [a7].

Pyramid representation [a8] is a predecessor to scale-space representation, constructed by simultaneously smoothing and subsampling a given signal. In this way, computationally highly efficient algorithms can be obtained. A problem noted with pyramid representations, however, is that it is usually algorithmically hard to relate structures at different scales, due to the discrete nature of the scale levels. In a scale-space representation, the existence of a continuous scale parameter makes it conceptually much easier to express this deep structure [a2]. For features defined as zero-crossings of differential invariants, the implicit function theorem (cf. Implicit function) directly defines trajectories across scales, and at those scales where a bifurcation occurs, the local behaviour can be modeled by singularity theory [a3], [a5].

Extensions of linear scale-space theory concern the formulation of non-linear scale-space concepts more committed to specific purposes [a9]. There are strong relations between scale-space theory and wavelet theory (cf. also Wavelet analysis), although these two notions of multi-scale representation have been developed from slightly different premises.

References

[a1] A.P. Witkin, "Scale-space filtering" , Proc. 8th Internat. Joint Conf. Art. Intell. Karlsruhe, West Germany Aug. 1983 (1983) pp. 1019–1022

[a2] J.J. Koenderink, "The structure of images" Biological Cybernetics , 50 (1984) pp. 363–370

[a3] T. Lindeberg, "Scale-space theory in computer vision" , Kluwer Acad. Publ. (1994)

[a4] L.M.J. Florack, "Image structure" , Kluwer Acad. Publ. (1997)[a5]J. Sporring, et al., "Gaussian scale-space theory" , Kluwer Acad. Publ. (1997)

[a6] B.M ter Haar Romeny, et al., "Proc. First Internat. Conf. scale-space" , Lecture Notes Computer Science , 1252 , Springer (1997)

[a7] R.A. Young, "The Gaussian derivative model for spatial vision: Retinal mechanisms" Spatial Vision , 2 (1987) pp. 273–293

[a8] P.J. Burt, E.H. Adelson, "The Laplacian Pyramid as a Compact Image Code" IEEE Trans. Commun. , 9 : 4 (1983) pp. 532–540

[a9] "Geometry-driven diffusion in computer vision" B.M ter Haar Romeny (ed.) , Kluwer Acad. Publ. (1994)

• 323.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Scale-Space Theory: A Basic Tool for Analysing Structures at Different Scales1994Ingår i: Journal of Applied Statistics, ISSN 0266-4763, E-ISSN 1360-0532, Vol. 21, s. 225-270Artikel i tidskrift (Refereegranskat)

An inherent property of objects in the world is that they only exist as meaningful entities over certain ranges of scale. If one aims at describing the structure of unknown real-world signals, then a multi-scale representation of data is of crucial importance.

This article gives a tutorial review of a special type of multi-scale representation, linear scale-space representation, which has been developed by the computer vision community in order to handle image structures at different scales in a consistent manner. The basic idea is to embed the original signal into a one-parameter family of gradually smoothed signals, in which the fine scale details are successively suppressed.

Under rather general conditions on the type of computations that are to performed at the first stages of visual processing, in what can be termed the visual front end, it can be shown that the Gaussian kernel and its derivatives are singled out as the only possible smoothing kernels. The conditions that specify the Gaussian kernel are, basically, linearity and shift-invariance combined with different ways of formalizing the notion that structures at coarse scales should correspond to simplifications of corresponding structures at fine scales --- they should not be accidental phenomena created by the smoothing method. Notably, several different ways of choosing scale-space axioms give rise to the same conclusion.

The output from the scale-space representation can be used for a variety of early visual tasks; operations like feature detection, feature classification and shape computation can be expressed directly in terms of (possibly non-linear) combinations of Gaussian derivatives at multiple scales. In this sense, the scale-space representation can serve as a basis for early vision.

During the last few decades a number of other approaches to multi-scale representations have been developed, which are more or less related to scale-space theory, notably the theories of pyramids, wavelets and multi-grid methods. Despite their qualitative differences, the increasing popularity of each of these approaches indicates that the crucial notion of scaleis increasingly appreciated by the computer vision community and by researchers in other related fields.

An interesting similarity with biological vision is that the scale-space operators closely resemble receptive field profiles registered in neurophysiological studies of the mammalian retina and visual cortex.

• 324.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Scale-space theory: A framework for handling image structures at multiple scales1996Ingår i: Proc. CERN School of Computing, Egmond aan Zee, The Netherlands, 8–21 September, 1996, 1996, Vol. 96, 8, s. 27-38Konferensbidrag (Refereegranskat)

This article gives a tutorial overview of essential components of scale-space theory --- a framework for multi-scale signal representation, which has been developed by the computer vision community to analyse and interpret real-world images by automatic methods.

