1 Introduction
When processing sensory data by automatic methods in areas of signal processing such as computer vision or audio processing or in computational modelling of biological perception, the notion of receptive field constitutes an essential concept (Hubel and Wiesel HubWie59Phys ; HubWie05book ; Aertsen and Johannesma AerJoh81BICY ; DeAngelis et al. DeAngOhzFre95TINS ; deAngAnz04VisNeuroSci ; Miller et al MilEscReaSch01JNeuroPhys ).
For sensory data as obtained from vision or hearing, or their counterparts in artificial perception, the measurement from a single light sensor in a video camera or on the retina, or the instantaneous sound pressure registered by a microphone is hardly meaningful at all, since any such measurement is strongly dependent on external factors such as the illumination of a visual scene regarding vision or the distance between the sound source and the microphone regarding hearing. Instead, the essential information is carried by the relative relations between local measurements at different points and temporal moments regarding vision or local measurements over different frequencies and temporal moments regarding hearing. Following this paradigm, sensory measurements should be performed over local neighbourhoods over spacetime regarding vision and over local neighbourhoods in the timefrequency domain regarding hearing, leading to the notions of spatiotemporal and spectrotemporal receptive fields.
Specifically, spatiotemporal receptive fields constitute a main class of primitives for expressing methods for video analysis (ZelnikManor and Irani ZelIra01CVPR , Laptev and Lindeberg LapLin04ECCVWS ; LapCapSchLin07CVIU ; Jhuang et al. JhuSerWolPog07ICCV ; Kläser et al. KlaMarSch08BMVC ; Niebles et al. NieWanFei08IJCV ; Wang et al. WanUllKlaLapSch09BMVC ; Poppe et al. Pop09IVC ; Shao and Mattivi ShaMatt10CIVR ; Weinland et al. WeiRonBoy11CVIU ; Wang et al. WanQiaTan15CVPR ), whereas spectrotemporal receptive fields constitute a main class of primitives for expressing methods for machine hearing (Patterson et al. PatRobHolMcKeoZhaAll92AudPhysPerc ; PatAllGig95JASA ; Kleinschmidt Kle02ActAcust ; Ezzat et al. EzzBouPog07InterSpeech ; Meyer and Kollmeier MeyKol08InterSpeech ; Schlute et al. SchBezWagNey07ICASSP ; Heckmann et al. HecDomJouGoe11SpeechComm ; Wu et al. WuZhaShi11ASLP ; Alias et al. AliSocJoaSev16ApplSci ).
A general problem when applying the notion of receptive fields in practice, however, is that the types of responses that are obtained in a specific situation can be strongly dependent on the scale levels at which they are computed. A spatiotemporal receptive field is determined by at least a spatial scale parameter and a temporal scale parameter, whereas a spectrotemporal receptive field is determined by at least a spectral and a temporal scale parameter. Beyond ensuring that local sensory measurements at different spatial, temporal and spectral scales are treated in a consistent manner, which by itself provides strong contraints on the shapes of the receptive fields (Lindeberg Lin13BICY ; Lin16JMIV ; Lindeberg and Friberg LinFri15PONE ; LinFri15SSVM ), it is necessary for computer vision or machine hearing algorithms to decide what responses within the families of receptive fields over different spatial, temporal and spectral scales they should base their analysis on.
Over the spatial domain, theoretically wellfounded methods have been developed for choosing spatial scale levels among receptive field responses over multiple spatial scales (Lindeberg Lin97IJCV ; Lin98IJCV ; Lin99CVHB ; Lin12JMIV ; Lin14EncCompVis ) leading to e.g. robust methods for imagebased matching and recognition (Lowe Low04IJCV ; Mikolajczyk and Schmid MikSch04IJCV ; Tuytelaars and van Gool TuyGoo04IJCV ; Bay et al. BayEssTuyGoo08CVIU ; Tuytelaars and Mikolajczyk TuyMik08Book ; van de Sande et al. SanGevSno10PAMI ; Larsen et al. LarDarDahPed12ECCV ) that are able to handle large variations of the size of the objects in the image domain and with numerous applications regarding object recognition, object categorization, multiview geometry, construction of 3D models from visual input, humancomputer interaction, biometrics and robotics.
Much less research has, however, been performed regarding the topic of choosing local appropriate scales in temporal data. While some methods for temporal scale selection have been developed (Lindeberg Lin97AFPAC ; Laptev and Lindeberg LapLin03ICCV ; Willems et al. WilTuyGoo08ECCV ), these methods suffer from either theoretical or practical limitations.
A main subject of this paper is present a theory for how to compare filter responses in terms of temporal derivatives that have been computed at different temporal scales, specifically with a detailed theoretical analysis of the possibilities of having temporal scale estimates as obtained from a temporal scale selection mechanism reflect the temporal duration of the underlying temporal structures that gave rise to the feature responses. Another main subject of this paper is to present a theoretical framework for temporal scale selection that leads to temporal scale invariance and enables the computation of scale covariant temporal scale estimates. While these topics can for a noncausal temporal domain be addressed by the noncausal Gaussian scalespace concept (Iijima Iij62 ; Witkin Wit83 ; Koenderink Koe84BC ; Koenderink and van Doorn KoeDoo92PAMI ; Lindeberg Lin93Dis ; Lin94SI ; Lin10JMIV ; Florack Flo97book ; ter Haar Romeny Haa04book ), the development of such a theory has been missing regarding a timecausal temporal domain.
1.1 Temporal scale selection
When processing timedependent signals in video or audio or more generally any temporal signal, special attention has to be put to the facts that:

the physical phenomena that generate the temporal signals may occur at different speed — faster or slower, and

the temporal signals may contain qualitatively different types of temporal structures at different temporal scales.
In certain controlled situations where the physical system that generates the temporal signals that is to be processed is sufficiently well known and if the variability of the temporal scales over time in the domain is sufficiently constrained, suitable temporal scales for processing the signals may in some situations be chosen manually and then be verified experimentally. If the sources that generate the temporal signals are sufficiently complex and/or if the temporal structures in the signals vary substantially in temporal duration by the underlying physical processes occurring significantly faster or slower, it is on the other hand natural to (i) include a mechanism for processing the temporal data at multiple temporal scales and (ii) try to detect in a bottomup manner at what temporal scales the interesting temporal phenomena are likely to occur.
