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On Witkin Range of Phase Congruency

Anonymous CVPR submission

Paper ID 648

Abstract

This article identifies a scale manifestative concept inlow-level vision, which we call Witkin range of phase con-gruency, and proposes a simple method for calculating thisimage feature. This concept is similar to the range of stabil-ity in Witkin’s scale space filtering, but we define it in termsof the phase congruency among Gabor-type wavelets of dif-ferent frequencies. The Witkin range of phase congruencyleads to a representational and computational scheme forcombining image information from multiple scales. In par-ticular, it adds two new dimensions to the traditional edgerepresentation produced by Canny edge detector, namely,the width and sharpness of the edge point. As a result, itcombines the edge representation and region representationinto an edged-region representation. In addition, this con-cept unifies two ubiquitous classes of visual phenomena,namely, geometric structures and stochastic textures, in ascale manifestative framework, which can account for thecontinuous transition from structures to textures in the pro-cess of image scaling or zooming. We illustrate our methodby a number of experiments on natural images.

1. Introduction

Scale is one of the most important issue in vision. Visualphenomena in natural scenes can appear at a wide rangeof scales in images, because of the variabilities in objectsizes, viewing distances, and camera resolution. Therefore,a meaningful interpretation of a natural image must be ei-ther scale invariant or scale manifestative. “Scale invariant”means that the interpretation will stay invariant under imagescaling. “Scale manifestative” means that the interpretationhas explicit scale parameters that follow simple and explicittransformations under image scaling.

There have been a number of multi-scale theories in vi-sion, most notably, the scale space theory [21, 8] and themulti-resolution wavelet analysis [11]. It has been a com-mon sense that we need to combine information from mul-tiple scales, mainly because some visual phenomena suchas edges can persist over a range of scales.

The scale persistency has long been observed. For in-stance, Marr [12] proposed the “coincidence assumption,”which holds that only those features that spatially coin-cident at all scales are meaningful. Witkin, in his paperon scale space filtering [21], investigated the persistencyof local maxima of Gaussian derivatives of 1D signal overscales, and explicitly identified the stability ranges of theselocal maxima in scale space. These ranges can then betranslated into flat intervals as basic elements for represent-ing the 1D signal. Witkin’s idea has been extended to 2Dby Lindeberg [8, 9] and other researchers. But the behaviorof the maxima of Gaussian derivatives in 2D is much morecomplicated than 1D, so that tracing maxima in 2D can bedifficult.

Parallel to scale space theory, the coincidence over scaleshas been extensively studied in the context of phase con-gruency of Fourier transform or Gabor wavelet transform.Morrone et al. [14] observed that image features appear atlocations where the Fourier components of the image at dif-ferent frequencies are maximally in phase. A phase congru-ency function is defined to measure the agreement amongthe phases at each position, and this function turns out to beequal to the local energy of the image [19, 16]. Kovesi [6]developed a computational method for phase congruencyfeature detection in 2D image, using log-Gabor wavelets tocompute local phases and energies. The phase congruencyfunction is defined on each pixel. Unlike Witkin’s scalespace filtering, it does not involve tracing over scales or fre-quencies, and is therefore simpler to implement in 2D.

The phase congruency function is elegant in terms of itsrelationship with image energy. However, it is not scalemanifestative, in the sense that it does not tell us the rangeof frequencies over which the phases are congruent. In thispaper, we introduce Witkin’s idea of range of stability tothe framework of phase congruency, and replace the phasecongruency function by the phase congruency range or whatwe call Witkin range.

The Witkin range gives us a more informative descrip-tion of image features. For instance, for an image structuresuch as an edge, the Witkin range can be translated into geo-metric scale parameters of the cross-section profile perpen-

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dicular to the edge elongation. In particular, the high fre-quency end of the Witkin range tells us the sharpness of thetransition of the image intensity across the edge, whereasthe low frequency end of the Witkin range tells us the widthor breadth of the two flat regions on the two sides of theedge. In other words, the Witkin range enables us to notonly detect the edge curves, but also recognize the edged-regions, so that we can form a representation that combinesboth edge-based concept and region-based concept.

