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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 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 Gaussian derivative networks achieve better scale generalisation than previously reported for these datasets for other types of deep networks. We first experimentally demonstrate that the Gaussian derivative networks 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 demonstrate that discretisations of Gaussian derivative networks, 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. Finally, by visualising the activation maps and the learned receptive fields, we demonstrate that the Gaussian derivative networks have very good explainability properties., Comment: 50 pages, 23 figures, 16 tables

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 order 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 kernels or integrated Gaussian 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. 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., Comment: 13 pages, 11 figures. arXiv admin note: text overlap with arXiv:2311.11317

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., Comment: Code available at https://github.com/jegp/nrf

This paper presents 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 have established that the directional selectivity becomes more narrow with increasing elongation of the receptive fields. By comparison with previously established biological results, concerning broad vs. sharp orientation tuning of visual neurons in the primary visual cortex, we demonstrate that those underlying theoretical predictions, in combination with these biological results, are consistent with a previously formulated biological hypothesis, stating that the biological receptive field shapes should span the degrees of freedom in affine image transformations, to support affine covariance over the population of receptive fields in the primary visual cortex. Based on this possible indirect support for the working hypothesis concerning affine covariance, we formulate a set of testable predictions that could be used to, with neurophysiological experiments, judge if the receptive fields in the primary visual cortex of higher mammals could be regarded as spanning a variability over the eccentricity or the elongation of the receptive fields, and, if so, then also characterize if such a variability would, in a structured way, be related to the pinwheel structure in the visual cortex., Comment: 25 pages, 13 figures. Note: Companion paper regarding theoretical analysis in arXiv:2304.11920

This paper develops an in-depth treatment concerning the problem of approximating the Gaussian smoothing and Gaussian derivative computations in scale-space theory for application on discrete data. With close connections to previous axiomatic treatments of continuous and discrete scale-space theory, we consider three main ways discretizing these scale-space operations in terms of explicit discrete convolutions, based on either (i) sampling the Gaussian kernels and the Gaussian derivative kernels, (ii) locally integrating the Gaussian kernels and the Gaussian derivative kernels over each pixel support region and (iii) basing the scale-space analysis on the discrete analogue of the Gaussian kernel, and then computing derivative approximations by applying small-support central difference operators to the spatially smoothed image data. We study the properties of these three main discretization methods both theoretically and experimentally, and characterize their performance by quantitative measures, including the results they give rise to with respect to the task of scale selection, investigated for four different use cases, and with emphasis on the behaviour at fine scales. The results show that the sampled Gaussian kernels and derivatives as well as the integrated Gaussian kernels and derivatives perform very poorly at very fine scales. At very fine scales, the discrete analogue of the Gaussian kernel with its corresponding discrete derivative approximations performs substantially better. The sampled Gaussian kernel and the sampled Gaussian derivatives do, on the other hand, lead to numerically very good approximations of the corresponding continuous results, when the scale parameter is sufficiently large, in the experiments presented in the paper, when the scale parameter is greater than a value of about 1, in units of the grid spacing., Comment: 40 pages, 21 figures

The influence of natural image transformations on receptive field responses is crucial for modelling visual operations in computer vision and biological vision. In this regard, covariance properties with respect to geometric image transformations in the earliest layers of the visual hierarchy are essential for expressing robust image operations, and for formulating invariant visual operations at higher levels. This paper defines and proves a set of joint covariance properties for spatio-temporal receptive fields in terms of spatio-temporal derivative operators applied to spatio-temporally smoothed image data under compositions of spatial scaling transformations, spatial affine transformations, Galilean transformations and temporal scaling transformations. Specifically, the derived relations show how the parameters of the receptive fields need to be transformed, in order to match the output from spatio-temporal receptive fields under composed spatio-temporal image transformations. For this purpose, we also fundamentally extend the notion of scale-normalized derivatives to affine-normalized derivatives, that are computed based on spatial smoothing with affine Gaussian kernels, and analyze the covariance properties of the resulting affine-normalized derivatives for the affine group as well as for important subgroups thereof. We conclude with a geometric analysis, showing how the derived joint covariance properties make it possible to relate or match spatio-temporal receptive field responses, when observing, possibly moving, local surface patches from different views, under locally linearized perspective or projective transformations, as well as when observing different instances of spatio-temporal events, that may occur either faster or slower between different views of similar spatio-temporal events., Comment: 41 pages, 13 figures. Note: From version 4, this paper considers a different form of joint composition of the geometric image transformations than in the earlier versions

