Generalisation across image scales remains a fundamental challenge for deep networks, which often fail to handle images at scales not seen during training (the out-of-distribution problem). In this paper, we present provably scale-invariant Gaussian derivative residual networks (GaussDerResNets), constructed out of scale-covariant Gaussian derivative residual blocks coupled in cascade, aimed at addressing this problem.

By adding residual skip connections to the previous notion of Gaussian derivative layers, deeper networks with substantially increased accuracy can be constructed, while preserving very good scale generalisation properties at the higher level of accuracy. Explicit proofs are provided regarding the underlying scale-covariant and scale-invariant properties in arbitrary dimensions. We also conceptually relate the functionality of the Gaussian derivative residual blocks to semi-discretisations of the velocity-adapted affine diffusion equation.

To analyse the ability of GaussDerResNets to generalise to new scales, we apply them on the new rescaled version of the STL-10 dataset, where training is done at a single fixed scale and evaluation is performed on multiple copies of the test set, each rescaled to a single distinct spatial scale, with scale factors extending over a range of 4. We also conduct similar systematic experiments on the rescaled versions of Fashion-MNIST and CIFAR-10 datasets introduced in our previous work.

Experimentally, we demonstrate that the GaussDerResNets have strong scale generalisation and scale selection properties, while also achieving good test accuracy, on all the three rescaled datasets. In our ablation studies, we investigate different architectural variants of GaussDerResNets, demonstrating that basing the architecture on depthwise-separable convolutions allows for decreasing both the number of parameters and the amount of computations, with reasonably maintained accuracy and scale generalisation. We also find that including a zero-order Gaussian term in the layer definition can sometimes be beneficial, as demonstrated for ourspatial-max-pooling-based networks trained on the rescaled STL-10 dataset.

In these ways, we demonstrate how deep networks can in a theoretically well-founded way handle variations in scale in the testing data that are not spanned by the training data.

This paper presents a time-causal analogue of the Gabor filter, as well as a both time-causal and time-recursive analogue of the Gabor transform, where the proposed time-causal representations obey both temporal scale covariance and a cascade property over temporal scales. The motivation behind these constructions is to enable theoretically well-founded time-frequency analysis over multiple temporal scales for real-time situations, or for physical or biological modelling situations, when the future cannot be accessed, and the non-causal access to the future in Gabor filtering is therefore not viable for a time-frequency analysis of the system.

We develop a principled axiomatically determined theory for formulating these time-causal time-frequency representations, obtained by replacing the Gaussian kernel in the Gabor filtering with a time-causal kernel, referred to as the time-causal limit kernel, and which guarantees simplification properties from finer to coarser levels of scales in a time-causal situation, similar as the Gaussian kernel can be shown to guarantee over a non-causal temporal domain. We do also develop an axiomatically determined theory for implementing a discrete analogue of the proposed time-causal frequency analysis method on discrete data, based on first-order recursive filters coupled in cascade, with provable variation-diminishing properties that strongly suppress the influence from local perturbations and noise, and with specially chosen time constants to achieve self-similarity over scales and temporal scale covariance.

In these ways, the proposed time-frequency representations guarantee well-founded treatment over multiple temporal scales, in situations when the characteristic scales in the signals, or physical or biological phenomena, to be analyzed may vary substantially, and additionally all steps in the time-frequency analysis have to be fully time-causal.

 This paper presents an analysis of properties of two hybrid discretisation methods for Gaussian derivatives, based on convolutions with either the normalised sampled Gaussian kernel or the integrated Gaussian kernel followed by central differences. The motivation for studying these discretisation methods is that in situations when multiple spatial derivatives of different orders are needed at the same scale level, they can be computed significantly more efficiently, compared to more direct derivative approximations based on explicit convolutions with either sampled Gaussian derivative kernels or integrated Gaussian derivative kernels. We characterise the properties of these hybrid discretisation methods in terms of quantitative performance measures, concerning the amount of spatial smoothing that they imply, as well as the relative consistency of the scale estimates obtained from scale-invariant feature detectors with automatic scale selection, with an emphasis on the behaviour for very small values of the scale parameter, which may differ significantly from corresponding results obtained from the fully continuous scale-space theory, as well as between different types of discretisation methods. The presented results are intended as a guide, when designing as well as interpreting the experimental results of scale-space algorithms that operate at very fine scale levels.

 This paper presents an in-depth theoretical analysis of the orientation selectivity properties of simple cells and complex cells, that can be well modelled by the generalized Gaussian derivative model for visual receptive fields, with the purely spatial component of the receptive fields determined by oriented affine Gaussian derivatives for different orders of spatial differentiation.

A detailed mathematical analysis is presented for the three different cases of either: (i) purely spatial receptive fields, (ii) space-time separable spatio-temporal receptive fields and (iii) velocity-adapted spatio-temporal receptive fields. Closed-form theoretical expressions for the orientation selectivity curves for idealized models of simple and complex cells are derived for all these main cases, and it is shown that the orientation selectivity of the receptive fields becomes more narrow, as a scale parameter ratio κ, defined as the ratio between the scale parameters in the directions perpendicular to vs. parallel with the preferred orientation of the receptive field, increases. It is also shown that the orientation selectivity becomes more narrow with increasing order of spatial differentiation in the underlying affine Gaussian derivative operators over the spatial domain.