In natural scenes, there are two ubiquitous classes of vi-sual phenomena. One is geometric structures that can berepresented by lines and regions. The other is stochastic tex-tures that are often characterized by some feature statistics.Although these two types of patterns often appear distinc-tively different, they are actually intrinsically connected:the same group of objects can be perceived as either ge-ometric structures or stochastic textures depending on theviewing distance and camera resolution. Due to this scal-ing connection, it is natural to believe that the visual sys-tem must estimate scale parameters explicitly, and trace thechange of the scale parameters over the image scaling pro-cess that can be caused by the change of viewing distance.We shall show that the Witkin range provides us with cru-cial scale information for describing large scale geometricstructures and small scale stochastic textures.

The Witkin range of phase congruency can be useful foredge representation, edge-based object recognition, track-ing and matching, and texture recognition. It also shedslight on low-level vision theories such as sparse coding,meaningful alignment, and natural image statistics.

2. Background

2.1. Witkin stability range

A key motivation for Witkin [21] to propose his theory ofscale space filtering is to combine visual information acrossdifferent scales. In particular, he studied the stability of thespatial locations of local maxima of Gaussian derivatives ofthe image data over scales.

See Fig.1 for an example. A 1D signal is taken froma slice of the image in Fig.1.a. Let’s denote this 1D sig-nal byu(x). Let Gσ(x) be a Gaussian kernel function (ordensity function) centered at 0 with standard deviationσ.Let uσ = u ∗ Gσ be the convolution ofu(x) with Gσ(x).Fig. 1.c displaysuσ(x) for a sample of scalesσ > 0. Foreachuσ(x), we can find the local maxima of its first deriva-tive ∂uσ(x)/∂x, or the zero-crossings of its second deriva-tive ∂2uσ(x)/∂x2. Fig.1.b plots the contours of these zero-crossings in the scale space. Clearly, the zero-crossings per-sist over a range of scales, until two zero-crossings mergeinto a singular point. The range of persistence or stabil-ity depends on the widths of the underlying intervals. As amatter of fact, one can recover these intervals based on the

Figure 1. (a) A 1D signal is obtained as a horizontal slice of thetoaster image. (b) The contour plot of the zero-crossings of thesecond derivatives in the joint spatial-scale domain. (c) The 1Dsignal at multiple resolutions, obtained by convolving the signalwith Gaussian kernels and sub-sampling the signal.

stability ranges.Lindeberg [8, 9] applied similar ideas to 2D images. But

the behavior of the maxima of Gaussian derivatives in 2D ismuch more complicated than 1D, so that tracing maxima in2D can be difficult.

2.2. Phase congruency function

In contrast to the scale space filtering based on localderivative operators, the phase congruency theory startedfrom global Fourier transform. Morrone et al. [14] observedthat for a signalu(x), the feature points correspond to thosepoints where the Fourier waves at different frequencies havecongruent phases. Specifically, let

u(x) =

∫A(ω) cos(ωx + φω)dω (1)

be the Fourier representation ofu(x). The phase of fre-quencyω at a pointx is ωx + φω mod2π. Thosex whereωx + φω are congruent acrossω are considered featurepoints. For instance, the top plot of Fig.2 shows a pe-riodic step function. The bottom plot displays several ofits Fourier componentsA(ω) cos(ωx + φω) (see eqn. (1)).Clearly, the edge points and the center points of the inter-vals correspond to those points where the Fourier waves ofdifferent frequencies are in phase.

A phase congruency function is defined as follows:

ϕ(x) = maxφ∈[0,2π]

∫A(ω) cos(ωx + φω − φ)G(ω)dω∫

A(ω)G(ω)dω, (2)

whereG(ω) is a window function in the frequency domain,e.g., a Gaussian kernel around a certain central frequency.Suppose the maximum is achieved atφ = φ(x). φ(x) maybe interpreted as the average phase across the frequencies

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Figure 2. A periodic step function and its several Fourier com-ponents. The waves at different frequencies are in phase at edgepoints and the center points of the intervals.

covered by the windowG(ω), andϕ(x) measures the vari-ation of phases within this window.

The phase congruency function (2) has an elegant con-nection with the local energy [19]. Specifically, let

G ∗ u(x) =

∫G(ω)A(ω) expi(ωx + φω)dω,

where G is the complex filter, which consists of a pairof filters of quadrature phase, whose Fourier transform isG(ω). Thenϕ(x) = |G ∗ u(x)| is the local energy, andφ(x) = arg[G ∗ u(x)] is the local phase. So the points ofmaximum phase congruency correspond to points of maxi-mum local energy.

