A Luminance- and Contrast-InvariantEdge-Similarity Measure

Saravana Kumar, Sim Heng Ong, Member, IEEE,Surendra Ranganath, and

Fook Tim Chew

Abslracl-A novel similarity measure for edge-detection that is robust to varyingluminance and contrast is presented. It incorporates a regularization term andemploys directional FIR edge filters with hyperbolic tangent profiles to ensureimproved noise performance and edge localization compared to classical methods.

THE accurate detection of edges is often not a straightforward taskdue to sensitivity to image contrast, noise and, to some extent,uneven illumination. Despite the importance of developing anedge detection method that is robust under these conditions,research toward this end [7], [8], [9], [11], [17] is limited.

Classical gradient-magnitude (GM) methods [1], [10], [12] areusually dependent on edge strength; hence, weaker edges such asthose due to texture may not be detected. Frei and Chen [5] haveproposed an alternative method of detecting valid edges regardlessof their magnitude. We term their approach an angle-based (AN)method since it is based on the computation of the cosine of theprojection angles between neighborhoods and predefined edgefilters. A problem with this technique is its sensitivity to noise anduneven illumination. Methods based on local thresholding of imagegradients are also sensitive to uneven illumination since they tend toinhibit edges in regions of low lurriinance [16] or, conversely,enhance them [6]. In general, edge detection methods that are robustunder varying contrast tend to be more affected by noise.

The spatial profile of the edge filter is another factor thatcontributes towards noise sensitivity. Canny's Gaussian firstderivative (GFD) filter [1] may be regarded as an optimal step-edgedetector. However, both the Canny [1] and Marr filters [10] requiresmoothing as a preprocessing step to reduce noise, and this blurringmakes weak edges harder to detect [19]. Canny's optimal filter isalso derived for an ideal step edge model [1], but in reality, imageedges have blurred profiles arising from the digital imageacquisition process.

Morrone et a1. [11] and later Kovesi [7] described a technique inwhich images are represented in the frequency domain and edgesoccur at points of maximum phase congruency. Such phasecongruency (PC) methods are invariant to changes in illuminationand contrast. Although they exhibit better contrast invariance thanGM methods, they give poorer edge localization in that false edgesare detected in the vicinity of sharp transitions. This is due to the

• S. Kumar, S.H. Ong, and S. Ranganath are with the Department ofElectrical and Computer Engineering, National University of Singapore,Block E4, Level 5, Room 48, 4 Engineering Drive 3, Singapore 117576.E-mail: lengp2453.eleongsh.elesr)@nus.edu.sg .

• F.T. Chew is with the Department of Biological Sciences, NationalUniversity of Singapore, 14 Science Drive 4, Singapore 117543.E-mail: dbscft@nus.edu.sg.

Manuscript received 23 Aug. 2005; revised 11 May 2006; accepted 11 May2006; published online 12 Oct. 2006.Recommended for acceptance by G. Sapiro.For information on obtaining reprints of this article, please send e-mail to:tpami@computer.org,and reference IfEECS Log Number TPAMI-0454-0805.

use of the log polar Gabor filter, which has multiple zero crossingsin addition to the main zero crossing at the filter's spatial origin.

More recently, Desolneux et a1. [3] proposed a contrast-invariant edge detection method based on the Helmholtz principle.It is a parameter-free method that defines edges as geometricstructures with large deviations from randomness. The detection ofa given edge is sensitive to the size of the windowed region whileedge localization is inferior to that of Canny's edge detector.

In this paper, we propose an edge detection algorithm that ismore robust than the GM and AN methods in the presence of varyingillumination, contrast, and noise level, and results in betterlocalizations of sharp transitions in an image compared toPC methods. It is based on a measure of edge similarity betweenimage neighborhoods and directional finite impulse response (FIR)edge filters with hyperbolic tangent (HBT) profiles. A balancebetween the conflicting requirements of contrast invariance andnoise tolerance is obtained by using a regularization term in thesimilarity measure.