A corresponding theoretical orientation selectivity analysis is also presented for purely spatial receptive fields according to an affine Gabor model, showing that: (i) the orientation selectivity becomes more narrow when making the receptive fields wider in the direction perpendicular to the preferred orientation of the receptive field; while (ii) an additional degree of freedom in the affine Gabor model does, however, also strongly affect the orientation selectivity properties.

When observing the surface patterns of objects delimited by smooth surfaces, the projections of the surface patterns to the image domain will be subject to substantial variabilities, as induced by variabilities in the geometric viewing conditions, and as generated by either monocular or binocular imaging conditions, or by relative motions between the object and the observer over time. To first order of approximation, the image deformations of such projected surface patterns can be modelled as local linearizations in terms of local 2-D spatial affine transformations. This paper presents a theoretical analysis of relationships between the degrees of freedom in 2-D spatial affine image transformations and the degrees of freedom in the affine Gaussian derivative model for visual receptive fields. For this purpose, we first describe a canonical decomposition of 2-D affine transformations on a product form, closely related to a singular value decomposition, while in closed form, and which reveals the degrees of freedom in terms of (i) uniform scaling transformations, (ii) an overall amount of global rotation, (iii) a complementary non-uniform scaling transformation and (iv) a relative normalization to a preferred symmetry orientation in the image domain. Then, we show how these degrees of freedom relate to the degrees of freedom in the affine Gaussian derivative model. Finally, we use these theoretical results to consider whether we could regard the biological receptive fields in the primary visual cortex of higher mammals as being able to span the degrees of freedom of 2-D spatial affine transformations, based on interpretations of existing neurophysiological experimental results.

Due to the variabilities in image structures caused by perspective scaling transformations, it is essential for deep networks to have an ability to generalise to scales not seen during training. This paper presents an in-depth analysis of the scale generalisation properties of the scale-covariant and scale-invariant Gaussian derivative networks, complemented with both conceptual and algorithmic extensions. For this purpose, Gaussian derivative networks (GaussDerNets) are evaluated on new rescaled versions of the Fashion-MNIST and the CIFAR-10 datasets, with spatial scaling variations over a factor of 4 in the testing data, that are not present in the training data. Additionally, evaluations on the previously existing STIR datasets show that the GaussDerNets achieve better scale generalisation than previously reported for these datasets for other types of deep networks.

We first experimentally demonstrate that the GaussDerNets have quite good scale generalisation properties on the new datasets, and that average pooling of feature responses over scales may sometimes also lead to better results than the previously used approach of max pooling over scales. Then, we demonstrate that using a spatial max pooling mechanism after the final layer enables localisation of non-centred objects in image domain, with maintained scale generalisation properties. We also show that regularisation during training, by applying dropout across the scale channels, referred to as scale-channel dropout, improves both the performance and the scale generalisation.

In additional ablation studies, we show that, for the rescaled CIFAR-10 dataset, basing the layers in the GaussDerNets on derivatives up to order three leads to better performance and scale generalisation for coarser scales, whereas networks based on derivatives up to order two achieve better scale generalisation for finer scales. Moreover, we demonstrate that discretisations of GaussDerNets based on the discrete analogue of the Gaussian kernel in combination with central difference operators perform best or among the best, compared to a set of other discrete approximations of the Gaussian derivative kernels. Furthermore, we show that the improvement in performance obtained by learning the scale values of the Gaussian derivatives, as opposed to using the previously proposed choice of a fixed logarithmic distribution of the scale levels, is usually only minor, thus supporting the previously postulated choice of using a logarithmic distribution as a very reasonable prior.

Finally, by visualising the activation maps and the learned receptive fields, we demonstrate that the GaussDerNets have very good explainability properties.

Due to the variabilities in image structures caused by perspective scaling transformations, it is essential for deep networks to have an ability to generalise to scales not seen during training. This paper presents an in-depth analysis of the scale generalisation properties of the scale-covariant and scale-invariant Gaussian derivative networks, complemented with both conceptual and algorithmic extensions. For this purpose, Gaussian derivative networks (GaussDerNets) are evaluated on new rescaled versions of the Fashion-MNIST and the CIFAR-10 datasets, with spatial scaling variations over a factor of 4 in the testing data, that are not present in the training data. Additionally, evaluations on the previously existing STIR datasets show that the GaussDerNets achieve better scale generalisation than previously reported for these datasets for other types of deep networks. We first experimentally demonstrate that the GaussDerNets have quite good scale generalisation properties on the new datasets and that average pooling of feature responses over scales may sometimes also lead to better results than the previously used approach of max pooling over scales. Then, we demonstrate that using a spatial max pooling mechanism after the final layer enables localisation of non-centred objects in the image domain, with maintained scale generalisation properties. We also show that regularisation during training, by applying dropout across the scale channels, referred to as scale-channel dropout, improves both the performance and the scale generalisation. In additional ablation studies, we show that, for the rescaled CIFAR-10 dataset, basing the layers in the GaussDerNets on derivatives up to order three leads to better performance and scale generalisation for coarser scales, whereas networks based on derivatives up to order two achieve better scale generalisation for finer scales. Moreover, we demonstrate that discretisations of GaussDerNets based on the discrete analogue of the Gaussian kernel in combination with central difference operators perform best or among the best, compared to a set of other discrete approximations of the Gaussian derivative kernels. Furthermore, we show that the improvement in performance obtained by learning the scale values of the Gaussian derivatives, as opposed to using the previously proposed choice of a fixed logarithmic distribution of the scale levels, is usually only minor, thus supporting the previously postulated choice of using a logarithmic distribution as a very reasonable prior. Finally, by visualising the activation maps and the learned receptive fields, we demonstrate that the GaussDerNets have very good explainability properties.