Kevosi [6] defines a phase congruency function by pool-ing the information from a bank of log-Gabor filters at dif-ferent scales and orientations, in the same spirit as function(2). Unlike scale space filtering, the phase congruency func-tion is defined for each pixel without tracing local maxima.

2.3. Gabor filters and edge detection

A biologically motivated class of image elements are Ga-bor wavelets [2], which are rotated, dilated, and translatedcopies of the following Gaussian modulated sine and cosinewaves [7]

G(x) ∝ exp−1

8(4x2

1 + x22)(e

iκx1 − eκ2/2), (3)

wherex = (x1, x2). Let’s denote a rotated, dilated, andtranslated copy of (3) by Gx,ω,θ, wherex is the center,ωis the frequency of the sine and cosine waves, andθ is theorientation. We normalize the Gabor wavelets over scale sothatGx,sω,θ = sGx,ω,θ, in order to maintain

〈f,Gx,ω,θ〉 = 〈fs, Gx,sω,θ〉, (4)

wherefs(x) = f(sx) is the scaled version off . The Gaborfilters can be replaced by other zero-mean filter pairs thatform Hilbert transforms of each other.

The Gabor filters can be used as edge detectors [20].For the dictionary of Gabor elementsGx,ω,θ, at each fre-quencyω, and at each pixelx, we find the optimal orienta-tion θ = arg maxθ |〈u,Gx,ω,θ〉|

2, where|〈u,Gx,ω,θ〉|2 =

〈u,G(0)x,ω,θ〉

2 + 〈u,G(1)x,ω,θ〉

2, with G(0)x,ω,θ and G

(1)x,ω,θ be-

ing cosine and sine components ofGx,ω,θ respectively,and 〈〉 denoting inner product. LetA = |〈u,Gx,ω,θ〉|,

φ = arctan[〈u,G(0)

x,ω,θ〉/〈u,G

(1)

x,ω,θ〉] be the magnitude (or

energy) and phase at the maximal orientation respectively.We can write

[∇ωu](x) = (Aω(x), θω(x), φω(x)) = (A, θ, φ) (5)

as a generalized version of the ubiquitous gradient operator∇u.

A point (x, ω) is an edge-ridge point if

Aω(x) ≥ Aω(x + t(sin θω(x), cos θω(x))), |t| < d, (6)

i.e., Aω(x) is maximal along the normal direction of theorientationθω(x) within a neighborhood of length2d [1].Edges and ridges can be discriminated by the correspondingphases [20].

3. Witkin range of phase congruency

In this section, we explain the basic idea of the Witkinrange. We also explain that it can be translated into anedged-region representation of the image. After that, wegive a precise definition of the Witkin range, and illustratethe edged-region representation by some examples.

3.1. Edged-region representation

Fig. 3 illustrates a fundamental observation. The ploton top shows a horizontal slice of an image of vertical baru(x). The second and the third plots display the magni-tude and phase of∇ωu(x) on this slice, where each curvecorresponds to a frequencyω. It is evident that an edge-ridge pointx (i.e., a local maximum in magnitude curves)can exist over a range of frequencies(ω0(x), ω1(x)), andwithin this range, the phase and orientation of∇ωu(x) re-mains constant forω ∈ (ω0(x), ω1(x)). For edge points, themagnitude of∇ωu(x) also remains constant (subject to dis-cretization error). In the bottom plot of Fig.3, we trace thetwo edge points over scales ∝ 1/ω. At a certain frequencyor scale, the two edge points merge into a ridge point.

Here comes the foundation of this paper: for an edgepointx, the rangeω ∈ (ω0(x), ω1(x)) in frequency domaincan be translated into spatial domain parameters about thecross-section profile of the edge. Specifically, the profile ofan edge is along the direction that is perpendicular to theedge elongation, and it can be modeled by a step functionblurred by a Gaussian kernel [3], whose bandwidth reflects

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Figure 3. Constancy of positions and phases of local energy max-ima across frequencies. The top plot depicts a horizontal slice ofan image of a vertical bar. The next two plots show local energyand phase, where each curve corresponds to a frequency. The bot-tom plot shows that the maxima corresponding to two edge pointsmerge into one ridge point over frequencies or scales.

(a) input image (b) edged-regions (c) ridged-region

Figure 4. Image interpretation by edged- and ridged- regions.