The subject of this article is to develop a theory for temporal scale selection in a timecausal temporal scale space as an extension of a previously developed theory for spatial scale selection in a spatial scale space (Lindeberg Lin97IJCV ; Lin98IJCV ; Lin99CVHB ; Lin12JMIV ; Lin14EncCompVis ), to generate bottomup hypotheses about characteristic temporal scales in timedependent signals, intended to serve as estimates of the temporal duration of local temporal structures in timedependent signals. Special focus will be on developing mechanisms analogous to scale selection in noncausal Gaussian scalespace, based on local extrema over scales of scalenormalized derivatives, while expressed within the framework of a timecausal and timerecursive temporal scale space in which the future cannot be accessed and the signal processing operations are thereby only allowed to make use of information from the present moment and a compact buffer of what has occurred in the past.
When designing and developing such scale selection mechanisms, it is essential that the computed scale estimates reflect the temporal duration of the corresponding temporal structures that gave rise to the feature responses. To understand the prerequisites for developing such temporal scale selection methods, we will in this paper perform an indepth theoretical analysis of the scale selection properties that such temporal scale selection mechanisms give rise to for different temporal scalespace concepts and for different ways of defining scalenormalized temporal derivatives.
Specifically, after an examination of the theoretical properties of different types of temporal scalespace concepts, we will focus on a class of recently extended timecausal temporal scalespace concepts obtained by convolution with truncated exponential kernels coupled in cascade (Lindeberg Lin90PAMI ; Lin15SSVM ; Lin16JMIV ; Lindeberg and Fagerström LF96ECCV ). For two natural ways of distributing the discrete temporal scale levels in such a representation, in terms of either a uniform distribution over the scale parameter corresponding to the variance of the composed scalespace kernel or a logarithmic distribution, we will study the scale selection properties that result from detecting local temporal scale levels from local extrema over scale of scalenormalized temporal derivatives. The motivation for studying a logarithmic distribution of the temporal scale levels, is that it corresponds to a uniform distribution in units of effective scale for some constants and , which has been shown to constitute the natural metric for measuring the scale levels in a spatial scale space (Koenderink Koe84BC ; Lindeberg Lin92PAMI ).
As we shall see from the detailed theoretical analysis that will follow, this will imply certain differences in scale selection properties of a temporally asymmetric timecausal scale space compared to scale selection in a spatially mirror symmetric Gaussian scale space. These differences in theoretical properties are in turn essential to take into explicit account when formulating algorithms for temporal scale selection in e.g. video analysis or audio analysis applications.
For the temporal scalespace concept based on a uniform distribution of the temporal scale levels in units of the variance of the composed scalespace kernel, it will be shown that temporal scale selection from local extrema over temporal scales will make it possible to estimate the temporal duration of local temporal structures modelled as local temporal peaks and local temporal ramps. For a dense temporal structure modelled as a temporal sine wave, the lack of true scale invariance for this concept will, however, imply that the temporal scale estimates will not be directly proportional to the wavelength of the temporal sine wave. Instead, the scale estimates are affected by a bias, which is not a desirable property.
For the temporal scalespace concept based on a logarithmic distribution of the temporal scale levels, and taken to the limit to scaleinvariant timecausal limit kernel (Lindeberg Lin16JMIV ) corresponding to an infinite number of temporal scale levels that cluster infinitely close near the temporal scale level zero, it will on the other hand be shown that the temporal scale estimates of a dense temporal sine wave will be truly proportional to the wavelength of the signal. By a general proof, it will be shown this scale invariant property of temporal scale estimates can also be extended to any sufficiently regular signal, which constitutes a general foundation for expressing scale invariant temporal scale selection mechanisms for timedependent video and audio and more generally also other classes of timedependent measurement signals.
As complement to this proposed overall framework for temporal scale selection, we will also present a set of general theoretical results regarding timecausal scalespace representations: (i) showing that previous application of the assumption of a semigroup property for timecausal scalespace concepts leads to undesirable temporal dynamics, which however can be remedied by replacing the assumption of a semigroup structure be a weaker assumption of a cascade property in turn based on a transitivity property, (ii) formulations of scalenormalized temporal derivatives for Koenderink’s timecausal scaletime model Koe88BC and (iii) ways of translating the temporal scale estimates from local extrema over temporal scales in the temporal scalespace representation based on the scaleinvariant timecausal limit kernel into quantitative measures of the temporal duration of the corresponding underlying temporal structures and in turn based on a scaletime approximation of the limit kernel.
In these ways, this paper is intended to provide a theoretical foundation for expressing theoretically wellfounded temporal scale selection methods for selecting local temporal scales over timecausal temporal domains, such as video and audio with specific focus on realtime image or sound streams. Applications of this scale selection methodology for detecting both sparse and dense spatiotemporal features in video are presented in a companion paper Lin16spattempscsel .
1.2 Structure of this article
As a conceptual background to the theoretical developments that will be performed, we will start in Section 2 with an overview of different approaches to handling temporal data within the scalespace framework including a comparison of relative advantages and disadvantages of different types of temporal scalespace concepts.
As a theoretical baseline for the later developments of methods for temporal scale selection in a timecausal scale space, we shall then in Section 3 give an overall description of basic temporal scale selection properties that will hold if the noncausal Gaussian scalespace concept with its corresponding selection methodology for a spatial image domain is applied to a onedimensional noncausal temporal domain, e.g. for the purpose of handling the temporal domain when analysing prerecorded video or audio in an offline setting.
In Sections 4–5 we will then continue with a theoretical analysis of the consequences of performing temporal scale selection in the timecausal scale space obtained by convolution with truncated exponential kernels coupled in cascade (Lindeberg Lin90PAMI ; Lin15SSVM ; Lin16JMIV ; Lindeberg and Fagerström LF96ECCV ). By selecting local temporal scales from the scales at which scalenormalized temporal derivatives assume local extrema over temporal scales, we will analyze the resulting temporal scale selection properties for two ways of defining scalenormalized temporal derivatives, by either variancebased normalization as determined by a scale normalization parameter or normalization for different values of the scale normalization power .
With the temporal scale levels required to be discrete because of the very nature of this temporal scalespace concept, we will specifically study two ways of distributing the temporal scale levels over scale, using either a uniform distribution relative to the temporal scale parameter corresponding to the variance of the composed temporal scalespace kernel in Section 4 or a logarithmic distribution of the temporal scale levels in Section 5.