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 with a simplifying kernel 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 future in Gabor filtering is therefore not viable for a time-frequency analysis of the system. We develop the theory for these representations, obtained by replacing the Gaussian kernel in Gabor filtering with a time-causal kernel, referred to as the time-causal limit kernel, 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. In these ways, the proposed time-frequency representations guarantee well-founded treatment over multiple 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., Comment: 31 pages, 7 figures, 7 tables, 1 algorithm

This paper presents a theoretical analysis of the orientation selectivity of simple 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 $\kappa$, 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. For comparison, we also present a corresponding theoretical orientation selectivity analysis for purely spatial receptive fields according to an affine Gabor model. The results from that analysis are consistent with the results obtained from the affine Gaussian derivative model,in the respect that the orientation selectivity becomes more narrow when making the receptive fields wider in the direction perpendicular to the preferred orientation of the receptive field., Comment: 30 pages, 12 figures, 1 table. Note: From version 3 of this preprint, the previous manuscript in 2304.11920v2 has been split into two papers, with the part regarding biological interpretations and biological hypotheses moved to arXiv:2404.04858

This paper presents a theory for how geometric image transformations can be handled by a first layer of linear receptive fields, in terms of true covariance properties, which, in turn, enable geometric invariance properties at higher levels in the visual hierarchy. Specifically, we develop this theory for a generalized Gaussian derivative model for visual receptive fields, which is derived in an axiomatic manner from first principles, that reflect symmetry properties of the environment, complemented by structural assumptions to guarantee internally consistent treatment of image structures over multiple spatio-temporal scales. It is shown how the studied generalized Gaussian derivative model for visual receptive fields obeys true covariance properties under spatial scaling transformations, spatial affine transformations, Galilean transformations and temporal scaling transformations, implying that a vision system, based on image and video measurements in terms of the receptive fields according to this model, can to first order of approximation handle the image and video deformations between multiple views of objects delimited by smooth surfaces, as well as between multiple views of spatio-temporal events, under varying relative motions between the objects and events in the world and the observer. We conclude by describing implications of the presented theory for biological vision, regarding connections between the variabilities of the shapes of biological visual receptive fields and the variabilities of spatial and spatio-temporal image structures under natural image transformations., Comment: 38 pages, 14 figures

This article presents an overview of a theory for performing temporal smoothing on temporal signals in such a way that: (i) temporally smoothed signals at coarser temporal scales are guaranteed to constitute simplifications of corresponding temporally smoothed signals at any finer temporal scale (including the original signal) and (ii) the temporal smoothing process is both time-causal and time-recursive, in the sense that it does not require access to future information and can be performed with no other temporal memory buffer of the past than the resulting smoothed temporal scale-space representations themselves. For specific subsets of parameter settings for the classes of linear and shift-invariant temporal smoothing operators that obey this property, it is shown how temporal scale covariance can be additionally obtained, guaranteeing that if the temporal input signal is rescaled by a uniform scaling factor, then also the resulting temporal scale-space representations of the rescaled temporal signal will constitute mere rescalings of the temporal scale-space representations of the original input signal, complemented by a shift along the temporal scale dimension. The resulting time-causal limit kernel that obeys this property constitutes a canonical temporal kernel for processing temporal signal in real-time scenarios when the regular Gaussian kernel cannot be used because of its non-causal access to information from the future and we cannot additionally require the temporal smoothing process to comprise a complementary memory of the past beyond the information contained in the temporal smoothing process itself, which in this way also serves as a multi-scale temporal memory of the past. This theory is generally applicable for both: (i) modelling continuous temporal phenomena over multiple temporal scales and (ii) digital processing of measured temporal signals in real time., Comment: 37 pages, 15 figures