Here, we briefly discuss the classical GM and AN methods andtheir shortcomings. Both methods are based on the measure ofsimilarity between an image neighborhood of size n x n and apredefined set of directional edge filters. However, the methodsdiffer in their definition of the similarity measure used. In the GMmethod, a normalized similarity measure 6i is defined as

- Ci

Ci = max{ICil, 1::; i ::;N}'

is the ilmer product of bi and g, which are the n2 x 1 columnrepresentations of the n x n image neighborhood centered at pixel iand the 2D edge filter, respectively, and N is the total number ofimage pixels. The drawback of this method is its sensitivity to edgestrength since weak edges tend to be suppressed in the presence ofsignificantly stronger edges in the image. Therefore, it may bedifficult to determine suitable thresholds to accurately detect bothstrong and weak edges.

Frei and Chen [5] introduced an angle-based method thatdetects edges based on the similarity of the image neighborhoodsto a predefined filter irrespective of edge strength. The measure ofsimilarity at each pixel i is the cosine of the projection angle, Pi,between bi and g:

with larger Pi implying greater similarity. However, there is anundesirable dependence on the average luminance, J.Li, of bi sincePi(bi) =F P;(bi - J.Li); larger values of Pi will be obtained for edgesin the darker regions of an image (where J.Li is low) compared tothose in the brighter regions (where J.Li is high).

The edge detection performances of the GM and AN methods arealso dependent on the choice of edge filter, which would require atrade-off to be made between noise suppression and edge localiza-tion. The AN method in [5]performs poorly in the presence of noisedue to the use of the highly localized 3 x 3 Sobel filter. Similarly, theGFD filter in Canny's GM method offers good edge localization buthas higher noise sensitivity compared to filters such as thedifference-of-boxes (DOB) [15] for the same filter width. The GFDfilter therefore requires smoothing as a preprocessing step to

Fig. 1. Least-squares estimates of PCA eigenvectors using FIR HBT filters. (a) and (c) PCA eigenvectors of second and third largest eigenvalues. (b) and

(d) Corresponding least-squares estimates using a linear combination of FIR HBT filters.

minimize noise, but this blurring may lead to weak edges beingdifficult to detect [9].

Petrou and Kittler [13] observe that image edges resemble rampprofiles rather than ideal step edges due' to the process of digitalimage acquisition, and they use the ramp profile with additive noiseas the edge model to derive their edge filter. This is obtained byoptimizing a performance measure that combines Canny's criteria[1] of accurate edge detection, edge localization, and minimizationof false edge responses. In this paper, we confirm their observationby obtaining an optimal estimate (in the least-squares sense) ofimage edges by using principal component analysis (PCA).However, unlike their approach, we base our edge detectors onthe actual profiles of image edges since our emphasis is onoptimizing two of Canny's criteria-accurate edge detection, ED,and localization, EL-without explicitly including the third criterionon suppression of false edge responses, SF, due to noise. Section 3.4will show that such responses to noise can be distinguished fromvalid edges since the separation between adjacent noise maxima inthe filter response exceeds the narrow widths considered for ouredge filters [2]. The criteria ED and EL are defined as

ED = r~:;S( -x)Gw(x)dx

nov J:: GW(x)dx '

EL = J:: S'( -x)G~(x)dx

noV J:: G~t(x)dx '

where S(x) is the natural edge model in an image centered at x = 0,Gw the impulse response of the edge filter, G~, its derivative, and nothe standard deviation of additive Gaussian white noise. From theSchwarz inequality, the upper bounds of both ED and EL arereached when Gw(x) = S( -x) in [-W, W], i.e., they are maximizedif the filter resembles the natural edges in an image [1].

3.1 Similarity to Natural EdgesWe now describe a method of obtaining 2D edge filters thatoptimally approximate these natural edges, in the least-squares

sense. A sliding window first extracts all n x n (n = 5) localneighborhoods in the image. PCA is then applied to this setof neighborhoods to generate n2 eigenvectors {ei, 1 :s: i :s: n2},

each of size n x n. A neighborhood bi can be expressed as,""n2

bi = ill + ~j~) Uijej, where the average over all local neighbor-hoods is ill = *=;:'1 bi and Uij is the projection of bi - ill

onto the jth eigenvector ej.