Because of the analytically simpler form for the timecausal scalespace kernels corresponding to a uniform distribution of the temporal scale levels, some theoretical scalespace properties will turn out to be easier to study in closed form for this temporal scalespace concept. We will specifically show that for a temporal peak modelled as the impulse response to a set of truncated exponential kernels coupled in cascade, the selected temporal scale level will serve as a good approximation of the temporal duration of the peak or be proportional to this measure depending on the value of the scale normalization parameter used for scalenormalized temporal derivatives based on variancebased normalization or the scale normalization power for scalenormalized temporal derivatives based on normalization. For a temporal onset ramp, the selected temporal scale level will on the other hand be either a good approximation of the time constant of the onset ramp or proportional to this measure of the temporal duration of the ramp. For a temporal sine wave, the selected temporal scale level will, however, not be directly proportional to the wavelength of the signal, but instead affected by a systematic bias. Furthermore, the corresponding scalenormalized magnitude measures will not be independent of the wavelength of the sine wave but instead show systematic wavelength dependent deviations. A main reason for this is that this temporal scalespace concept does not guarantee temporal scale invariance if the temporal scale levels are distributed uniformly in terms of the temporal scale parameter corresponding to the temporal variance of the temporal scalespace kernel.
With a logarithmic distribution of the temporal scale levels, we will on the other hand show that for the temporal scalespace concept defined by convolution with the timecausal limit kernel (Lindeberg Lin16JMIV ) corresponding to an infinitely dense distribution of the temporal scale levels towards zero temporal scale, the temporal scale estimates will be perfectly proportional to the wavelength of a sine wave for this temporal scalespace concept. It will also be shown that this temporal scalespace concept leads to perfect scale invariance in the sense that (i) local extrema over temporal scales are preserved under temporal scaling factors corresponding to integer powers of the distribution parameter of the timecausal limit kernel underlying this temporal scalespace concept and are transformed in a scalecovariant way for any temporal input signal and (ii) if the scale normalization parameter or equivalently if the scale normalization power , the magnitude values at the local extrema over scale will be equal under corresponding temporal scaling transformations. For this temporal scalespace concept we can therefore fulfil basic requirements to achieve temporal scale invariance also over a timecausal and timerecursive temporal domain.
To simplify the theoretical analysis we will in some cases temporarily extend the definitions of temporal scalespace representations over discrete temporal scale levels to a continuous scale variable, to make it possible to compute local extrema over temporal scales from differentiation with respect to the temporal scale parameter. Section 6 discusses the influence that this approximation has on the overall theoretical analysis.
Section 7 then illustrates how the proposed theory for temporal scale selection can be used for computing local scale estimates from 1D signals with substantial variabilities in the characteristic temporal duration of the underlying structures in the temporal signal.
In Section 8, we analyse how the derived scale selection properties carry over to a set of spatiotemporal feature detectors defined over both multiple spatial scales and multiple temporal scales in a timecausal spatiotemporal scalespace representation for video analysis. Section 9 then outlines how corresponding selection of local temporal and logspectral scales can be expressed for audio analysis operations over a timecausal spectrotemporal domain. Finally, Section 10 concludes with a summary and discussion.
To simplify the presentation, we have put some derivations and theoretical analysis in the appendix. Appendix A presents a general theoretical argument of why a requirement about a semigroup property over temporal scales will lead to undesirable temporal dynamics for a timecausal scale space and argue that the essential structure of noncreation of new image structures from any finer to any coarser temporal scale can instead nevertheless be achieved with the less restrictive assumption about a cascade smoothing property over temporal scales, which then allows for better temporal dynamics in terms of e.g. shorter temporal delays.
In relation to Koenderink’s scaletime model Koe88BC , Appendix B shows how corresponding notions of scalenormalized temporal derivatives based on either variancebased normalization or normalization can be defined also for this timecausal temporal scalespace concept.
Appendix C shows how the temporal duration of the timecausal limit kernel proposed in (Lindeberg Lin16JMIV ) can be estimated by a scaletime approximation of the limit kernel via Koenderink’s scaletime model leading to estimates of how a selected temporal scale level from local extrema over temporal scale can be translated into a estimates of the temporal duration of temporal structures in the temporal scalespace representation obtained by convolution with the timecausal limit kernel. Specifically, explicit expressions are given for such temporal duration estimates based on first and secondorder temporal derivatives.
2 Theoretical background and related work
2.1 Temporal scalespace concepts
For processing temporal signals at multiple temporal scales, different types of temporal scalespace concepts have been developed in the computer vision literature (see Figure 1):
For offline processing of prerecorded signals, a noncausal Gaussian temporal scalespace concept may in many situations be sufficient. A Gaussian temporal scalespace concept is constructed over the 1D temporal domain in a similar manner as a Gaussian spatial scalespace concept is constructed over a Ddimensional spatial domain (Iijima Iij62 ; Witkin Wit83 ; Koenderink Koe84BC ; Koenderink and van Doorn KoeDoo92PAMI ; Lindeberg Lin93Dis ; Lin94SI ; Lin10JMIV ; Florack Flo97book ; ter Haar Romeny Haa04book ), with or without the difference that a model for temporal delays may or may not be additionally included (Lindeberg Lin10JMIV ).
When processing temporal signals in real time, or when modelling sensory processes in biological perception computationally, it is on the other hand necessary to base the temporal analysis on timecausal operations.
The first timecausal temporal scalespace concept was developed by Koenderink Koe88BC , who proposed to apply Gaussian smoothing on a logarithmically transformed time axis with the present moment mapped to the unreachable infinity. This temporal scalespace concept does, however, not have any known timerecursive formulation. Formally, it requires an infinite memory of the past and has therefore not been extensively applied in computational applications.
Lindeberg Lin90PAMI ; Lin15SSVM ; Lin16JMIV and Lindeberg and Fagerström LF96ECCV proposed a timecausal temporal scalespace concept based on truncated exponential kernels or equivalently firstorder integrators coupled in cascade, based on theoretical results by Schoenberg Sch50 (see also Schoenberg Sch88book and Karlin Kar68 ) implying that such kernels are the only variationdiminishing kernels over a 1D temporal domain that guarantee noncreation of new local extrema or equivalently zerocrossings with increasing temporal scale. This temporal scalespace concept is additionally timerecursive and can be implemented in terms of computationally highly efficient firstorder integrators or recursive filters over time. This theory has been recently extended into a scaleinvariant timecausal limit kernel (Lindeberg Lin16JMIV ), which allows for scale invariance over the temporal scaling transformations that correspond to exact mappings between the temporal scale levels in the temporal scalespace representation based on a discrete set of logarithmically distributed temporal scale levels.