• 325.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Scale-Space Theory in Computer Vision1993Bok (Övrigt vetenskapligt)

A basic problem when deriving information from measured data, such as images, originates from the fact that objects in the world, and hence image structures, exist as meaningful entities only over certain ranges of scale. "Scale-Space Theory in Computer Vision" describes a formal theory for representing the notion of scale in image data, and shows how this theory applies to essential problems in computer vision such as computation of image features and cues to surface shape. The subjects range from the mathematical foundation to practical computational techniques. The power of the methodology is illustrated by a rich set of examples.

This book is the first monograph on scale-space theory. It is intended as an introduction, reference, and inspiration for researchers, students, and system designers in computer vision as well as related fields such as image processing, photogrammetry, medical image analysis, and signal processing in general.

The presentation starts with a philosophical discussion about computer vision in general. The aim is to put the scope of the book into its wider context, and to emphasize why the notion of scaleis crucial when dealing with measured signals, such as image data. An overview of different approaches to multi-scale representation is presented, and a number special properties of scale-space are pointed out.

Then, it is shown how a mathematical theory can be formulated for describing image structures at different scales. By starting from a set of axioms imposed on the first stages of processing, it is possible to derive a set of canonical operators, which turn out to be derivatives of Gaussian kernels at different scales.

The problem of applying this theory computationally is extensively treated. A scale-space theory is formulated for discrete signals, and it demonstrated how this representation can be used as a basis for expressing a large number of visual operations. Examples are smoothed derivatives in general, as well as different types of detectors for image features, such as edges, blobs, and junctions. In fact, the resulting scheme for feature detection induced by the presented theory is very simple, both conceptually and in terms of practical implementations.

Typically, an object contains structures at many different scales, but locally it is not unusual that some of these "stand out" and seem to be more significant than others. A problem that we give special attention to concerns how to find such locally stable scales, or rather how to generate hypotheses about interesting structures for further processing. It is shown how the scale-space theory, based on a representation called the scale-space primal sketch, allows us to extract regions of interest from an image without prior information about what the image can be expected to contain. Such regions, combined with knowledge about the scales at which they occur constitute qualitative information, which can be used for guiding and simplifying other low-level processes.

Experiments on different types of real and synthetic images demonstrate how the suggested approach can be used for different visual tasks, such as image segmentation, edge detection, junction detection, and focus-of-attention. This work is complemented by a mathematical treatment showing how the behaviour of different types of image structures in scale-space can be analysed theoretically.

It is also demonstrated how the suggested scale-space framework can be used for computing direct cues to three-dimensional surface structure, using in principle only the same types of visual front-end operations that underlie the computation of image features.

Although the treatment is concerned with the analysis of visual data, the general notion of scale-space representation is of much wider generality and arises in several contexts where measured data are to be analyzed and interpreted automatically.

• 326.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Separable time-causal and time-recursive spatio-temporal receptive fields2015Ingår i: Scale Space and Variational Methods in Computer Vision: 5th International Conference, SSVM 2015, Lège-Cap Ferret, France, May 31 - June 4, 2015, Proceedings / [ed] J.-F. Aujol et al., Springer, 2015, s. 90-102Konferensbidrag (Refereegranskat)

We present an improved model and theory for time-causal and time-recursive spatio-temporal receptive fields,obtained by a combination of Gaussian receptive fields over the spatial domain and first-order integrators or equivalently truncated exponential filters coupled in cascade over the temporal domain. Compared to previous spatio-temporal scale-space formulations in terms of non-enhancement of local extrema or scale invariance, these receptive fields are based on different scale-space axiomatics over time by ensuring non-creation of new local extrema or zero-crossings with increasing temporal scale. Specifically, extensions are presented about parameterizing the intermediate temporal scale levels, analysing the resulting temporal dynamics and transferring the theory to a discrete implementation in terms of recursive filters over time.

• 327.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Time-causal and time-recursive spatio-temporal receptive fields2016Ingår i: Journal of Mathematical Imaging and Vision, ISSN 0924-9907, E-ISSN 1573-7683, Vol. 55, nr 1, s. 50-88Artikel i tidskrift (Refereegranskat)

We present an improved model and theory for time-causal and time-recursive spatio-temporal receptive fields, obtained by a combination of Gaussian receptive fields over the spatial domain and first-order integrators or equivalently truncated exponential filters coupled in cascade over the temporal domain.

Compared to previous spatio-temporal scale-space formulations in terms of non-enhancement of local extrema or scale invariance, these receptive fields are based on different scale-space axiomatics over time by ensuring non-creation of new local extrema or zero-crossings with increasing temporal scale. Specifically, extensions are presented about (i) parameterizing the intermediate temporal scale levels, (ii) analysing the resulting temporal dynamics, (iii) transferring the theory to a discrete implementation in terms of recursive filters over time, (iv) computing scale-normalized spatio-temporal derivative expressions for spatio-temporal feature detection and (v) computational modelling of receptive fields in the lateral geniculate nucleus (LGN) and the primary visual cortex (V1) in biological vision.