(a) input image (b) edged-regions

Figure 5. Edged-region representation.

the sharpness of the edge. For the Witkin range, the highfrequency endω1(x) tells us how sharp the intensity transi-tion is across the edge. The low frequency endω0(x) tellsus how wide the two flat pieces of the step function canextend. Asx runs on the one dimensional edge curve, theresulting cross-section profile sweeps an edged-region withthe edge curve being the mid-axis.

See Fig.4 and Fig.5 for illustrations of edged-region

representation. An edged-region is composed of two seg-ments of smooth sub-regions, colored by grey and greenrespectively in the above two figures. The two sub-regionsare separated by an edge curve. The widths or breadths ofthe two sub-regions are decided by the low frequency endω0(x) of the Witkin range. The sharpness of the segmenta-tion is decided by the high frequency endω1(x). That is, bycombining the Gabor edge information across frequencies,we essentially perform a local image segmentation, wherethe Gabor filters at different frequencies explore the twosub-regions being segmented by the edge. This enables usto not only detect the edge, but also recognize the edged-region. The edged-region representation combines both theedge concept and the region concept, which are two mostprominent representations in low- and mid- level vision.

Viewed in frequency domain, an edged-region is a com-position of Gabor wavelets across the Witkin range of fre-quencies, or an edged-region spans a range of frequencies.

The Witkin range transforms in a simple way during thescaling process. When we zoom out the image by a factor ofs, the Witkin range(ω0, ω1) will be scaled to(sω0, sω1). Sothe edged-region becomes thinner and sharper by a factor ofs.

An interesting observation is that, due to finite resolutionof the image, the Witkin range(sω0, sω1) will eventually gobeyond the frequency limit of the camera resolution ass in-creases, and the edged-region will be shredded and leakedout. This can explain the transition from geometric struc-tures to stochastic textures, as we will study later.

3.2. Definition of Witkin range of phase congruency

In 2D images, tracing the edge points can be difficult.Recall that the phase congruency function (2) is defined foreach pixel without tracing. Similarly, we can also define theWitkin range of phase congruency without tracing the edgepoint. The following is our version of definition.

The Witkin range is defined on scale-maximum edgepoints. An edge point(x, ω) is a scale-maximum edge pointif 1) (x, ω) is an edge point at frequencyω in the sense ofinequality (6); 2) it is also a local maxima in scale or fre-quency domain:

Aω(x) ≥ Aω(x), ∀ω ∈ (ω − δ, ω + δ), (7)

i.e., a small neighborhood ofω. 3) Its phaseφω(x) is dom-inated by sine component of the Gabor filter. The scale-maximum edge points transform in a simple way duringimage scaling.

For a scale-maximum edge point(x, ω), we define itsWitkin range(ω0, ω1) as a continuous range of frequenciesω aroundω so that∇ωu(x) has almost constant magnitude,phase and orientation. Specifically, let

Ω = ω : ∇ωu(x) ∈ ∂(∇ωu(x)), (8)

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where∂(∇ωu(x)) is a small neighborhood of∇ωu(x). Re-call that∇ωu(x) = [Aω(x), θω(x), φω(x)], i.e., magnitude(energy), orientation, and phase, see eqn. (5). In our imple-mentation,∇ωu(x) ∈ ∂(∇ωu(x)) if

Aω(x)/Aω(x) ≥ fA,

|θω(x) − θω(x)| ≤ ǫθ,

|φω(x) − φω(x)| ≤ ǫφ, (9)

that is, the magnitudeAω(x) should be within a factor (e.g.,fA = .8) of Aω(x), the orientation and phase should beclose to those of(ω, x) (e.g.,ǫθ = π/12, andǫφ = π/6).Then

ω0 = maxω ≤ ω, ω /∈ Ω,

ω1 = minω ≥ ω, ω /∈ Ω. (10)

We can translate(ω0, ω1) to Witkin width s0 ∝ 1/ω0, andWitkin sharpnesss1 ∝ 1/ω1. The proportion factor canbe chosen so that when applied to a bar structure with twoparallel edges (see Fig.3), the Witkin width should agreewith the half-width of the bar. It is clear that this proportionfactor depends onfA, ǫθ, andǫφ in (9).