Eigenvector el corresponds to the largest eigenvalue A), whereAJ, ... An2 are in decreasing order of magnitude. This eigenvectorbehaves as a low pass filter where Uilel approximates the averageluminance f.Li of bi - ill given by f.Li = r& =';:1(bi(j) - mU))· Thelocal gray-level variation in bi is therefore given by

n'Si = bi - ill - f.Li = 2.:= Uijej'

j=2

In the set of eigenvectors {ej : 2 :s: j :s: n2} in (5), e2 is the basisfunction that yields the best mean square error approximation forSi as Ui2e2 since it gives the smallest mean square error, Jmin,

over all local neighborhoods compared to the other eigenvectors.PCA is a successive approximation scheme which ensures that

the approximation error Si - Ui2e2 is best approximated by using e3,

the subsequent error Si - Ui2e2 - Ui3e3 is best approximated by e4,

etc. Figs. 1a and 1c show the 5 x 5 eigenvectors e2 and e3 extractedfrom the Lena image. The eigenvector pair have similar profiles butare orthogonal. The number of zero crossings in the eigenvectorsincreases from eigenvector pair e2 - e3 to e6 - e7 and beyond,indicating that the eigenvectors corresponding to smaller values ofeigenvalues capture higher frequency information of the localneighborhoods and, hence, are more susceptible to noise. Weconsider the two eigenvectors e2 and e3 for edge detection sincethey most accurately approximate the gray-level variation in localneighborhoods.

IG,(OJ)l u

G.u(W '4

Fig. 2. Spatial and frequency properties of HBT filter. (a) 1D spatial profile of filterG2 (0" = 0.5 (dark gray'). 1.0 (medium gray') and 2.0 (light gray'»). (b) Comparisonbetween frequency responses of 1D FIR HBT filters after normalization by theirrespective maximum values (0" = 0.5 (dark gray). 1.0 (medium gray) and 2.0 (lightgray)) and Gaussian filter (8 = 1.0 (dashed lines)).

We note from Fig. 1 that both e2 and e3 have blurred step edgeprofiles and can be approximated by e2 and e3 (Figs. Ib and Id)given by e2 = 0<21h,+ Q22h2and e3 = 0<3,h, + 0<32h2,where {Q} areweights, h2 is orthogonal to hi, and hi has the HBT profile (withinits region of support)

1 - e-all"(x+y)

GIl'(x. y) = ---- for Ixl, lyl ::; W, and 0 otherwise. (7), 1+ e-all"(x+y)

The region of support for GII" is limited by W to ensure edgelocalization (IV = 2 in Fig. 1).

The orthogonal FIR filter pair, hi and h2, is obtained by samplingGII" at integer locations (Xd,Yd) within [-W, W]. The parameter owdefines the steepness of the profile at the zero crossing and itsrelationship to the filter support W is described in Section 3. Theweights O<ij are determined by projecting both e2 and e3 onto hi andh2, i.e., O<ij = (ei' hj)/(hj. hj). In thisexample, the weights 0<21,0<22,0<31,and 0<32have values -0.98, -0.18, 0.18, and -0.98, respectively. Theerror Ei defined by Ei = Ilei - ej II/Iledl is only 10.3 percent for e2 and11.5 percent for e3. This indicates that the eigenvectors e2 and e3obtained from the set of local image neighborhoods can beaccurately approximated by the orthogonal pair hi and h2.

3.2 Properties of HBT FiltersWe discuss the influence of 0"11"on the spatial and frequencycharacteristics of the continuous FIR HBT filters using theirID representation

1 - e-awT

Gw(x) = 1 + e~all"x for Ixl ::; W, and 0 otherwise (8)

since the ID case may be extended in a straightforward manner to2D. The filter GII" is odd-symmetric with a single zero-crossing atthe origin with the slope at the zero crossing point given by 0"11'/2.The spatial profile of filter GII' is plotted in Fig. 2a for W = 2. Thefilter resembles a ramp for 0"2 ::; 0.5, a blurred edge for largervalues of 0"2, and approaches a DOB filter for O"\V > 5.0.