We show that by distributing the intermediate temporal scale levels according to a logarithmic distribution, we obtain a new family of temporal scale-space kernels with better temporal characteristics compared to a more traditional approach of using a uniform distribution of the intermediate temporal scale levels. Specifically, the new family of time-causal kernels has much faster temporal response properties (shorter temporal delays) compared to the kernels obtained from a uniform distribution. When increasing the number of temporal scale levels, the temporal scale-space kernels in the new family do also converge very rapidly to a limit kernel possessing true self-similar scale-invariant properties over temporal scales. Thereby, the new representation allows for true scale invariance over variations in the temporal scale, although the underlying temporal scale-space representation is based on a discretized temporal scale parameter.

We show how scale-normalized temporal derivatives can be defined for these time-causal scale-space kernels and how the composed theory can be used for computing basic types of scale-normalized spatio-temporal derivative expressions in a computationally efficient manner.

• 328.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Time-causal and time-recursive spatio-temporal receptive fields2015Rapport (Övrigt vetenskapligt)

We present an improved model and theory for time-causal and time-recursive spatio-temporal receptive fields, obtained by a combination of Gaussian receptive fields over the spatial domain and first-order integrators or equivalently truncated exponential filters coupled in cascade over the temporal domain.

Compared to previous spatio-temporal scale-space formulations in terms of non-enhancement of local extrema or scale invariance, these receptive fields are based on different scale-space axiomatics over time by ensuring non-creation of new local extrema or zero-crossings with increasing temporal scale. Specifically, extensions are presented about (i) parameterizing the intermediate temporal scale levels, (ii) analysing the resulting temporal dynamics, (iii) transferring the theory to a discrete implementation in terms of recursive filters over time, (iv) computing scale-normalized spatio-temporal derivative expressions for spatio-temporal feature detection and (v) computational modelling of receptive fields in the lateral geniculate nucleus (LGN) and the primary visual cortex (V1) in biological vision.

We show that by distributing the intermediate temporal scale levels according to a logarithmic distribution, we obtain a new family of temporal scale-space kernels with better temporal characteristics compared to a more traditional approach of using a uniform distribution of the intermediate temporal scale levels. Specifically, the new family of time-causal kernels has much faster temporal response properties (shorter temporal delays) compared to the kernels obtained from a uniform distribution. When increasing the number of temporal scale levels, the temporal scale-space kernels in the new family do also converge very rapidly to a limit kernel possessing true self-similar scale-invariant properties over temporal scales. Thereby, the new representation allows for true scale invariance over variations in the temporal scale, although the underlying temporal scale-space representation is based on a discretized temporal scale parameter.

We show how scale-normalized temporal derivatives can be defined for these time-causal scale-space kernels and how the composed theory can be used for computing basic types of scale-normalized spatio-temporal derivative expressions in a computationally efficient manner.

• 329.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Time-recursive velocity-adapted spatio-temporal scale-space filters2002Ingår i: : ECCV'02 published in Springer Lecture Notes in Computer Science, volume 2350, 2002, Vol. 2350, s. 52-67Konferensbidrag (Refereegranskat)

This paper presents a theory for constructing and computing velocity-adapted scale-space filters for spatio-temporal image data. Starting from basic criteria in terms of time-causality, time-recursivity, locality and adaptivity with respect to motion estimates, a family of spatio-temporal recursive filters is proposed and analysed. An important property of the proposed family of smoothing kernels is that the spatio-temporal covariance matrices of the discrete kernels obey similar transformation properties under Galilean transformations as for continuous smoothing kernels on continuous domains. Moreover, the proposed theory provides an efficient way to compute and generate nonseparable scale-space representations without need for explicit external warping mechanisms or keeping extended temporal buffers of the past. The approach can thus be seen as a natural extension of recursive scale-space filters from pure temporal data to spatio-temporal domains.

• 330.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
KTH, Tidigare Institutioner, Numerisk analys och datalogi, NADA.
Galilean-corrected spatio-temporal interest operators2004Rapport (Övrigt vetenskapligt)

This paper presents a set of image operators for detecting regions in space-time where interesting events occur. To define such regions of interest, we compute a spatio-temporal secondmoment matrix from a spatio-temporal scale-space representation, and diagonalize this matrix locally, using a local Galilean transformation in space-time, optionally combined with a spatial rotation, so as to make the Galilean invariant degrees of freedom explicit. From the Galilean-diagonalized descriptor so obtained, we then formulate different types of space-time interest operators, and illustrate their properties on different types of image sequences.