Our definition of Witkin width generalizes the traditionaldefinition of width for bar structures to any geometric struc-tures. For instance, Fig.6 shows the edged-region plots ofa triangle and a circle. Here we only plot the darker sub-region of the edged-region. Specifically, at each edge pointx, we plot a black bar (1 pixel wide) of lengths0, i.e., theWitkin width of this edge point. The bar is perpendicularto the edge elongation, and extends to the darker segmentof the edged-region. Then the bars for all the edge pointsmake up the darker sub-region of the edged-region.

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Figure 6. Edged-region representations of triangle and circle. Onlythe dark sides of the edges are plotted.

Fig. 7, Fig. 8, and Fig.9 show the edge-region repre-sentations of three natural scene images. Note that in our

implementation, there is an upper bound on the scale ofthe Gabor filters (or equivalently, a lower bound on the fre-quency), so there is an upper bound on Witkin width. That iswhy at some points, the Witkin edges are not wide enough.

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Figure 7. A natural scene image and its edged-region representa-tion.

The above definition of Witkin range is clearly scalemanifestative. If we scale the image by a factor, then theWitkin width and sharpness should scale in the same way,as long as they are above the camera resolution.

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Figure 8. A natural scene image and its edged-region representa-tion.

3.3. Ridged-region representation

The top plot of Fig.3 displays the cross-section profileof a bar structure. A key point is that a bar structure isnot only described by the width of the central flat interval,but also the widths of two flat wings on the two sides ofthe central interval. So a ridged-region should have threesegments, corresponding to the central piece and the two

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Figure 9. A natural scene image and its edged-region representa-tion.

wings respectively. For an ridge point, we can also definea Witkin range that can be translated to the width of thecentral piece and the width of the two flat wings. Due tospace limit, we shall not elaborate on this. We just wantto point out that the width of the central piece is intimatelyrelated to the Witkin width of the two parallel edges of thebar structure. But the Witkin width of an edge is a far moregeneral definition, because many edges are not edges of abar structure. For instance, the base line of the triangle inFig. 4, or the edge on the circle in Fig6, or the shorter sideof a rectangle.

The width of the edge structure and the width of the twoflat wings of the bar structure go beyond the important workof Lindeberg on scale selection for edges and bars [10]. Itappears that Lindeberg’s scales correspond to the sharpnessof the edge and the width of the central piece of the bar.

4. Unifying structures and textures

Geometric structures and stochastic textures are oftentreated separately in computer vision. Structures are usu-ally obtained by edge detection and image segmentation,while textures are mostly characterized by feature statisticssuch as histograms of filter responses [4]. However, struc-tures and textures are intrinsically connected by image scal-ing. Fig.10displays a sequence of images of an ivy wall of

leaves taken at increasingly far distances. At near distance,the geometric shapes of individual leaves are perceptible.But as the viewing distance increases, the image becomesmore complex and the individual shapes become impercep-tible, and the image can only be described by a collectivetexture summary.

Figure 10. A sequence of ivy wall images taken at increasingly fardistances.

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Figure 11. Conditional expectations of Witkin width, sharpness,and range over scale. Plain curve shows starting scale or Witkinsharpness, circled curve shows ending scale or Witkin width, andcrossed curve shows the difference between starting and endingscales, or the size of Witkin range. Each plot is conditioned on aninterval of magnitudes of scale-maximum edge points. The mag-nitudes of the plots are in increasing order from left to right.

This suggests that geometric structures and stochastictextures should be treated in a unified framework. The dis-tinction between structures and textures is an artificial one,because the transition from structures to textures is a con-tinuous process caused by the continuous image scaling orzooming. It is therefore desirable to have a scale manifesta-tive quantity to trace this transition.

It has been a mystery how human being perceives thewide variety of texture patterns. Julesz [5], in his study ofhuman texture perception over nearly three decades, pro-posed two famous conjectures. The first conjecture is abouttexture statistics, and Julesz proposed co-occurrence statis-tics of image intensities. The second conjecture is abouttextons, which are considered as basic elements for texture

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perception. We believe that image scaling holds the secretto this puzzle, and scale manifestative quantities must be acrucial ingredient in texture perception.

This can be seen even more evidently in Fig.12, wherethe texture surfaces of flower leaves and pebbles are slantsurfaces that appear in perspective. Clearly, our perceptionof the texture surfaces is not homogenous and we perceivea gradual change over distance.

The Witkin range of phase congruency can be used totrace the transition from structure to texture, as well as thechange of texture information over distance.