Based on semigroups that guarantee either selfsimilarity over temporal scales or nonenhancement of local extrema with increasing temporal scales, Fagerström Fag05IJCV and Lindeberg Lin10JMIV have derived timecausal semigroups that allow for a continuous temporal scale parameter and studied theoretical properties of these kernels.
Concerning temporal processing over discrete time, Fleet and Langley FleLan95PAMI performed temporal filtering for optic flow computations based on recursive filters over time. Lindeberg Lin90PAMI ; Lin15SSVM ; Lin16JMIV and Lindeberg and Fagerström LF96ECCV showed that firstorder recursive filters coupled in cascade constitutes a natural timecausal scalespace concept over discrete time, based on the requirement that the temporal filtering over a 1D temporal signal must not increase the number of local extrema or equivalently the number of zerocrossings in the signal. In the specific case when all the time constants in this model are equal and tend to zero while simultaneously increasing the number of temporal smoothing steps in such a way that the composed temporal variance is held constant, these kernels can be shown to approach the temporal Poisson kernel LF96ECCV . If on the other hand the time constants of the firstorder integrators are chosen so that the temporal scale levels become logarithmically distributed, these temporal smoothing kernels approach a discrete approximation of the timecausal limit kernel Lin16JMIV .
Applications of using these linear temporal scalespace concepts for modelling the temporal smoothing step in visual and auditory receptive fields have been presented by Lindeberg Lin97ICSSTCV ; CVAP257 ; Lin10JMIV ; Lin13BICY ; Lin13PONE ; Lin15SSVM ; Lin16JMIV , ter Haar Romeny et al. RomFloNie01SCSP , Lindeberg and Friberg LinFri15PONE ; LinFri15SSVM and Mahmoudi Mah16JMIV . Nonlinear spatiotemporal scalespace concepts have been proposed by Guichard Gui98TIP . Applications of the noncausal Gaussian temporal scalespace concept for computing spatiotemporal features have been presented by Laptev and Lindeberg LapLin03ICCV ; LapLin04ECCVWS ; LapCapSchLin07CVIU , Kläser et al. KlaMarSch08BMVC , Willems et al. WilTuyGoo08ECCV , Wang et al. WanUllKlaLapSch09BMVC , Shao and Mattivi ShaMatt10CIVR and others, see specifically Poppe Pop09IVC for a survey of early approaches to visionbased human human action recognition, Jhuang et al. JhuSerWolPog07ICCV and Niebles et al. NieWanFei08IJCV for conceptually related noncausal Gabor approaches, Adelson and Bergen AdeBer85JOSA and Derpanis and Wildes DerWil12PAMI for closely related spatiotemporal orientation models and Han et al. HanXuZhu15JMIV for a related midlevel temporal representation termed the video primal sketch.
Applications of the temporal scalespace model based on truncated exponential kernels with equal time constants coupled in cascade and corresponding to Laguerre functions (Laguerre polynomials multiplied by a truncated exponential kernel) for computing spatiotemporal features have presented by RiveroMoreno and Bres RivBre04ImAnalRec , Shabani et al. ShaClaZel12BMVC and Berg et al. BerReyRid14SensMEMSElOptSyst as well as for handling time scales in video surveillance (Jacob and Pless JacPle08CircSystVidTech ), for performing edge preserving smoothing in video streams (Paris Par08ECCV ) and is closely related to Tikhonov regularization as used for image restoration by e.g. Surya et al. SurVorPelJosSeePal15JMIV . A general framework for performing spatiotemporal feature detection based on the temporal scalespace model based on truncated exponential kernels coupled in cascade with specifically the both theoretical and practical advantages of using logarithmic distribution of the intermediated temporal scale levels in terms of temporal scale invariance and better temporal dynamics (shorter temporal delays) has been presented in Lindeberg Lin16JMIV .
2.2 Relative advantages of different temporal scale spaces
When developing a temporal scale selection mechanism over a timecausal temporal domain, a first problem concerns what timecausal scalespace concept to base the multiscale temporal analysis upon. The above reviewed temporal scalespace concepts have different relative advantages from a theoretical and computational viewpoint. In this section, we will perform an indepth examination of the different temporal scalespace concepts that have been developed in the literature, which will lead us to a class of timecausal scalespace concepts that we argue is particularly suitable with respect to the set of desirable properties we aim at.
The noncausal Gaussian temporal scale space is in many cases the conceptually easiest temporal scalespace concept to handle and to study analytically (Lindeberg Lin10JMIV ). The corresponding temporal kernels are scale invariant, have compact closedform expressions over both the temporal and frequency domains and obey a semigroup property over temporal scales. When applied to prerecorded signals, temporal delays can if desirable be disregarded, which eliminates any need for temporal delay compensation. This scalespace concept is, however, not timecausal and not timerecursive, which implies fundamental limitations with regard to realtime applications and realistic modelling of biological perception.
Koenderink’s scaletime kernels Koe88BC
are truly timecausal, allow for a continuous temporal scale parameter, have good temporal dynamics and have a compact explicit expression over the temporal domain. These kernels are, however, not timerecursive, which implies that they in principle require an infinite memory of the past (or at least extended temporal buffers corresponding to the temporal extent to which the infinite support temporal kernels are truncated at the tail). Thereby, the application of Koenderink’s scaletime model to video analysis implies that substantial temporal buffers are needed when implementing this nonrecursive temporal scalespace in practice. Similar problems with substantial need for extended temporal buffers arise when applying the noncausal Gaussian temporal scalespace concept to offline analysis of extended video sequences. The algebraic expressions for the temporal kernels in the scaletime model are furthermore not always straightforward to handle and there is no known simple expression for the Fourier transform of these kernels or no known simple explicit cascade smoothing property over temporal scales with respect to the regular (untransformed) temporal domain. Thereby, certain algebraic calculations with the scaletime kernels may become quite complicated.
The temporal scalespace kernels obtained by coupling truncated exponential kernels or equivalently firstorder integrators in cascade are both truly timecausal and truly timerecursive (Lindeberg Lin90PAMI ; Lin15SSVM ; Lin16JMIV ; Lindeberg and Fagerström LF96ECCV ). The temporal scale levels are on the other hand required to be discrete. If the goal is to construct a realtime signal processing system that analyses continuous streams of signal data in real time, one can however argue that a restriction of the theory to a discrete set of temporal scale levels is less of a contraint, since the signal processing system anyway has to be based on a finite amount of sensors and hardware/wetware for sampling and processing the continuous stream of signal data.
