Temporal Scale Spaces

Fagerström, Daniel

doi:10.1007/s11263-005-1837-8

Cited by 11 publications

(26 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The one in space can be computed with an explicit scheme for the heat equation with sub sampling as described in [13]. For the spatio-temporal part the scheme in [4] can be used. This makes the proposed scale-space highly efficient as only a two or three point derivative kernel needs to be applied in the temporal direction instead of a full temporal convolution kernel.…”

Section: Discussionmentioning

confidence: 99%

“…Symmetric stable densities, θ = 0 was the result of the scale space axiomatization in [15,2]. The maximally asymmetric, extremal stable density functions are one sided for 0 < α < 1 and θ = ±α, these where the result of an axiomatization of scale spaces with temporal causality in [4]. In [15,2] these result are extended to the Euclidean similarity group, which on R 2 consists of translation in the plane, rotation and scaling.…”

Section: The Affine Linementioning

confidence: 99%

“…But as discussed in [4] defining temporal measurement in terms of a convolution or a evaluation equation on the history is to beg the question. It would require the measurement device to already have access to what it is supposed to measure.…”

Section: Time Causal Galilean Scale-spacesmentioning

confidence: 99%

“…The form of the generators of the gl(1) scale-space wedge is given in In [4] it is shown that (15), with B = D α τ − is equivalent to ∂ τ u = −D 1/α t+ u with initial value u(t, 0) = f (t), where 0 < 1/α < 1 that is α > 1 for Φ τ to be a time causal scale space.…”

Section: Definitionmentioning

confidence: 99%

“…And observe that in addition to that a real time system cannot be dependent on future information it cannot be directly dependent on past input either, only its representation of past input, its memory [4]. Based on this insight, we instead define a temporally causal spatio-temporal scale space as a semigroups in space and memory, and derive a family of scale-spaces fulfilling this.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Spatio-temporal Scale-Spaces

Fagerström

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

Abstract. A family of spatio-temporal scale-spaces suitable for a moving observer is developed. The scale-spaces are required to be time causal for being usable for real time measurements, and to be "velocity adapted", i.e. to have Galilean covariance to avoid favoring any particular motion. Furthermore standard scale-space axioms: linearity, positivity, continuity, translation invariance, scaling covariance in space and time, rotational invariance in space and recursivity are used. An infinitesimal criterion for scale-spaces is developed, which simplifies calculations and makes it possible to define scale spaces on bounded regions. We show that there are no temporally causal Galilean scale-spaces that are semigroups acting on space and time, but that there are such scale-spaces that are semigroups acting on space and memory (where the memory is the scale-space). The temporally causal scale-space is a time-recursive process using current input and the scale-space as state, i.e. there is no need for storing earlier input. The diffusion equation acting on the memory with the input signal as boundary condition, is a member of this family of scale spaces and is special in the sense that its generator is local.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: The Affine Linementioning

confidence: 99%

Section: Time Causal Galilean Scale-spacesmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Spatio-temporal Scale-Spaces

Fagerström

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

A computational theory of visual receptive fields

2013

View full text Add to dashboard Cite

A receptive field constitutes a region in the visual field where a visual cell or a visual operator responds to visual stimuli. This paper presents a theory for what types of receptive field profiles can be regarded as natural for an idealized vision system, given a set of structural requirements on the first stages of visual processing that reflect symmetry properties of the surrounding world. These symmetry properties include (i) covariance properties under scale changes, affine image deformations, and Galilean transformations of space–time as occur for real-world image data as well as specific requirements of (ii) temporal causality implying that the future cannot be accessed and (iii) a time-recursive updating mechanism of a limited temporal buffer of the past as is necessary for a genuine real-time system. Fundamental structural requirements are also imposed to ensure (iv) mutual consistency and a proper handling of internal representations at different spatial and temporal scales. It is shown how a set of families of idealized receptive field profiles can be derived by necessity regarding spatial, spatio-chromatic, and spatio-temporal receptive fields in terms of Gaussian kernels, Gaussian derivatives, or closely related operators. Such image filters have been successfully used as a basis for expressing a large number of visual operations in computer vision, regarding feature detection, feature classification, motion estimation, object recognition, spatio-temporal recognition, and shape estimation. Hence, the associated so-called scale-space theory constitutes a both theoretically well-founded and general framework for expressing visual operations. There are very close similarities between receptive field profiles predicted from this scale-space theory and receptive field profiles found by cell recordings in biological vision. Among the family of receptive field profiles derived by necessity from the assumptions, idealized models with very good qualitative agreement are obtained for (i) spatial on-center/off-surround and off-center/on-surround receptive fields in the fovea and the LGN, (ii) simple cells with spatial directional preference in V1, (iii) spatio-chromatic double-opponent neurons in V1, (iv) space–time separable spatio-temporal receptive fields in the LGN and V1, and (v) non-separable space–time tilted receptive fields in V1, all within the same unified theory. In addition, the paper presents a more general framework for relating and interpreting these receptive fields conceptually and possibly predicting new receptive field profiles as well as for pre-wiring covariance under scaling, affine, and Galilean transformations into the representations of visual stimuli. This paper describes the basic structure of the necessity results concerning receptive field profiles regarding the mathematical foundation of the theory and outlines how the proposed theory could be used in further studies and modelling of biological vision. It is also shown how receptive field responses can be interpreted physically,...

show abstract

A time-causal and time-recursive scale-covariant scale-space representation of temporal signals and past time

Lindeberg

2023

Biol Cybern

View full text Add to dashboard Cite

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 temporal 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 signals 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. We describe how the time-causal limit kernel relates to previously used temporal models, such as Koenderink’s scale-time kernels and the ex-Gaussian kernel. We do also give an overview of how the time-causal limit kernel can be used for modelling the temporal processing in models for spatio-temporal and spectro-temporal receptive fields, and how it more generally has a high potential for modelling neural temporal response functions in a purely time-causal and time-recursive way, that can also handle phenomena at multiple temporal scales in a theoretically well-founded manner. We detail how this theory can be efficiently implemented for discrete data, in terms of a set of recursive filters coupled in cascade. Hence, the 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. We conclude by stating implications of the theory for modelling temporal phenomena in biological, perceptual, neural and memory processes by mathematical models, as well as implications regarding the philosophy of time and perceptual agents. Specifically, we propose that for A-type theories of time, as well as for perceptual agents, the notion of a non-infinitesimal inner temporal scale of the temporal receptive fields has to be included in representations of the present, where the inherent nonzero temporal delay of such time-causal receptive fields implies a need for incorporating predictions from the actual time-delayed present in the layers of a perceptual hierarchy, to make it possible for a representation of the perceptual present to constitute a representation of the environment with timing properties closer to the actual present.

show abstract

Temporal Scale Spaces

Cited by 11 publications

References 8 publications

Spatio-temporal Scale-Spaces

Spatio-temporal Scale-Spaces

A computational theory of visual receptive fields

A time-causal and time-recursive scale-covariant scale-space representation of temporal signals and past time

Contact Info

Product

Resources

About