• 331.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
KTH, Tidigare Institutioner, Numerisk analys och datalogi, NADA.
Real-time scale selection in hybrid multi-scale representations2003Ingår i: Proc. Scale-Space’03, Springer Berlin/Heidelberg, 2003, Vol. 2695, s. 148-163Konferensbidrag (Refereegranskat)

Local scale information extracted from visual data in a bottom-up manner constitutes an important cue for a large number of visual tasks. This article presents a framework for how the computation of such scale descriptors can be performed in real time on a standard computer.

The proposed scale selection framework is expressed within a novel type of multi-scale representation, referred to as hybrid multi-scale representation, which aims at integrating and providing variable trade-offs between the relative advantages of pyramids and scale-space representation, in terms of computational efficiency and computational accuracy. Starting from binomial scale-space kernels of different widths, we describe a family pyramid representations, in which the regular pyramid concept and the regular scale-space representation constitute limiting cases. In particular, the steepness of the pyramid as well as the sampling density in the scale direction can be varied.

It is shown how the definition of gamma-normalized derivative operators underlying the automatic scale selection mechanism can be transferred from a regular scale-space to a hybrid pyramid, and two alternative definitions are studied in detail, referred to as variance normalization and l(p)-normalization. The computational accuracy of these two schemes is evaluated, and it is shown how the choice of sub-sampling rate provides a trade-off between the computational efficiency and the accuracy of the scale descriptors. Experimental evaluations are presented for both synthetic and real data. In a simplified form, this scale selection mechanism has been running for two years, in a real-time computer vision system.

• 332.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
KTH, Skolan för datavetenskap och kommunikation (CSC), Datorseende och robotik, CVAP.
Analysis of aerosol images using the scale-space primal sketch1991Ingår i: Machine Vision and Applications, ISSN 0932-8092, E-ISSN 1432-1769, Vol. 4, nr 3, s. 135-144Artikel i tidskrift (Refereegranskat)

We outline a method to analyze aerosol images using the scale-space representation. The pictures, which are photographs of an aerosol generated by a fuel injector, contain phenomena that by a human observer are perceived as periodic or oscillatory structures. The presence of these structures is not immediately apparent since the periodicity manifests itself at a coarse level of scale while the dominating objects inthe images are small dark blobs, that is, fine scale objects. Experimentally, we illustrate that the scale-space theory provides an objective method to bring out these events. However, in this form the method still relies on a subjective observer in order to extract and verify the existence of the periodic phenomena.Then we extend the analysis by adding a recently developed image analysis concept called the scale-space primal sketch. With this tool, we are able to extract significant structures from a grey-level image automatically without any strong a priori assumptions about either the shape or the scale (size) of the primitives. Experiments demonstrate that the periodic drop clusters we perceived in the image are detected by the algorithm as significant image structures. These results provide objective evidence verifying the existence of oscillatory phenomena.

• 333.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
KTH, Skolan för datavetenskap och kommunikation (CSC), Datorseende och robotik, CVAP.
Construction of a Scale-Space Primal Sketch1990Ingår i: Proceedings of the British Machine Vision Conference 1990: BMVC'90 (Oxford, England), The British Machine Vision Association and Society for Pattern Recognition , 1990, s. 97-102Konferensbidrag (Refereegranskat)

We present a multi-scale representation of grey-level shape, called scale-space primal sketch, that makes explicit features in scale-space as well as the relations between features at different levels of scale. The representation gives a qualitative description of the image structure that allows for extraction of significant image structure --- stable scales and regions of interest --- in a solely bottom-up data-driven manner. Hence, it can be seen as preceding further processing, which can then be properly tuned. Experiments on real imagery demonstrate that the proposed theory gives perceptually intuitive results.

• 334.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
KTH, Skolan för datavetenskap och kommunikation (CSC), Datorseende och robotik, CVAP.
On the Computation of a Scale-Space Primal Sketch1991Ingår i: Journal of Visual Communication and Image Representation, ISSN 1047-3203, E-ISSN 1095-9076, Vol. 2, nr 1, s. 55-78Artikel i tidskrift (Refereegranskat)

Scale-space theory provides a well-founded framework for dealing with image structures that naturally occur at different scales. According to this theory one can from a given signal derive a family of signals by successively removing features when moving from fine to coarse scale. In contrast to other multiscale representations, scale-space is based on a precise mathematical definition of causality, and the behavior of structure as scale changes can be analytically described. However, the information in the scale-space embedding is only implicit. There is no explicit representation of features or the relations between features at different levels of scale. In this paper we present a theory for constructing such an explicit representation on the basis of formal scale-space theory. We treat gray-level images, but the approach is valid for any bounded function, and can therefore be used to derive properties of, e.g., spatial derivatives. Hence it is useful for studying representations based on intensity discontinuities as well. The representation is obtained in a completely data-driven manner, without relying on any specific parameters. It gives a description of the image structure that is rather coarse. However, since significant scales and regions are actually determined from the data, our approach can be seen as preceding further processing, which can then be properly tuned. An important problem in deriving the representation concerns measuring structure in such a way that the significance over scale can be established. This problem and the problem of proper parameterization of scale are given a careful analysis. Experiments on real imagery demonstrate that the proposed theory gives perceptually intuitive results.