In the special case when all the time constants are equal, the corresponding temporal kernels in the temporal scalespace model based on truncated exponential kernels coupled in cascade have compact explicit expressions that are easy to handle both in the temporal domain and in the frequency domain, which simplifies theoretical analysis. These kernels obey a semigroup property over temporal scales, but are not scale invariant and lead to slower temporal dynamics when a larger number of primitive temporal filters are coupled in cascade (Lindeberg Lin15SSVM ; Lin16JMIV ).
In the special case when the temporal scale levels in this scalespace model are logarithmically distributed, these kernels have a manageable explicit expression over the Fourier domain that enables some closedform theoretical calculations. Deriving an explicit expression over the temporal domain is, however, harder, since the explicit expression then corresponds to a linear combination of truncated exponential filters for all the time constants, with the coefficients determined from a partial fraction expansion of the Fourier transform, which may lead to rather complex closedform expressions. Thereby certain analytical calculations may become harder to handle. As shown in Lin16JMIV and Appendix C, some such calculations can on the other hand be well approximated via a scaletime approximation of the timecausal temporal scalespace kernels. When using a logarithmic distribution of the temporal scales, the composed temporal kernels do however have very good temporal dynamics and much better temporal dynamics compared to corresponding kernels obtained by using truncated exponential kernels with equal time constants coupled in cascade. Moreover, these kernels lead to a computationally very efficient numerical implementation. Specifically, these kernels allow for the formulation of a timecausal limit kernel that obeys scale invariance under temporal scaling transformations, which cannot be achieved if using a uniform distribution of the temporal scale levels (Lindeberg Lin15SSVM ; Lin16JMIV ).
The temporal scalespace representations obtained from the selfsimilar timecausal semigroups have a continuous scale parameter and obey temporal scale invariance (Fagerström Fag05IJCV ; Lindeberg Lin10JMIV ). These kernels do, however, have less desirable temporal dynamics (see Appendix A for a general theoretical argument about undesirable consequences of imposing a temporal semigroup property on temporal kernels with temporal delays) and/or lead to pseudodifferential equations that are harder to handle both theoretically and in terms of computational implementation. For these reasons, we shall not consider those timecausal semigroups further in this treatment.
2.3 Previous work on methods for scale selection
A general framework for performing scale selection for local differential operations was proposed in Lindeberg Lin93SCIA ; Lin93Dis based on the detection of local extrema over scale of scalenormalized derivative expressions and then refined in Lindeberg Lin97IJCV ; Lin98IJCV — see Lindeberg Lin99CVHB ; Lin14EncCompVis for tutorial overviews.
This scale selection approach has been applied to a large number of feature detection tasks over spatial image domains including detection of scaleinvariant interest points (Lindeberg Lin97IJCV ; Lin12JMIV , Mikolajczyk and Schmid MikSch04IJCV ; Tuytelaars and Mikolajczyk TuyMik08Book ), performing feature tracking (Bretzner and Lindeberg BL97CVIU ), computing shape from texture and disparity gradients (Lindeberg and Gårding LG93ICCV ; Gårding and Lindeberg GL94IJCV ), detecting 2D and 3D ridges (Lindeberg Lin98IJCV ; Sato et al. SatNakShiAtsYouKolGerKik98MIA ; Frangi et al. FraNieHooWalVie00MED ; Krissian et al. KriMalAyaValTro00CVIU ), computing receptive field responses for object recognition (Chomat et al. ChoVerHalCro00ECCV ; Hall et al. HalVerCro00ECCV ), performing hand tracking and hand gesture recognition (Bretzner et al. BreLapLin02FG ) and computing timetocollision (Negre et al. NegBraCroLau08ExpRob ).
Specifically, very successful applications have been achieved in the area of imagebased matching and recognition (Lowe Low04IJCV ; Bay et al. BayEssTuyGoo08CVIU ; Lindeberg Lin12Scholarpedia ; Lin15JMIV ). The combination of local scale selection from local extrema of scalenormalized derivatives over scales (Lindeberg Lin93Dis ; Lin97IJCV ) with affine shape adaptation (Lindeberg and Garding LG96IVC ) has made it possible to perform multiview image matching over large variations in viewing distances and viewing directions (Mikolajczyk and Schmid MikSch04IJCV ; Tuytelaars and van Gool TuyGoo04IJCV ; Lazebnik et al. LazSchPon05PAMI ; Mikolajczyk et al. MikTuySchZisMatSchKadGoo05IJCV ; Rothganger et al. RotLazSchPon06IJCV ). The combination of interest point detection from scalespace extrema of scalenormalized differential invariants (Lindeberg Lin93Dis ; Lin97IJCV ) with local image descriptors (Lowe Low04IJCV ; Bay et al. BayEssTuyGoo08CVIU ) has made it possible to design robust methods for performing object recognition of natural objects in natural environments with numerous applications to object recognition (Lowe Low04IJCV ; Bay et al. BayEssTuyGoo08CVIU ), object category classification (Bosch et al. BosZisMun07ICCV ; Mutch and Lowe MutLow08IJCV ), multiview geometry (Hartley and Zisserman HarZis04Book ), panorama stitching (Brown and Lowe BroLow07IJCV ), automated construction of 3D object and scene models from visual input (Brown and Lowe BroLow053DIM ; Agarwal et al. AgaSnaSimSeiSze09ICCV ), synthesis of novel views from previous views of the same object (Liu LiuYueTor11PAMI ), visual search in image databases (Lew et al. LewSebDjeJai06ACMMulti ; Datta et al. DatJosLiWan08CompSurv ), human computer interaction based on visual input (Porta Por02HumCompStud ; Jaimes and Sebe JaiSeb07CVIU ), biometrics (Bicego et al. BicLagGroTis06CVPRW ; Li Li09EncBiometr ) and robotics (Se et al. SeLowLit05TROB ; Siciliano and Khatib SicKha08HandBookRob ).