Fig. 3. Influence of O"w on (a) £,0101 and (b) ell for II' = 1 (light gray'). 2 (medium

gray') and 3 (dark gray).

Fig. 2b compares the frequency responses of ID FIR HBT filterswiththatofCanny'sGFDfilter,F(x) = xe-x'j2s' (withs = 1.0 to limitthe truncation at the tail ends of the Gaussian function). Both filtersare discretized by sampling them atxd = {-2. -1. 0.1. 2}. It is clearlyseen that the family of FIR HBT filters has a narrower bandwidth,indicating better noise reduction compared to Canny's GFD.

3.3 Tuning of HBT Filter ParametersWe determine the parameter value of o"w for a given filter width Wsuch that the HBT filter pair can best approximate the natural stepedges in an image and therefore ensure that the filter pair meetsthe objectives of good SNR and accurate edge localization [1], [2].This is done by selecting 0"11"for a given W such that itscorresponding approximation error E is the smallest. For theeigenvectors extracted from the Lena image, the relationshipbetween Etvtat (= E2 + E3) and 0"11'for W = 1,2.3 are shown inFig. 3a. The optimal values of 0"1, 0"2, and 0"3 are 2.5, 1.0, and 0.7,respectively. Table 1 lists the optimal values obtained for fiveimages from the USC-SIPI image database [18].

3.4 Average Distance between Adjacent Noise MaximaCanny's third criterion, SF, aims to limit the detection of falseedges due to noise in the vicinity of a valid step edge. Ideally, itrequires the mean distance between adjacent noise maxima toapproximate the width of the filter response to a single step edge,i.e., 2IV, although a fraction of this may suffice. Demigny andKamle [2] define Canny's third criterion in the discrete domain as

271"CII'=----

arccos (-p) ,

where CII' is the average distance between adjacent noise maximafor filter width Wand p is expressed as

LZ:~~(GII"(k) - GII"(k - 1))(GI\(k) - GII"(k + 1))P = ",k-+oc ?L-k:_oo(GII"(k) - GII'(k - l)t

Fig. 3b shows the relationship between CIV and O"w for filter widthsW = 1,2,3. The maximum values of CI (;:::;3.4) and C2 (;:::; 4.8) are

TABLE 1Optimal ow, \If = I, Two and Three for Standard Images from use-slPI Image Database

No. Images 0"1 0"2 0"3 No. Images 0"1 0"2 0"3

1 Aerial 2.3 1.1 0.8 6 Stream and Bridge 2.6 1.4 0.92 Airplane(F-J6) 3.1 1.2 0.8 7 Level Step Wedges 2.3 1.4 0.93 Baboon 2.0 1.4 1.0 8 Man 2.5 1.2 0.84 Couple 3.1 1.4 1.0 9 Moon Swjace 2.2 1.1 0.85 House 3.4 1.2 0.8 10 Boat 3.2 1.4 0.4

greater thanrespectively), 1

shows that COI

prevent the del

The undesirabmay beamelioin (3) to

TABLE 2Influence of HBT Filter's (72 on the Noise Response

(72 2:,;RUN (xIO-3)0.5 2.241.0 2.241.5 2.292.0 2.35

greater than the corresponding values of 2TV (2.0 and 4.0,respectively), while C3 (~4.8) is slightly smaller than 6.0. Thisshows that compact filters with widths up to 1\1 = 3 effectivelyprevent the detection of false edges in the vicinity of the true edge.

4 EDGE DETECTION SCHEME INCORPORATING NEWSIMilARITY MEASURE

The undesirable sensitivity to illumination in the AN method [5)may be ameliorated by modifying their edge-similarity measure Piin (3) to

pi = (b; - bi,g), lib; - billllgll'

where bi is the average ofb;. The use of P: ensures that the detectionscheme is invariant to the average luminance and contrast of theneighborhood. However, some smooth regions may yield a high P(if they approximate the form of the attenuated spatial profile of theedge filter and, thereby, lead to erroneous edge points. We presentan AN method that is robust under varying illumination, contrast,and noise. Our goals are 1) to detect meaningful edges regardless ofillumination, and 2) to reduce significantly noise sensitivity whilepreserving responses to true edges. .