We pool the following scale manifestative texture statis-tics. Let (x, ω) be a randomly selected scale-maximumedge point. LetA be the magnitude of∇ωu(x). Let (s1, s0)be its Witkin sharpness and width defined in previous sec-tion. We experiment with the following three statisticalproperties.E[s0|A], E[s1|A], andE[s0 − s1|A], that is, theconditional expectations of Witkin sharpness, width, andrange. These three statistics can be estimated as follows.We divide the range ofA into several intervals. We collectthe scale-maximum edge points whose magnitudes fall intoeach interval, and then estimate the conditional expectationsfor this interval by corresponding averages.

Fig. 11 shows the change of conditional expectationsover the scaling process. We choose 8 images of the ivywall taken at increasingly far distances. We divide the mag-nitudes of scale-maximum edge points into three intervals.In Fig. 11, the three plots correspond to the three intervalsof magnitudes in increasing order, from left to right. We cansee that as the viewing distance increases, the Witkin width,sharpness and range decrease in general.

These plots also trace the transition from structures totextons and texture statistics. At near distance, we see rel-atively large edged-regions. As the distance increases, theedged-regions become smaller. This roughly correspond tothe texton regime in Julesz’s second conjecture. If the view-ing distance increases still further, the edged-regions willbe smaller than the pixel resolution, and they will be shred-ded and leaked out. Then there are no significant align-ments among filter responses, and we may just pool somemarginal statistics from the filter responses [4], since thejoint patterns have been largely destroyed by image scaling.This roughly corresponds to the regime of texture statisticsin Julesz’s first conjecture.

The Witkin width also indicates the size of the neighbor-hood that we should use to pool the texture statistics. Thelarger the Witkin width, the larger the local window shouldbe for spatial pooling.

The above statistics are also crucial for perceiving slantsurfaces such as those in Fig.12. Fig. 13 show the changeof conditional expectations of Witkin width, sharpness, andrange over vertical axis of the two images. It is possible thatsuch statistics can be used for re-constructing 3D informa-

Figure 12. Slant texture surfaces of flower leaves and stones.

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Figure 13. Conditional expectations of Witkin width, sharpnessand range over the vertical axis in the two images in Fig12.

tion of slant texture surfaces.

5. Discussion

5.1. Contributions and open ends

The following are contributions of this article.1) Identify and define the Witkin range for phase con-

gruency, as a substitute for scale space tracing and phasecongruency function.

2) Define the Witkin width of edge point, and proposethe edged-region (as well as ridged-region) representationthat combines both edge concept and region concept.

3) Study geometric structures and stochastic textures ina unified scale manifestative framework, and define a set ofscale manifestative texture statistics.

The following are two major open ends of our work.1) The current version of Witkin range may not be the-

oretically or empirically superior to other possible alterna-tives. We hope this work will stimulate more researchers to

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experiment with this concept and search for better versions.2) There are other image structures such as roofs, ramps,

as well as topological structures such as corners and junc-tions. The geometric scale parameters of these structuresshould also be estimated based on similar ideas.

5.2. Potential applications

1) Edge feature. Our method is not in competition withCanny [1] or other edge detection methods. Instead, itequips each edge point with two important scale parame-ters, namely, sharpness and width.

2) Edge-based object recognition. For instance, for anobject like a tree, the Witkin width is useful for identifyingtree trunk, branches and twigs, without resorting to sophis-ticated region-based analysis.

3) Tracking and matching. In real life, objects canchange distances from the viewer rapidly, e.g., a ball is com-ing, a dog is running away, or the scene outside the windowof a moving train. The changes of Witkin ranges help usperceive the change of viewing distances.

4) Texture recognition and shape from texture.

5.3. Connections to other vision theories

1) Sparse coding. Olshausen and Field [15] proposedsparse coding as a strategy for V1. Our work suggeststhat the sparse coding elements are compositions of phase-congruent Gabor wavelets or edged-regions.

2) Meaningful alignment. Moisan, Desolneux, andMorel [13] proposed meaningful alignment as a statisticalprinciple for perceptual grouping. Our work can be consid-ered as identifying meaningful alignment over scales or infrequency domain.

3) Natural image statistics. Portilla and Simoncelli [17]proposed a class of joint statistics of filter responses to char-acterize texture patterns. The Witkin range can be consid-ered a scale explicit characterization of the joint distribu-tion. Ruderman and Bialek [18] studied the scaling of im-age statistics of natural scenes. The Witkin range statisticsare worth of being investigated.

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