• 335.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
KTH, Skolan för datavetenskap och kommunikation (CSC), Datorseende och robotik, CVAP.
Scale detection and region extraction from a scale-space primal sketch1990Ingår i: Computer Vision, 1990. Proceedings, Third International Conference on, IEEE Computer Society, 1990, s. 416-426Konferensbidrag (Refereegranskat)

We present: (1) a multi-scale representation of gray-level shape, called a scale-space primal sketch, which makes explicit both features in scale-space and the relations between features at different levels of scales; (2) a theory for extraction of significant image structure from this representation; and (3) applications to edge detection, histogram analysis and junction classification demonstrating how the proposed method can be used for guiding later stage processing. The representation gives a qualitative description of the image structure that allows for detection of stable scales and regions of interest in a solely bottom-up data-driven way. In other words, it generates coarse segmentation cues and can be hence seen as preceding further processing, which can then be properly tuned. We argue that once such information is available many other processing tasks can become much simpler. Experiments on real imagery demonstrate that the proposed theory gives perceptually intuitive results.

• 336.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
KTH, Skolan för datavetenskap och kommunikation (CSC), Datorseende och robotik, CVAP.
The Scale-Space Primal Sketch: Construction and Experiments1992Ingår i: Image and Vision Computing, ISSN 0262-8856, E-ISSN 1872-8138, Vol. 10, nr 1, s. 3-18Artikel i tidskrift (Refereegranskat)

We present a multi-scale representation of grey-level shape, called the scale-space primal sketch, that makes explicit features in scale-space as well as the relations between features at different levels of scale. The representation gives a qualitative description of the image structure that allows for extraction of significant image structure — stable scales and regions of interest-in a solely bottom-up data-driven manner. Hence, it can be seen as preceding further processing, which can then be properly tuned. Experiments on real imagery demonstrate that the proposed theory gives intuitively reasonable results.

• 337.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
KTH, Skolan för datavetenskap och kommunikation (CSC), Datorseende och robotik, CVAP.
Scale-space with causal time direction1996Ingår i: : ECCV'96 (Cambridge, U.K.) published in Springer Lecture Notes in Computer Science, vol 1064, Berlin / Heidelberg: Springer , 1996, Vol. 1064, s. 229-240Konferensbidrag (Refereegranskat)

This article presents a theory for multi-scale representation of temporal data. Assuming that a real-time vision system should represent the incoming data at different time scales, an additional causality constraint arises compared to traditional scale-space theory—we can only use what has occurred in the past for computing representations at coarser time scales. Based on a previously developed scale-space theory in terms of noncreation of local maxima with increasing scale, a complete classification is given of the scale-space kernels that satisfy this property of non-creation of structure and respect the time direction as causal. It is shown that the cases of continuous and discrete time are inherently different.

• 338.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Utrecht University.
Foveal scale-space and the linear increase of receptive field size as a function of eccentricity1994Rapport (Övrigt vetenskapligt)

This paper addresses the formulation of a foveal scale-space and its relation to the scaling property of receptive field sizes with eccentricity. It is shown how the notion of a fovea can be incorporated into conventional scale-space theory leading to a foveal log-polar scale-space. Natural assumptions about uniform treatment of structures over scales and finite processing capacity imply a linear increase of minimum receptive field size as a function of eccentricity. These assumptions are similar to the ones used for deriving linear scale-space theory and the Gaussian receptive field model for an idealized visual front-end.

• 339.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Utrecht University.
On the decrease of resolution as a function of eccentricity for a foveal vision system1992Rapport (Övrigt vetenskapligt)
• 340.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
KTH, Skolan för datavetenskap och kommunikation (CSC), Tal, musik och hörsel, TMH.
Idealized computational models for auditory receptive fields2015Ingår i: PLoS ONE, ISSN 1932-6203, E-ISSN 1932-6203, Vol. 10, nr 3, artikel-id e0119032Artikel i tidskrift (Refereegranskat)

We present a theory by which idealized models of auditory receptive fields can be derived in a principled axiomatic manner, from a set of structural properties to (i) enable invariance of receptive field responses under natural sound transformations and (ii) ensure internal consistency between spectro-temporal receptive fields at different temporal and spectral scales.