We propose the use of the following cosine measure R; witha regularization parameter 1 and an empirically determinedconstant c:

R; = (bi - bi + C"Y, g)IIbi - bi + C"YlIllgll

An estimate, i,of 1 may be obtained by using the median absolutedeviation (MAD) [4] of the n x n image neighborhoods:

, median(Y;: 1 :::::i :::::N)1 = 0.6745 ' where Y; =

1 n' - 22" "2)bi (j) - b;] ,n j=1

Neighborhoods corresponding to smooth regions have smaller Y;compared to those containing edges. A small value of i increasesrobustness to varying edge strength but results in heightenedsensitivity to noise, and vice versa.

Table 2 shows the influence of (72 on the 5 x 5 HBT filter'sresponse to random noise. The energy of the response is given bythe mean sum squared filter response, LiR~/ N, averaged over10 different random noise records, where each record is a 512 x512 image with zero mean and unit standard deviation. The resultsagree with the qualitative evaluation of Fig. 2b, which shows thenoise response decreasing as (72 increases. Similar results areobtained for (72 = 0.5 and 1.0 since (72 = 0.5 gives rise to a largerside lobe although its main lobe is smaller than that of (72 = 1.0. Allfour HBT filters also exhibit a smaller noise response compared tothe GFD filter's value of 2.41 x 10-3 obtained with s = 0.8.

TABLE 3Influence of Parameter c on the Noise Response

c "DRUN (xlO-2)o 7.88I 2.472 0.823 0.394 0.225 0.14

Table 3 shows the influence of parameter c in (12). As observed,the filter response to noise decreases as the parameter c increases.Typically, we set c > 1 in order to effectively suppress noise fromsmooth regions such as those observed when using (11). However,large values of c may result in (12) being insensitive to the presenceof weak edges. We have empirically determined that c = 4 offers agood compromise value.

The details of our method are as follows:Determine HBT parameter, (7\\".

Step 1. For a given !II x N image, apply peA to find theeigenvectors e2 and e:j.

Step 2. From a set of four 2D HBT filters oriented along 0 degrees,45 degrees, 90 degrees, and 135 degrees and defined by a given Wand (7\\", find the orthogonal HBT filter pair hi and hj that bestapproximates e2 and e3, i.e., yields the smallest Ctotal (Section 3.3).The filter width W ranges from 1 to 3. A smaller width achievesbetter edge localiza lion bu t poorer noise tolerance and vice versa.

Step 3. Determine the (7\\" corresponding to the smallest Ctatal byusing a simple 1D bisection method, where Step 2 is repeatedfor specific (711' values within the interval [0.01, 5.00]. This ispossible since the error curves as shown in Fig. 3 have only oneminima and the corresponding (711' offers the ideal HBT slope tooptimally approximate the natural image edges.

Determine ii.Step 4. Obtain an estimate of 1; (13) and set c = 4.Compute similarity maps

Step 5. Using 1i and c from Step 4, apply R; to the given image tocompute four similarity maps where each map corresponds toone of the four HBT filters defined by Wand (7\V from Step 3.The use of these four filters ensures a more robust detection ofedges that are diagonally oriented.

Determine equivalent similarity map.

Step 6. At each pixel location, compare the corresponding valuesfrom the four similarity maps; the largest magnitude is selectedas the pixel value at this point.

Detect edge pixels.

Step 7. Determine the local maximas in the equivalent map andapply a suitable threshold on the local maxima to determine theedge pixels.

The robustness of the proposed method is compared against theAN [5] and GM methods [1], [10], [12] under varying illumination,contrast, and noise. The noise reduction property of the FIR HBTfilter is also compared with the GFD filter. In the results shown inthis section, pixel values are scaled to the range [0, 255] and imageintensities are inverted so that pixels with strong edge responsesare shown as dark points.