The ability to handle large scale variations is crucial for many real world visual tasks. A straightforward approach for handling scale in a deep network is to process an image at several scales simultaneously in a set of scale channels. Scale invariance can then, in principle, be achieved by using weight sharing between the scale channels together with max or average pooling over the outputs from the scale channels. The ability of such scale channel networks to generalise to scales not present in the training set over significant scale ranges has, however, not previously been explored. In this paper, we present a systematic study of this methodology by implementing different types of scale channel networks and evaluating their ability to generalise to previously unseen scales. We develop a formalism for analysing the covariance and invariance properties of scale channel networks, and explore how different design choices, unique to scaling transformations, affect the overall performance of scale channel networks. We first show that two previously proposed scale channel network designs do not generalise well to scales not present in the training set. We explain theoretically and demonstrate experimentally why generalisation fails in these cases. We then propose a new type of foveated scale channel architecture}, where the scale channels process increasingly larger parts of the image with decreasing resolution. This new type of scale channel network is shown to generalise extremely well, provided sufficient image resolution and the absence of boundary effects. Our proposed FovMax and FovAvg networks perform almost identically over a scale range of 8, also when training on single scale training data, and do also give improved performance when learning from datasets with large scale variations in the small sample regime., Comment: 31 pages, 15 figures, 6 tables. arXiv admin note: substantial text overlap with arXiv:2004.01536

This paper presents a hybrid approach between scale-space theory and deep learning, where a deep learning architecture is constructed by coupling parameterized scale-space operations in cascade. By sharing the learnt parameters between multiple scale channels, and by using the transformation properties of the scale-space primitives under scaling transformations, the resulting network becomes provably scale covariant. By in addition performing max pooling over the multiple scale channels, a resulting network architecture for image classification also becomes provably scale invariant. We investigate the performance of such networks on the MNISTLargeScale dataset, which contains rescaled images from original MNIST over a factor of 4 concerning training data and over a factor of 16 concerning testing data. It is demonstrated that the resulting approach allows for scale generalization, enabling good performance for classifying patterns at scales not present in the training data., Comment: 21 pages, 10 figures

A large number of deep learning architectures use spatial transformations of CNN feature maps or filters to better deal with variability in object appearance caused by natural image transformations. In this paper, we prove that spatial transformations of CNN feature maps cannot align the feature maps of a transformed image to match those of its original, for general affine transformations, unless the extracted features are themselves invariant. Our proof is based on elementary analysis for both the single- and multi-layer network case. The results imply that methods based on spatial transformations of CNN feature maps or filters cannot replace image alignment of the input and cannot enable invariant recognition for general affine transformations, specifically not for scaling transformations or shear transformations. For rotations and reflections, spatially transforming feature maps or filters can enable invariance but only for networks with learnt or hardcoded rotation- or reflection-invariant features, Comment: 22 pages, 3 figures

Spatial transformer networks (STNs) were designed to enable convolutional neural networks (CNNs) to learn invariance to image transformations. STNs were originally proposed to transform CNN feature maps as well as input images. This enables the use of more complex features when predicting transformation parameters. However, since STNs perform a purely spatial transformation, they do not, in the general case, have the ability to align the feature maps of a transformed image with those of its original. STNs are therefore unable to support invariance when transforming CNN feature maps. We present a simple proof for this and study the practical implications, showing that this inability is coupled with decreased classification accuracy. We therefore investigate alternative STN architectures that make use of complex features. We find that while deeper localization networks are difficult to train, localization networks that share parameters with the classification network remain stable as they grow deeper, which allows for higher classification accuracy on difficult datasets. Finally, we explore the interaction between localization network complexity and iterative image alignment., Comment: 13 pages, 7 figures, 6 tables

The ability to handle large scale variations is crucial for many real world visual tasks. A straightforward approach for handling scale in a deep network is to process an image at several scales simultaneously in a set of scale channels. Scale invariance can then, in principle, be achieved by using weight sharing between the scale channels together with max or average pooling over the outputs from the scale channels. The ability of such scale channel networks to generalise to scales not present in the training set over significant scale ranges has, however, not previously been explored. We, therefore, present a theoretical analysis of invariance and covariance properties of scale channel networks and perform an experimental evaluation of the ability of different types of scale channel networks to generalise to previously unseen scales. We identify limitations of previous approaches and propose a new type of foveated scale channel architecture, where the scale channels process increasingly larger parts of the image with decreasing resolution. Our proposed FovMax and FovAvg networks perform almost identically over a scale range of 8, also when training on single scale training data, and do also give improvements in the small sample regime., Comment: 14 pages, 6 figures, 3 tables

