Tony Lindeberg

Tony Lindeberg

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.

Tony Lindeberg

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.

Tony Lindeberg

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.

Jens Egholm Pedersen, Tony Lindeberg, Peter Gerstoft

Spike-based encodings are sparse and energy-efficient, but have largely been formulated probabilistically, disconnected from most signal processing literature. We recast spike encoders as time-causal wavelet frames with quantitative bandwidths and reconstruction error bounds. The proposed wavelets preserve the sparsity and locality of spiking representations, with reconstruction up to spike quantization and time discretization. We demonstrate reconstruction on ECG and audio datasets, achieving a normalized RMSE comparable to continuous wavelet transforms. The spiking wavelets map directly to neuromorphic hardware.

Andrzej Perzanowski, Tony Lindeberg

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

Tony Lindeberg

This paper presents a framework for time-causal wavelet analysis. It targets real-time processing of temporal signals, where data from the future are not available. The study builds upon temporal scale-space theory, originating from a complete classification of temporal smoothing kernels that guarantee non-creation of new structures from finer to coarser temporal scale levels. We construct temporal wavelets from the temporal derivatives of a special time-causal smoothing kernel, referred to as the time-causal limit kernel, as arising from the classification of variation-diminishing smoothing transformations with the complementary requirement of temporal scale covariance, to guarantee self-similar handling of structures in the input signal at different temporal scales. This enables decomposition of the signal into different components at different scales, while adhering to temporal causality. The paper establishes theoretical foundations for these time-causal wavelet representations, and maps structural relationships to the non-causal Ricker or Mexican hat wavelets. We also describe how efficient discrete approximations of the presented theory can be performed in terms of first-order recursive filters coupled in cascade, which enables numerically well-conditioned real-time processing with low resource usage. We characterize and quantify how the continuous scaling properties transfer to the discrete implementation, demonstrating how the proposed time-causal wavelet representation can reflect the duration of locally dominant temporal structures in the input signal.

Andrzej Perzanowski, Tony Lindeberg

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. 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. Experimentally, we demonstrate that the GaussDerResNets have strong scale generalisation and scale selection properties 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.

Jens Egholm Pedersen, Tony Lindeberg, Peter Gerstoft

We establish a theoretical connection between wavelet transforms and spiking neural networks through scale-space theory. We rely on the scale-covariant guarantees in the leaky integrate-and-fire neurons to implement discrete mother wavelets that approximate continuous wavelets. A reconstruction experiment demonstrates the feasibility of the approach and warrants further analysis to mitigate current approximation errors. Our work suggests a novel spiking signal representation that could enable more energy-efficient signal processing algorithms.

Tony Lindeberg

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.

Tony Lindeberg

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.

Tony Lindeberg

Because of the variabilities of real-world image structures under the natural image transformations that arise when observing similar objects or spatio-temporal events under different viewing conditions, the receptive field responses computed in the earliest layers of the visual hierarchy may be strongly influenced by such geometric image transformations. One way of handling this variability is by basing the vision system on covariant receptive field families, which expand the receptive field shapes over the degrees of freedom in the image transformations. This paper addresses the problem of deriving relationships between spatial and spatio-temporal receptive field responses obtained for different values of the shape parameters in the resulting multi-parameter families of receptive fields. For this purpose, we derive both (i) infinitesimal relationships, roughly corresponding to a combination of notions from semi-groups and Lie groups, as well as (ii) macroscopic cascade smoothing properties, which describe how receptive field responses at coarser spatial and temporal scales can be computed by applying smaller support incremental filters to the output from corresponding receptive fields at finer spatial and temporal scales, structurally related to the notion of Lie algebras, although with directional preferences. The presented results provide (i) a deeper understanding of the relationships between spatial and spatio-temporal receptive field responses for different values of the filter parameters, which can be used for both (ii) designing more efficient schemes for computing receptive field responses over populations of multi-parameter families of receptive fields, as well as (iii)~formulating idealized theoretical models of the computations of simple cells in biological vision.

Tony Lindeberg, Zahra Babaiee, Peyman M. Kiasari

This paper presents the results of analysing and modelling a set of 8 ``master key filters'', which have been extracted by applying a clustering approach to the receptive fields learned in depthwise-separable deep networks based on the ConvNeXt architecture. For this purpose, we first compute spatial spread measures in terms of weighted mean values and weighted variances of the absolute values of the learned filters, which support the working hypotheses that: (i) the learned filters can be modelled by separable filtering operations over the spatial domain, and that (ii) the spatial offsets of the those learned filters that are non-centered are rather close to half a grid unit. Then, we model the clustered ``master key filters'' in terms of difference operators applied to a spatial smoothing operation in terms of the discrete analogue of the Gaussian kernel, and demonstrate that the resulting idealized models of the receptive fields show good qualitative similarity to the learned filters. This modelling is performed in two different ways: (i) using possibly different values of the scale parameters in the coordinate directions for each filter, and (ii) using the same value of the scale parameter in both coordinate directions. Then, we perform the actual model fitting by either (i) requiring spatial spread measures in terms of spatial variances of the absolute values of the receptive fields to be equal, or (ii) minimizing the discrete $l_1$- or $l_2$-norms between the idealized receptive field models and the learned filters. Complementary experimental results then demonstrate the idealized models of receptive fields have good predictive properties for replacing the learned filters by idealized filters in depthwise-separable deep networks, thus showing that the learned filters in depthwise-separable deep networks can be well approximated by discrete scale-space filters.

Jens Egholm Pedersen, Jörg Conradt, Tony Lindeberg

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.

Ylva Jansson, Tony Lindeberg

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.

Tony Lindeberg

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.

Ylva Jansson, Maksim Maydanskiy, Lukas Finnveden et al.

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

Lukas Finnveden, Ylva Jansson, Tony Lindeberg

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.

Ylva Jansson, Tony Lindeberg

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.

Lukas Finnveden, Ylva Jansson, Tony Lindeberg

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.

Tony Lindeberg

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.

Tony Lindeberg

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.

Ylva Jansson, Tony Lindeberg

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.

Tony Lindeberg

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.

Tony Lindeberg

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.

Tony Lindeberg

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.

Tony Lindeberg

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.

Tony Lindeberg

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.

Tony Lindeberg

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.

Tony Lindeberg, Anders Friberg

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.

Tony Lindeberg

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.