Alternative approaches for performing scale selection over spatial image domains have also been proposed in terms of (i) detecting peaks of weighted entropy measures (Kadir and Brady KadBra01IJCV ) or Lyaponov functionals (Sporring et al. SpoCoilTra00ICIP ) over scales, (ii) minimising normalized error measures over scale (Lindeberg Lin97IVC ), (iii) determining minimum reliable scales for edge detection based on a noise suppression model (Elder and Zucker EldZuc98PAMI ), (iv) determining at what scale levels to stop in nonlinear diffusionbased image restoration methods based on similarity measurements relative to the original image data (Mrázek and Navara MraNav03IJCV
), (v) by comparing reliability measures from statistical classifiers for texture analysis at multiple scales (Kang
et al. KanMorNag05ScSp ), (vi) by computing image segmentations from the scales at which a supervised classifier delivers class labels with the highest reliability measure (Loog et al. LooLiTax09LNCS ; Li et al. LiTaxLoo11ScSp ), (vii) selecting scales for edge detection by estimating the saliency of elongated edge segments (Liu et al. LiuWanYaoZha12CVPR ) or (viii) considering subspaces generated by local image descriptors computed over multiple scales (Hassner et al. HasMayZel12CVPR ).More generally, spatial scale selection can be seen as a specific instance of computing invariant receptive field responses under natural image transformations, to (i) handle objects in the world of different physical size and to account for scaling transformations caused by the perspective mapping, and with extensions to (ii) affine image deformations to account for variations in the viewing direction and (iii) Galilean transformations to account for relative motions between objects in the world and the observer as well as to (iv) illumination variations (Lindeberg Lin13PONE ).
Early theoretical work on temporal scale selection in a timecausal scale space was presented in Lindeberg Lin97AFPAC with primary focus on the temporal Poisson scalespace, which possesses a temporal semigroup structure over a discrete timecausal temporal domain while leading to long temporal delays (see Appendix A for a general theoretical argument). Temporal scale selection in noncausal Gaussian spatiotemporal scale space has been used by Laptev and Lindeberg LapLin03ICCV and Willems et al. WilTuyGoo08ECCV for computing spatiotemporal interest points, however, with certain theoretical limitations that are explained in a companion paper Lin16spattempscsel .^{1}^{1}1The spatiotemporal scale selection method in (Laptev and Lindeberg LapLin03ICCV ) is based on a spatiotemporal Laplacian operator that is not scale covariant under independent relative scaling transformations of the spatial vs. the temporal domains Lin16spattempscsel , which implies that the spatial and temporal scale estimate will not be robust under independent variabilities of the spatial and temporal scales in video data. The spatiotemporal scale selection method applied to the determinant of the spatiotemporal Hessian in (Willems et al. WilTuyGoo08ECCV ) does not make use of the full flexibility of the notion of normalized derivative operators Lin16spattempscsel and has not previously been developed over a timecausal spatiotemporal domain. The purpose of this article is to present a much further developed and more general theory for temporal scale selection in timecausal scale spaces over continuous temporal domains and to analyse the theoretical scale selection properties for different types of model signals.
3 Scale selection properties for the noncausal Gaussian temporal scale space concept
In this section, we will present an overview of theoretical properties that will hold if the Gaussian temporal scalespace concept is applied to a noncausal temporal domain, if additionally the scale selection mechanism that has been developed for a noncausal spatial domain is directly transferred to a noncausal temporal domain. The set of temporal scalespace properties that we will arrive at will then be used as a theoretical baseline for developing temporal scalespace properties over a timecausal temporal domain.
3.1 Noncausal Gaussian temporal scalespace
Over a onedimensional temporal domain, axiomatic derivations of a temporal scalespace representation based on the assumptions of (i) linearity, (ii) temporal shift invariance, (iii) semigroup property over temporal scale, (iv) sufficient regularity properties over time and temporal scale and (v) nonenhancement of local extrema imply that the temporal scalespace representation
(1) 
should be generated by convolution with possibly timedelayed temporal kernels of the form (Lindeberg Lin10JMIV )
(2) 
where is a temporal scale parameter corresponding to the variance of the Gaussian kernel and is a temporal delay. Differentiating the kernel with respect to time gives
(3)  
(4) 
see the top row in Figure 1 for graphs. When analyzing prerecorded temporal signals, it can be preferable to set the temporal delay to zero, leading to temporal scalespace kernels having a similar form as spatial Gaussian kernels:
(5) 
3.2 Temporal scale selection from scalenormalized derivatives
As a conceptual background to the treatments that we shall later develop regarding temporal scale selection in timecausal temporal scale spaces, we will in this section describe the theoretical structure that arises by transferring the theory for scale selection in a Gaussian scale space over a spatial domain to the noncausal Gaussian temporal scale space:
Given the temporal scalespace representation of a temporal signal obtained by convolution with the Gaussian kernel according to (1), temporal scale selection can be performed by detecting local extrema over temporal scales of differential expressions expressed in terms of scalenormalized temporal derivatives at any scale according to (Lindeberg Lin97IJCV ; Lin98IJCV ; Lin99CVHB ; Lin14EncCompVis )
(6) 
where is the scalenormalized temporal variable, is the order of temporal differentiation and is a free parameter. It can be shown (Lin97IJCV, , Section 9.1) that this notion of normalized derivatives corresponds to normalizing the th order Gaussian derivatives over a onedimensional domain to constant norms over scale
(7) 
with
(8) 
where the perfectly scale invariant case corresponds to normalization for all orders of temporal differentiation.
Temporal scale invariance.
A general and very useful scale invariant property that results from this construction of the notion of scalenormalized temporal derivatives can be stated as follows: Consider two signals and that are related by a temporal scaling transformation
(9) 
and assume that there is a local extremum over scales at in a differential expression defined as a homogeneous polynomial of Gaussian derivatives computed from the scalespace representation of the original signal . Then, there will be a corresponding local extremum over scales at in the corresponding differential expression computed from the scalespace representation of the rescaled signal (Lin97IJCV, , Section 4.1).
This scaling result holds for all homogeneous polynomial differential expression and implies that local extrema over scales of normalized derivatives are preserved under scaling transformations. Specifically, this scale invariant property implies that if a local scale temporal level level in dimension of time is selected to be proportional to the temporal scale estimate such that , then if the temporal signal is transformed by a temporal scale factor , the temporal scale estimate and therefore also the selected temporal scale level will be transformed by a similar temporal factor , implying that the selected temporal scale levels will automatically adapt to variations in the characteristic temporal scale of the signal. Thereby, such local extrema over temporal scale provide a theoretically wellfounded way to automatically adapt the scale levels to local scale variations.