For defining a time-frequency transformation of a purely temporal sound signal, it is shown that the framework allows for a new way of deriving the Gabor and Gammatone filters as well as a novel family of generalized Gammatone filters, with additional degrees of freedom to obtain different trade-offs between the spectral selectivity and the temporal delay of time-causal temporal window functions.

When applied to the definition of a second-layer of receptive fields from a spectrogram, it is shown that the framework leads to two canonical families of spectro-temporal receptive fields, in terms of spectro-temporal derivatives of either spectro-temporal Gaussian kernels for non-causal time or a cascade of time-causal first-order integrators over the temporal domain and a Gaussian filter over the logspectral domain. For each filter family, the spectro-temporal receptive fields can be either separable over the time-frequency domain or be adapted to local glissando transformations that represent variations in logarithmic frequencies over time. Within each domain of either non-causal or time-causal time, these receptive field families are derived by uniqueness from the assumptions.

It is demonstrated how the presented framework allows for computation of basic auditory features for audio processing and that it leads to predictions about auditory receptive fields with good qualitative similarity to biological receptive fields measured in the inferior colliculus (ICC) and primary auditory cortex (A1) of mammals.

• 341.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
KTH, Skolan för datavetenskap och kommunikation (CSC), Tal, musik och hörsel, TMH.
Scale-space theory for auditory signals2015Ingår i: Scale Space and Variational Methods in Computer Vision: 5th International Conference, SSVM 2015, Lège-Cap Ferret, France, May 31 - June 4, 2015, Proceedings / [ed] J.-F. Aujol et al., Springer, 2015, Vol. 9087, s. 3-15Konferensbidrag (Refereegranskat)

We show how the axiomatic structure of scale-space theory can be applied to the auditory domain and be used for deriving idealized models of auditory receptive fields via scale-space principles. For defining a time-frequency transformation of a purely temporal signal, it is shown that the scale-space framework allows for a new way of deriving the Gabor and Gammatone filters as well as a novel family of generalized Gammatone filters with additional degrees of freedom to obtain different trade-offs between the spectral selectivity and the temporal delay of time-causal window functions. Applied to the definition of a second layer of receptive fields from the spectrogram, it is shown that the scale-space framework leads to two canonical families of spectro-temporal receptive fields, using a combination of Gaussian filters over the logspectral domain with either Gaussian filters or a cascade of first-order integrators over the temporal domain. These spectro-temporal receptive fields can be either separable over the time-frequency domain or be adapted to local glissando transformations that represent variations in logarithmic frequencies over time. Such idealized models of auditory receptive fields respect auditory invariances, can be used for computing basic auditory features for audio processing and lead to predictions about auditory receptive fields with good qualitative similarity to biological receptive fields in the inferior colliculus (ICC) and the primary auditory cortex (A1).

• 342.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Shape from Texture from a Multi-Scale Perspective1993Ingår i: Fourth International Conference on Computer Vision, 1993. Proceedings: ICCV'93 / [ed] H.-H. Nagel, IEEE conference proceedings, 1993, s. 683-691Konferensbidrag (Refereegranskat)

The problem of scale in shape from texture is addressed. The need for (at least) two scale parameters is emphasized; a local scale describing the amount of smoothing used for suppressing noise and irrelevant details when computing primitive texture descriptors from image data, and an integration scale describing the size of the region in space over which the statistics of the local descriptors is accumulated.

A novel mechanism for automatic scale selection is used, based on normalized derivatives. It is used for adaptive determination of the two scale parameters in a multi-scale texture descriptor, thewindowed second moment matrix, which is defined in terms of Gaussian smoothing, first order derivatives, and non-linear pointwise combinations of these. The same scale-selection method can be used for multi-scale blob detection without any tuning parameters or thresholding.

The resulting texture description can be combined with various assumptions about surface texture in order to estimate local surface orientation. Two specific assumptions, weak isotropy'' and constant area'', are explored in more detail. Experiments on real and synthetic reference data with known geometry demonstrate the viability of the approach.

• 343.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
KTH, Skolan för datavetenskap och kommunikation (CSC), Datorseende och robotik, CVAP.
Automatic generation of break points for MDL based curve classification1995Ingår i: Scandinavian Conference on Image Analysis: SCIA'95 / [ed] G. Borgefors, 1995, s. 767-776Konferensbidrag (Refereegranskat)

This article presents a method for segmenting and classifying edges using minimum description length (MDL) approximation with automatically generated break points. A scheme is proposed where junction candidates are first detected in a multi-scale pre-processing step, which generates junction candidates with associated regions of interest. These junction features are matched to edges based on spatial coincidence. For each matched pair, a tentative break point is introduced at the edge point closest to the junction. Finally, these feature combinations serve as input for an MDL approximation method which tests the validity of the break point hypotheses and classifies the resulting edge segments as either straight'' or curved''. Experiments on real world image data demonstrate the viability of the approach.