Spatial transformer networks (STNs) were designed to enable CNNs to learn invariance to image transformations. STNs were originally proposed to transform CNN feature maps as well as input images. This enables the use of more complex features when predicting transformation parameters. However, since STNs perform a purely spatial transformation, they do not, in the general case, have the ability to align the feature maps of a transformed image and its original. We present a theoretical argument for this and investigate the practical implications, showing that this inability is coupled with decreased classification accuracy. We advocate taking advantage of more complex features in deeper layers by instead sharing parameters between the classification and the localisation network., Comment: Accepted to Northern Lights Deep Learning Workshop 2020, Troms{\o}, 2 pages, 3 figures

This article presents a theory for constructing hierarchical networks in such a way that the networks are guaranteed to be provably scale covariant. We first present a general sufficiency argument for obtaining scale covariance, which holds for a wide class of networks defined from linear and non-linear differential expressions expressed in terms of scale-normalized scale-space derivatives. Then, we present a more detailed development of one example of such a network constructed from a combination of mathematically derived models of receptive fields and biologically inspired computations. Based on a functional model of complex cells in terms of an oriented quasi quadrature combination of first- and second-order directional Gaussian derivatives, we couple such primitive computations in cascade over combinatorial expansions over image orientations. Scale-space properties of the computational primitives are analysed and we give explicit proofs of how the resulting representation allows for scale and rotation covariance. A prototype application to texture analysis is developed and it is demonstrated that a simplified mean-reduced representation of the resulting QuasiQuadNet leads to promising experimental results on three texture datasets., Comment: 29 pages, 16 figures, 3 tables. arXiv admin note: substantial text overlap with arXiv:1903.00289

This article presents a continuous model for hierarchical networks based on a combination of mathematically derived models of receptive fields and biologically inspired computations. Based on a functional model of complex cells in terms of an oriented quasi quadrature combination of first- and second-order directional Gaussian derivatives, we couple such primitive computations in cascade over combinatorial expansions over image orientations. Scale-space properties of the computational primitives are analysed and it is shown that the resulting representation allows for provable scale and rotation covariance. A prototype application to texture analysis is developed and it is demonstrated that a simplified mean-reduced representation of the resulting QuasiQuadNet leads to promising experimental results on three texture datasets., Comment: 12 pages, 3 figures, 1 table

This work presents a first evaluation of using spatio-temporal receptive fields from a recently proposed time-causal spatio-temporal scale-space framework as primitives for video analysis. We propose a new family of video descriptors based on regional statistics of spatio-temporal receptive field responses and evaluate this approach on the problem of dynamic texture recognition. Our approach generalises a previously used method, based on joint histograms of receptive field responses, from the spatial to the spatio-temporal domain and from object recognition to dynamic texture recognition. The time-recursive formulation enables computationally efficient time-causal recognition. The experimental evaluation demonstrates competitive performance compared to state-of-the-art. Especially, it is shown that binary versions of our dynamic texture descriptors achieve improved performance compared to a large range of similar methods using different primitives either handcrafted or learned from data. Further, our qualitative and quantitative investigation into parameter choices and the use of different sets of receptive fields highlights the robustness and flexibility of our approach. Together, these results support the descriptive power of this family of time-causal spatio-temporal receptive fields, validate our approach for dynamic texture recognition and point towards the possibility of designing a range of video analysis methods based on these new time-causal spatio-temporal primitives., Comment: 29 pages, 16 figures

Scale selection methods based on local extrema over scale of scale-normalized derivatives have been primarily developed to be applied sparsely --- at image points where the magnitude of a scale-normalized differential expression additionally assumes local extrema over the domain where the data are defined. This paper presents a methodology for performing dense scale selection, so that hypotheses about local characteristic scales in images, temporal signals and video can be computed at every image point and every time moment. A critical problem when designing mechanisms for dense scale selection is that the scale at which scale-normalized differential entities assume local extrema over scale can be strongly dependent on the local order of the locally dominant differential structure. To address this problem, we propose a methodology where local extrema over scale are detected of a quasi quadrature measure involving scale-space derivatives up to order two and propose two independent mechanisms to reduce the phase dependency of the local scale estimates by: (i) introducing a second layer of post-smoothing prior to the detection of local extrema over scale and (ii) performing local phase compensation based on a model of the phase dependency of the local scale estimates depending on the relative strengths between first- vs. second-order differential structure. This general methodology is applied over three types of domains: (i) spatial images, (ii) temporal signals and (iii) spatio-temporal video. Experiments show that the proposed methodology leads to intuitively reasonable results with local scale estimates that reflect variations in the characteristic scales of locally dominant structures over space and time., Comment: 43 pages, 14 figures, 3 tables