TABLE 2Influence of HBT Filter's 172 on the Noise Response

172 I:, R; / N (x 10-3)0.5 2.241.0 2.241.5 2.292.0 2.35

greater than the corresponding values of 2W (2.0 and 4.0,respectively), while C3 ("" 4.8) is slightly smaller than 6.0. Thisshows that compact filters with widths up to W = 3 effectivelyprevent the detection of false edges in the vicinity of the true edge.

4 EDGE DETECTION SCHEME INCORPORATING NEWSIMilARITY MEASURE

The undesirable sensitivity to illumination in the AN method [5]may be ameliorated by modifying their edge-similarity measure Piin (3) to

pi = (bi - bi,g), Ilbi - billllgll'

where bi is the average of bi' The use of PI ensures that the detectionscheme is invariant to the average luminance and contrast of theneighborhood. However, some smooth regions may yield a high PIif they approximate the form of the attenuated spatial profile of theedge filter and, thereby, lead to erroneous edge points. We presentan AN method that is robust under varying illumination, contrast,and noise. Our goals are 1) to detect meaningful edges regardless ofillumination, and 2) to reduce significantly noise sensitivity whilepreserving responses to true edges.

We propose the use of the following cosine measure R. witha regularization parameter I and an empirically determinedconstant c:

R. = (bi - bi +c"g) .Ijbi - bi + qllllgil

An estimate, i,of I may be obtained by using the median absolutedeviation (MAD) [4] of the n x n image neighborhoods:

, median(Y;: 1 :::::i :::::N),= 0.6745 ' where Y; = 1 n' - 22 'L[bi(j) - bi] ,n j=1

Neighborhoods corresponding to smooth regions have smaller Y;compared to those containing edges. A small value of i increasesrobustness to varying edge strength but results in heightenedsensitivity to noise, and vice versa.

Table 2 shows the influence of 172 on the 5 x 5 HBT filter'sresponse to random noise. The energy of the response is given bythe mean sum squared filter response, Li R;/ N, averaged over10 different random noise records, where each record is a 512 x512 image with zero mean and unit standard deviation. The resultsagree with the qualitative evaluation of Fig. 2b, which shows thenoise response decreasing as 172 increases. Similar results areobtained for 172 = 0.5 and 1.0 since 172 = 0.5 gives rise to a largerside lobe although its main lobe is smaller than that of 172 = 1.0. Allfour HBT filters also exhibit a smaller noise response compared tothe GFD filter's value of 2.41 x 10-3 obtained with s = 0.8.

TABLE 3Influence of Parameter c on the Noise Response

C I:iRf! N (x 10-2)o 7.88J 2.472 0.823 0.394 0.225 0.14

Table 3 shows the influence of parameter c in (12). As observed,the filter response to noise decreases as the parameter c increases.Typically, we set c > 1 in order to effectively suppress noise fromsmooth regions such as those observed when using (11). However,large values of c may result in (12) being insensitive to the presenceof weak edges. We have empirically determined that c = 4 offers agood compromise value.

The details of our method are as follows:Determine HBT parameter, iJw.

Step 1. For a given M x N image, apply PCA to find theeigenvectors e2 and e3.

Step 2. From a set of four 2D HBT filters oriented along a degrees,45 degrees, 90 degrees, and 135 degrees and defined by a given Wand iJIV, find the orthogonal HBT filter pair hi and hj that bestapproximates e2 and e3, i.e., yields the smallest Etotal (Section 3.3).The filter width W ranges from 1 to 3. A smaller width achievesbetter edge localization but poorer noise tolerance and vice versa.

Step 3. Determine the iJIV corresponding to the smallest Etotal byusing a simple 1D bisection method, where Step 2 is repeatedfor specific iJw values within the interval [0.01, 5.00]. This ispossible since the error curves as shown in Fig. 3 have only oneminima and the corresponding iJIV offers the ideal HBT slope tooptimally approximate the natural image edges.

Determine ii'Step 4. Obtain an estimate of Ii (13) and set c = 4.Compute similarity maps

Step 5. Using Ii and c from Step 4, apply R. to the given image tocompute four similarity maps where each map corresponds toone of the four HBT filters defined by Wand iJw from Step 3.The use of these four filters ensures a more robust detection ofedges that are diagonally oriented.