• 344.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
KTH, Skolan för datavetenskap och kommunikation (CSC), Datorseende och robotik, CVAP.
Segmentation and classification of edges using minimum description length approximation and complementary junction cues1997Ingår i: Computer Vision and Image Understanding, ISSN 1077-3142, E-ISSN 1090-235X, Vol. 67, nr 1, s. 88-98Artikel i tidskrift (Refereegranskat)

This article presents a method for segmenting and classifying edges using minimum description length (MDL) approximation with automatically generated break points. A scheme is proposed where junction candidates are first detected in a multiscale preprocessing step, which generates junction candidates with associated regions of interest. These junction features are matched to edges based on spatial coincidence. For each matched pair, a tentative break point is introduced at the edge point closest to the junction. Finally, these feature combinations serve as input for an MDL approximation method which tests the validity of the break point hypotheses and classifies the resulting edge segments as either “straight” or “curved.” Experiments on real world image data demonstrate the viability of the approach.

• 345.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
KTH, Skolan för datavetenskap och kommunikation (CSC), Datorseende och robotik, CVAP.
Segmentation and classification of edges using minimum description length approximation and complementary junction cues1995Ingår i: Theory and Applications of Image Analysis II: Selected Papers from the 9th Scandinavian Conference on Image Analysis, Uppsala, Sweden, 1995 / [ed] Gunilla Borgefors, World Scientific, 1995Kapitel i bok, del av antologi (Refereegranskat)

This article presents a method for segmenting and classifying edges using minimum description length (MDL) approximation with automatically generated break points. A scheme is proposed where junction candidates are first detected in a multi-scale pre-processing step, which generates junction candidates with associated regions of interest. These junction features are matched to edges based on spatial coincidence. For each matched pair, a tentative break point is introduced at the edge point closest to the junction. Finally, these feature combinations serve as input for an MDL approximation method which tests the validity of the break point hypotheses and classifies the resulting edge segments as either straight'' or curved''. Experiments on real world image data demonstrate the viability of the approach.

• 346.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Karolinska Institutet.
Analysis of Brain Activation Patterns Using A 3-D Scale-Space Primal Sketch1997Ingår i: : HBM'97, published in Neuroimage, volume 5, number 4, 1997, s. 393-393Konferensbidrag (Refereegranskat)

This paper presents a method for automatically determining the spatial extent and the significance ofrCBF changes.

• 347.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Analysis of brain activation patterns using a 3-D scale-space primal sketch1999Ingår i: Human Brain Mapping, ISSN 1065-9471, E-ISSN 1097-0193, Vol. 7, nr 3, s. 166-94Artikel i tidskrift (Refereegranskat)

A fundamental problem in brain imaging concerns how to define functional areas consisting of neurons that are activated together as populations. We propose that this issue can be ideally addressed by a computer vision tool referred to as the scale-space primal sketch. This concept has the attractive properties that it allows for automatic and simultaneous extraction of the spatial extent and the significance of regions with locally high activity. In addition, a hierarchical nested tree structure of activated regions and subregions is obtained. The subject in this article is to show how the scale-space primal sketch can be used for automatic determination of the spatial extent and the significance of rCBF changes. Experiments show the result of applying this approach to functional PET data, including a preliminary comparison with two more traditional clustering techniques. Compared to previous approaches, the method overcomes the limitations of performing the analysis at a single scale or assuming specific models of the data.

• 348.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Utrecht University,.
Linear Scale-Space II: Early visual operations1994Ingår i: Geometry-Driven Diffusion in Vision, Kluwer Academic Publishers, 1994, s. 43-77Kapitel i bok, del av antologi (Övrigt vetenskapligt)

Vision deals with the problem of deriving information about the world from the light reflected from it. Although the active and task-oriented nature of vision is only implicit in this formulation, this view captures several of the essential aspects of vision. As Marr (1982) phrased it in his book Vision, vision is an information processing task, in which an internal representation of information is of utmost importance. Only by representation information can be captured and made available to decision processes. The purpose of a representation is to make certain aspects of the information content explicit, that is, immediately accessible without any need for additional processing.