Specifically, scalenormalized scalespace derivatives of order at corresponding temporal moments will be related according to
(10) 
which means that implies perfect scaleinvariance in the sense that the normalized derivatives at corresponding points will be equal. If , the difference in magnitude can on the other hand be easily compensated for using the scale values of the corresponding scaleadaptive image features (see below).
3.3 Temporal peak
For a temporal peak modelled as a Gaussian function with variance
(11) 
it can be shown that scale selection from local extrema over scale of secondorder scalenormalized temporal derivatives
(12) 
implies that the scale estimate at the position of the peak will be given by (Lindeberg (Lin98IJCV, , Equation (56)) (Lin12JMIV, , Equation (212)))
(13) 
If we require the scale estimate to reflect the temporal duration of the peak such that
(14) 
then this implies
(15) 
which in the specific case of corresponds to (Lin98IJCV, , Section 5.6.1)
(16) 
and in turn corresponding to normalization for according to (8).
If we additionally renormalize the original Gaussian peak to having maximum value equal to one
(17) 
then if using the same value of for computing the magnitude response as for selecting the temporal scale, the maximum magnitude value over scales will be given by
(18) 
and will not be independent of the temporal scale of the original peak unless . If on the other hand using as motivated by requirements of scale calibration (14) for , the scale dependency will for a Gaussian peak be of the form
(19) 
To get a scaleinvariant magnitude measure for comparing the responses of secondorder temporal derivative responses at different temporal scales for the purpose of scale calibration, we should therefore consider a scaleinvariant magnitude measure for peak detection of the form
(20) 
which for a Gaussian temporal peak will assume the value
(21) 
Specifically, this form of postnormalization corresponds to computing the scalenormalized derivatives for at the selected scale (14) of the temporal peak, which according to (8) corresponds to normalization of the secondorder temporal derivative kernels.
3.4 Temporal onset ramp
If we model a temporal onset ramp with temporal duration as the primitive function of the Gaussian kernel with variance
(22) 
it can be shown that scale selection from local extrema over scale of firstorder scalenormalized temporal derivatives
(23) 
implies that the scale estimate at the central position will be given by (Lin98IJCV, , Equation (23))
(24) 
If we require this scale estimate to reflect the temporal duration of the ramp such that
(25) 
then this implies
(26) 
which in the specific case of corresponds to (Lin98IJCV, , Section 4.5.1)
(27) 
and in turn corresponding to normalization for according to (8).
If using the same value of for computing the magnitude response as for selecting the temporal scale, the maximum magnitude value over scales will be given by
(28) 
which is not independent of the temporal scale of the original onset ramp unless . If using for temporal scale selection, the selected temporal scale according to (24) would, however, become infinite. If on the other hand using as motivated by requirements of scale calibration (25) for , the scale dependency will for a Gaussian onset ramp be of the form
(29) 
To get a scaleinvariant magnitude measure for comparing the responses of firstorder temporal derivative responses at different temporal scales, we should therefore consider a scaleinvariant magnitude measure for ramp detection of the form
(30) 
which for a Gaussian onset ramp will assume the value
(31) 
Specifically, this form of postnormalization corresponds to computing the scalenormalized derivatives for at the selected scale (25) of the onset ramp and thus also to normalization of the firstorder temporal derivative kernels for .
3.5 Temporal sine wave
For a signal defined as a temporal sine wave
(32) 
it can be shown that there will be a peak over temporal scales in the magnitude of the th order temporal derivative at temporal scale (Lin97IJCV, , Section 3)
(33) 
If we define a temporal scale parameter of dimension according to , then this implies that the scale estimate is proportional to the wavelength of the sine wave according to (Lin97IJCV, , Equation (9))
(34) 
and does in this respect reflect a characteristic time constant over which the temporal phenomena occur. Specifically, the maximum magnitude measure over scale (Lin97IJCV, , Equation (10))
(35) 
is for independent of the angular frequency of the sine wave and thereby scale invariant.
In the following, we shall investigate how these scale selection properties can be transferred to two types of timecausal temporal scalespace concepts.
4 Scale selection properties for the timecausal temporal scale space concept based on firstorder integrators with equal time constants
In this section, we will present a theoretical analysis of the scale selection properties that are obtained in the timecausal scalespace based on truncated exponential kernels coupled in cascade, for the specific case of a uniform distribution of the temporal scale levels in units of the composed variance of the composed temporal scalespace kernels, and corresponding to the timeconstants of all the primitive truncated exponential kernels being equal.
We will study three types of idealized model signals for which closedform theoretical analysis is possible: (i) a temporal peak modelled as a set of truncated exponential kernels with equal time constants coupled in cascade, (ii) a temporal onset ramp modelled as the primitive function of the temporal peak model and (iii) a temporal sine wave. Specifically, we will analyse how the selected scale levels obtained from local extrema of temporal derivatives over scale relate to the temporal duration of a temporal peak or a temporal onset ramp alternatively how the selected scale levels depends on the the wavelength of a sine wave.
We will also study how good approximation the scalenormalized magnitude measure at the maximum over temporal scales is compared to the corresponding fully scaleinvariant magnitude measures that are obtained from the noncausal temporal scale concept as listed in Section 3.
4.1 Timecausal scale space based on truncated exponential kernels with equal time constants coupled in cascade
Given the requirements that the temporal smoothing operation in a temporal scalespace representation should obey (i) linearity, (ii) temporal shift invariance, (iii) temporal causality and (iv) guarantee noncreation of new local extrema or equivalently new zerocrossings with increasing temporal scale for any onedimensional temporal signal, it can be shown (Lindeberg Lin90PAMI ; Lin15SSVM ; Lin16JMIV ; Lindeberg and Fagerström LF96ECCV ) that the temporal scalespace kernels should be constructed as a cascade of truncated exponential kernels of the form
(36) 
If we additionally require the time constants of all such primitive kernels that are coupled in cascade to be equal, then this leads to a composed temporal scalespace kernel of the form
(37) 
corresponding to Laguerre functions (Laguerre polynomials multiplied by a truncated exponential kernel) and also equal to the probability density function of the Gamma distribution having a Laplace transform of the form
(38) 
Differentiating the temporal scalespace kernel with respect to time gives
(39)  
(40) 
see the second row in Figure 1 for graphs. The norms of these kernels are given by
(41)  
(42) 
The temporal scale level at level corresponds to temporal variance
and temporal standard deviation
.4.2 Temporal peak
Consider an input signal defined as a timecausal temporal peak corresponding to filtering a delta function with firstorder integrators with time constants coupled in cascade:
(43) 
With regard to the application area of vision, this signal can be seen as an idealized model of an object with temporal duration that first appears and then disappears from the field of view, and modelled on a form to be algebraically compatible with the algebra of the temporal receptive fields. With respect to the application area of hearing, this signal can be seen as an idealized model of a beat sound over some frequency range of the spectrogram, also modelled on a form to be compatible with the algebra of the temporal receptive fields.