This paper presents an analysis of properties of two hybrid discretization methods for Gaussian derivatives, based on convolutions with either the normalized sampled Gaussian kernel or the integrated Gaussian kernel followed by central differences. The motivation for studying these discretization methods is that in situations when multiple spatial derivatives of different orders are needed at the same scale level, they can be computed significantly more efficiently compared to more direct derivative approximations based on explicit convolutions with either sampled Gaussian derivative kernels or integrated Gaussian derivative kernels.  

While these computational benefits do also hold for the genuinely discrete approach for computing discrete analogues of Gaussian derivatives, based on convolution with the discrete analogue of the Gaussian kernel followed by central differences, the underlying mathematical primitives for the discrete analogue of the Gaussian kernel, in terms of modified Bessel functions of integer order, may not be available in certain frameworks for image processing, such as when performing deep learning based on scale-parameterized filters in terms of Gaussian derivatives, with learning of the scale levels. The hybrid discretizations studied in this paper do, from this perspective, offer a computationally more efficient way of implementing deep networks based on Gaussian derivatives for such use cases.  

In this paper, we present a characterization of the properties of these hybrid discretization methods, in terms of quantitative performance measures concerning the amount of spatial smoothing that they imply, as well as the relative consistency of scale estimates obtained from scale-invariant feature detectors with automatic scale selection, with an emphasis on the behaviour for very small values of the scale parameter, which may differ significantly from corresponding results obtained from the fully continuous scale-space theory, as well as between different types of discretization methods.

The presented results are intended as a guide, when designing as well as interpreting the experimental results of scale-space algorithms that operate at very fine scale levels.

Biological nervous systems constitute important sources of inspiration towards computers that are faster, cheaper, and more energy efficient. Neuromorphic disciplines view the brain as a coevolved system, simultaneously optimizing the hardware and the algorithms running on it. There are clear efficiency gains when bringing the computations into a physical substrate, but we presently lack theories to guide efficient implementations. Here, we present a principled computational model for neuromorphic systems in terms of spatio-temporal receptive fields, based on affine Gaussian kernels over space and leaky-integrator and leaky integrate-and-fire models over time. Our theory is provably covariant to spatial affine and temporal scaling transformations, and with close similarities to the visual processing in mammalian brains. We use these spatio-temporal receptive fields as a prior in an event-based vision task, and show that this improves the training of spiking networks, which otherwise is known as problematic for event-based vision. This work combines efforts within scale-space theory and computational neuroscience to identify theoretically well-founded ways to process spatio-temporal signals in neuromorphic systems. Our contributions are immediately relevant for signal processing and event-based vision, and can be extended to other processing tasks over space and time, such as memory and control.

This paper presents the results of combining (i) theoretical analysis regarding connections between the orientation selectivity and the elongation of receptive fields for the affine Gaussian derivative model with (ii) biological measurements of orientation selectivity in the primary visual cortex to investigate if (iii) the receptive fields can be regarded as spanning a variability in the degree of elongation.

From an in-depth theoretical analysis of idealized models for the receptive fields of simple and complex cells in the primary visual cortex, we established that the orientation selectivity becomes more narrow with increasing elongation of the receptivefields. Combined with previously established biological results, concerning broad vs. sharp orientation tuning of visual neurons in the primary visual cortex, as well as previous experimental results concerning distributions of the resultant of the orientation selectivity curves for simple and complex cells, we show that these results are consistent with the receptive fields spanning a variability over the degree of elongation of the receptive fields. We also show that our principled theoretical model for visual receptive fields leads to qualitatively similar types of deviations from a uniform histogram of the resultant descriptor of the orientation selectivity curves for simple cells, as can be observed in the results from biological experiments.

To firmly determine if the underlying working hypothesis, regarding the receptive fields spanning a variability in the degree of elongation, would truly hold for the receptive fields in the primary visual cortex of higher mammals, we formulate a set of testable predictions, that can be used for investigate this property experimentally, and, if applicable, then also characterize if such a variability would, in a structured way, be related to the pinwheel structure in the visual cortex.