This introductory chapter deals with a fundamental aspect of early image representation---the notion of scale. As Koenderink (1984) emphasizes, the problem of scale must be faced in any imaging situation. An inherent property of objects in the world and details in images is that they only exist as meaningful entities over certain ranges of scale. A simple example of this is the concept of a branch of a tree, which makes sense only at a scale from, say, a few centimeters to at most a few meters. It is meaningless to discuss the tree concept at the nanometer or the kilometer level. At those scales it is more relevant to talk about the molecules that form the leaves of the tree, or the forest in which the tree grows. Consequently, a multi-scale representation is of crucial importance if one aims at describing the structure of the world, or more specifically the structure of projections of the three-dimensional world onto two-dimensional images.

The need for multi-scale representation is well understood, for example, in cartography; maps are produced at different degrees of abstraction. A map of the world contains the largest countries and islands, and possibly, some of the major cities, whereas towns and smaller islands appear at first in a map of a country. In a city guide, the level of abstraction is changed considerably to include streets and buildings etc. In other words, maps constitute symbolic multi-scale representations of the world around us, although constructed manually and with very specific purposes in mind.

To compute any type of representation from image data, it is necessary to extract information, and hence interact with the data using certain operators. Some of the most fundamental problems in low-level vision and image analysis concern: what operators to use, where to apply them, and how large they should be. If these problems are not appropriately addressed, the task of interpreting the output results can be very hard. Ultimately, the task of extracting information from real image data is severely influenced by the inherent measurement problem that real-world structures, in contrast to certain ideal mathematical entities, such as points'' or lines'', appear in different ways depending upon the scale of observation.

Phrasing the problem in this way shows the intimate relation to physics. Any physical observation by necessity has to be done through some finite aperture, and the result will, in general, depend on the aperture of observation. This holds for any device that registers physical entities from the real world including a vision system based on brightness data. Whereas constant size aperture functions may be sufficient in many (controlled) physical applications, e.g., fixed measurement devices, and also the aperture functions of the basic sensors in a camera (or retina) may have to determined a priori because of practical design constraints, it is far from clear that registering data at a fixed level of resolution is sufficient. A vision system for handling objects of different sizes and at difference distances needs a way to control the scale(s) at which the world is observed.

The goal of this chapter is to review some fundamental results concerning a framework known as scale-space that has been developed by the computer vision community for controlling the scale of observation and representing the multi-scale nature of image data. Starting from a set of basic constraints (axioms) on the first stages of visual processing it will be shown that under reasonable conditions it is possible to substantially restrict the class of possible operations and to derive a (unique) set of weighting profiles for the aperture functions. In fact, the operators that are obtained bear qualitative similarities to receptive fields at the very earliest stages of (human) visual processing (Koenderink 1992). We shall mainly be concerned with the operations that are performed directly on raw image data by the processing modules are collectively termed the visual front-end. The purpose of this processing is to register the information on the retina, and to make important aspects of it explicit that are to be used in later stage processes. If the operations are to be local, they have to preserve the topology at the retina; for this reason the processing can be termed retinotopic processing.

Early visual operationsAn obvious problem concerns what information should be extracted and what computations should be performed at these levels. Is any type of operation feasible? An axiomatic approach that has been adopted in order to restrict the space of possibilities is to assume that the very first stages of visual processing should be able to function without any direct knowledge about what can be expected to be in the scene. As a consequence, the first stages of visual processing should be as uncommitted and make as few irreversible decisions or choices as possible.

The Euclidean nature of the world around us and the perspective mapping onto images impose natural constraints on a visual system. Objects move rigidly, the illumination varies, the size of objects at the retina changes with the depth from the eye, view directions may change etc. Hence, it is natural to require early visual operations to be unaffected by certain primitive transformations (e.g. translations, rotations, and grey-scale transformations). In other words, the visual system should extract properties that are invariant with respect to these transformations.

As we shall see below, these constraints leads to operations that correspond to spatio-temporal derivatives which are then used for computing (differential) geometric descriptions of the incoming data flow. Based on the output of these operations, in turn, a large number of feature detectors can be expressed as well as modules for computing surface shape.

The subject of this chapter is to present a tutorial overview on the historical and current insights of linear scale-space theories as a paradigm for describing the structure of scalar images and as a basis for early vision. For other introductory texts on scale-space; see the monographs by Lindeberg (1991, 1994) and Florack (1993) as well as the overview articles by ter Haar Romeny and Florack (1993) and Lindeberg (1994).

• 349. Lindskog, M
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
A kinetic model on DARPP-32 and the PKA/PP1 cascade2002Konferensbidrag (Refereegranskat)
• 350. Lindskog, M.
KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB. KTH, Skolan för datavetenskap och kommunikation (CSC), Beräkningsbiologi, CB.
Biochemical Networks in Psychiatric Disease2010Ingår i: Systems Biology in Psychiatric Research: From High-Throughput Data to Mathematical Modeling, Wiley-VCH Verlagsgesellschaft, 2010, s. 301-320Kapitel i bok, del av antologi (Refereegranskat)
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