This article gives an overview of a normative computational theory of visual receptive fields, by which idealized functional models of early spatial, spatio-chromatic and spatio-temporal receptive fields can be derived in an axiomatic way based on structural properties of the environment in combination with assumptions about the internal structure of a vision system to guarantee consistent handling of image representations over multiple spatial and temporal scales. Interestingly, this theory leads to predictions about visual receptive field shapes with qualitatively very good similarity to biological receptive fields measured in the retina, the LGN and the primary visual cortex (V1) of mammals., Comment: 15 pages, 17 figures. arXiv admin note: text overlap with arXiv:1210.0754

When designing and developing scale selection mechanisms for generating hypotheses about characteristic scales in signals, it is essential that the selected scale levels reflect the extent of the underlying structures in the signal. This paper presents a theory and in-depth theoretical analysis about the scale selection properties of methods for automatically selecting local temporal scales in time-dependent signals based on local extrema over temporal scales of scale-normalized temporal derivative responses. Specifically, this paper develops a novel theoretical framework for performing such temporal scale selection over a time-causal and time-recursive temporal domain as is necessary when processing continuous video or audio streams in real time or when modelling biological perception. For a recently developed time-causal and time-recursive scale-space concept defined by convolution with a scale-invariant limit kernel, we show that it is possible to transfer a large number of the desirable scale selection properties that hold for the Gaussian scale-space concept over a non-causal temporal domain to this temporal scale-space concept over a truly time-causal domain. Specifically, we show that for this temporal scale-space concept, it is possible to achieve true temporal scale invariance although the temporal scale levels have to be discrete, which is a novel theoretical construction., Comment: 44 pages, 15 figures, 10 tables in Journal of Mathematical Imaging and Vision, 2017

The affine Gaussian derivative model can in several respects be regarded as a canonical model for receptive fields over a spatial image domain: (i) it can be derived by necessity from scale-space axioms that reflect structural properties of the world, (ii) it constitutes an excellent model for the receptive fields of simple cells in the primary visual cortex and (iii) it is covariant under affine image deformations, which enables more accurate modelling of image measurements under the local image deformations caused by the perspective mapping, compared to the more commonly used Gaussian derivative model based on derivatives of the rotationally symmetric Gaussian kernel. This paper presents a theory for discretizing the affine Gaussian scale-space concept underlying the affine Gaussian derivative model, so that scale-space properties hold also for the discrete implementation. Two ways of discretizing spatial smoothing with affine Gaussian kernels are presented: (i) by solving semi-discretized affine diffusion equation, which has derived by necessity from the requirements of a semi-group structure over scale as parameterized by a family of spatial covariance matrices and obeying non-creation of new structures from any finer to any coarser scale in terms of non-enhancement of local extrema and (ii) approximating these semi-discrete affine receptive fields by parameterized families of 3x3-kernels as obtained from an additional discretization along the scale direction. The latter discrete approach can be optionally complemented by spatial subsampling at coarser scales, leading to the notion of affine hybrid pyramids. Using these theoretical results, we outline hybrid architectures for discrete approximations of affine covariant receptive field families, to be used as a first processing layer for affine covariant and affine invariant visual operations at higher levels., Comment: 41 pages, 11 figures

We present an improved model and theory for time-causal and time-recursive spatio-temporal receptive fields, based on 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, (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 much faster temporal response properties (shorter temporal delays) compared to a uniform distribution. Specifically, these kernels converge very rapidly to a limit kernel possessing true self-similar scale-invariant properties over temporal scales, thereby allowing 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., Comment: 39 pages, 12 figures, 5 tables in Journal of Mathematical Imaging and Vision, published online Dec 2015

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., Comment: 12 pages, 2 figures, 2 tables. arXiv admin note: substantial text overlap with arXiv:1404.2037

This paper presents 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 enable invariance of receptive field responses under natural sound transformations and 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 the combination of a time-causal generalized Gammatone filter 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., Comment: 55 pages, 22 figures, 3 tables

Receptive field profiles registered 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. This article presents a theoretical model 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. 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. 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., Comment: 40 pages, 17 figures