Scale estimate and maximum magnitude from temporal peak (uniform distr)  

(var, )  (var, )  (var, )  (, )  
4  3.1  0.504  6.1  10.3 
8  7.1  0.502  14.1  18.3 
16  15.1  0.501  30.1  34.3 
32  31.1  0.500  62.1  66.3 
64  63.1  0.500  126.1  130.3 
Scale estimate and maximum magnitude from temporal ramp (uniform distr)  

(var, )  (, )  (var, )  (, )  
4  3.2  3.6  0.282  0.254 
8  7.2  7.7  0.282  0.272 
16  15.2  15.8  0.282  0.277 
32  31.2  31.8  0.282  0.279 
64  63.2  64.0  0.282  0.281 
Define the temporal scalespace representation by convolving this signal with the temporal scalespace kernel (43) corresponding to firstorder integrators having the same time constants
(44) 
where we have applied the semigroup property that follows immediately from the corresponding Laplace transforms
(45) 
By differentiating the temporal scalespace representation (44) with respect to time we obtain
(46)  
(47) 
implying that the maximum point is assumed at
(48) 
and the inflection points at
(49)  
(50) 
This form of the expression for the time of the temporal maximum implies that the temporal delay of the underlying peak and the temporal delay of the temporal scalespace kernel are not fully additive, but instead composed according to
(51) 
If we define the temporal duration of the peak as the distance between the inflection points, if furthermore follows that this temporal duration is related to the temporal duration of the original peak and the temporal duration of the temporal scalespace kernel according to
(52) 
Notably these expressions are not scale invariant, but instead strongly dependent on a preferred temporal scale as defined by the time constant of the primitive firstorder integrators that define the uniform distribution of the temporal scales.
Scalenormalized temporal derivatives.
When using temporal scale normalization by variancebased normalization, the first and secondorder scalenormalized derivatives are given by
(53)  
(54) 
When using temporal scale normalization by normalization, the first and secondorder scalenormalized derivatives are on the other hand given by (Lindeberg (Lin16JMIV, , Equation (75)))
(55)  
(56) 
with the scalenormalization factors determined such that the norm of the scalenormalized temporal derivative computation kernel
(57) 
equals the norm of some other reference kernel, where we here take the norm of the corresponding Gaussian derivative kernels (Lindeberg (Lin16JMIV, , Equation (76)))
(58) 
for , thus implying
(59)  
(60) 
where and denote the norms (7) of corresponding Gaussian derivative kernels for the value of at which they become constant over scales by normalization, and the norms and of the temporal scalespace kernels and for the specific case of are given by (41) and (4.1).
Temporal scale selection.
Let us assume that we want to register that a new object has appeared by a scalespace extremum of the scalenormalized secondorder derivative response.
To determine the temporal moment at which the temporal event occurs, we should formally determine the time where , which by our model (54) would correspond to solving a thirdorder algebraic equation. To simplify the problem, let us instead approximate the temporal position of the peak in the secondorder derivative by the temporal position of the peak according to (48) in the signal and study the evolution properties over scale of
(61) 
In the case of variancebased normalization for a general value of , we have
(62) 
and in the case of normalization for
(63) 
To determine the scale at which the local maximum is assumed, let us temporarily extend this definition to continuous values of and differentiate the corresponding expressions with respect to . Solving the equation
(64) 
numerically for different values of then gives the dependency on the scale estimate as function of shown in Table 2 for variancebased normalization with either or and normalization for .
As can be seen from the results in Table 2, when using variancebased scale normalization for , the scale estimate closely follows the scale of the temporal peak and does therefore imply a good approximate transfer of the scale selection property (14) to this temporal scalespace concept. If one would instead use variancebased normalization for or normalization for , then that would, however, lead to substantial overestimates of the temporal duration of the peak.
Furthermore, if we additionally normalize the input signal to having unit contrast, then the corresponding timecausal correspondence to the postnormalized magnitude measure (20)
(65) 
is for scale estimates proportional to the temporal duration of the underlying temporal peak very close to constant under variations of the temporal duration of the underlying temporal peak as determined by the parameter , thus implying a good approximate transfer of the scale selection property (21).
4.3 Temporal onset ramp
Consider an input signal defined as a timecausal onset ramp corresponding to the primitive function of firstorder integrators with time constants coupled in cascade:
(66) 
With respect to the application area of vision, this signal can be seen as an idealized model of a new object with temporal diffuseness that appears in the field of view and modelled on a form to be algebraically compatible with the algebra of the temporal receptive fields. With respect to the application area of hearing, this signal can be seen as an idealized model of the onset of a new sound in some frequency band of the spectrogram, also modelled on a form to be compatible with the algebra of the temporal receptive fields.
Define the temporal scalespace representation of the signal by convolution with the temporal scalespace kernel (43) corresponding to firstorder integrators having the same time constants
(67) 
Then, the firstorder temporal derivative is given by
(68) 
which assumes its temporal maximum at .
Temporal scale selection.
Let us assume that we are going to detect a new appearing object from a local maximum in the firstorder derivative over both time and temporal scales. When using variancebased normalization for a general value of , the scalenormalized response at the temporal maximum in the firstorder derivative is given by
(69) 
When using normalization for a general value of , the corresponding scalenormalized response is
(70) 
where the norm of the firstorder scalespace derivative kernel can be expressed in terms of exponential functions, the Gamma function and hypergeometric functions, but is too complex to be written out here. Extending the definition of these expressions to continuous values of and solving the equation
(71) 
numerically for different values of then gives the dependency on the scale estimate as function of shown in Table 2 for variancebased normalization with or normalization for .
As can be seen from the numerical results, for both variancebased normalization and normalization with corresponding values of and , the numerical scale estimates in terms of closely follow the diffuseness scale of the temporal ramp as parameterized by . Thus, for both of these scale normalization models, the numerical results indicate an approximate transfer of the scale selection property (14) to this temporal scalespace model. Additionally, the maximum magnitude values according to (
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