Determine equivalent similarity map.

Step 6. At each pixel location, compare the corresponding valuesfrom the four similarity maps; the largest magnitude is selectedas the pixel value at this point.

Detect edge pixels.

Step 7. Determine the local maxim as in the equivalent map andapply a suitable threshold on the local maxima to determine theedge pixels.

The robustness of the proposed method is compared against theAN [5] and GM methods [1], [10], [12] under varying illumination,contrast, and noise. The noise reduction property of the FIR HBTfilter is also compared with the GFD filter. In the results shown inthis section, pixel values are scaled to the range [0, 255] and imageintensities are inverted so that pixels with strong edge responsesare shown as dark points.

r

Fig. 5. Comparison between the C, and R, measures. (a) Light microscope image of infected red blood cells. (b) Similarity map, measure C,. (c) Similarity map, measure

R,. (d) Edge map, measure C,. (e) Edge map, measure R,.

5.1 Uneven IlluminationFig. 4a shows a 512 x 512 Lena image with an artificially imposedillumination gradient. Figs. 4b and 4c are similarity maps obtainedusing Steps 1-6 of the algorithm in Section 4 with W = 2 andow = 1.0. Fig. 4b is computed using the similarity measure Pi [5]and Fig. 4c using R;. As expected, the similarity map of Fig. 4b ismore distinct in the darker rather than in the brighter regions of theimage. The similarity map of Fig. 4c is equally distinct regardlessof image illumination. This example illustrates the fact that Pi,unlike R;, is sensitive to variation in illumination.

5.2 Contrast VariationFig. Sa shows a 570 x 760 light microscope image of malaria-infectedred blood cells. The malaria parasites appear as tiny black spotswithin the red blood cells while the relatively larger black blob justto the right of the image center is a white blood cell. Figs. 5b and 5care similarity maps obtained using the algorithm in Section 2.4 withIV = 2 and ITII" = 0.5; Fig. 5b is computed using the similaritymeasure 6; from (1) and Fig. 5c computed using R; from (12). We seein Fig. 5b that using 6; gives rise to large values of the similaritymeasure at the boundaries of the white blood cell and parasiteregions at the cost of suppressing the valid edges at the boundariesof the red blood cells. In contrast, both strong and weak edges areaccurately detected in the similarity map of Fig. 5c. Ri ensures therobustness of the proposed method in handling a wide range of edgestrength values. This is a significant advantage in edge detection

since a fixed threshold can be easily determined for automaticallythresholding the similarity maps of a large set of images, whereasthe large difference in magnitude between the strong and weakedges in Fig. 5b complicates the search for a suitable edge threshold.The edge maps in Figs. 5d and 5e correspond to 5b and 5c,respectively, and they are obtained via Canny's nonmaximasuppression and hysteresis thresholding [1]. As observed inFigs. 5d and 5e, some cell edges are undetected in the classicalGM method whereas our method accurately detects all edges.

5.3 NoiseWe compare the performances of the FIR HBT filter and the GFDfilter using Steps 1 and 4-6 of the proposed algorithm. Both filtershave a spatial dimension of 5 x 5 (W = 2), parameter 8 of the GFDfilter is fixed at 0.8 [1], and IT2 of the HBT filter is found to be 1.11from Steps 2 and 3. Fig. 6a shows an image of an outdoor scenecontaining a wide range of edge strengths. The image is corruptedby Gaussian noise with an image SNR of approximately 10 dB. It isevident that the similarity map using the FIR HBT filter (Fig. 6c)gives superior edge response and noise performance compared totheGFD filter (Fig. 6b). This is explained by the careful selection of IT2

to ensure that HBT has a high similarity to the edges in the image aswell as a narrower bandwidth (IT2 = 1.11 from Steps 2 and 3 ofSection 4) compared to GFD (8 = 0.80). Figs. 6d and 6e are theresultant edge maps corresponding to the GFD and HBT filters,respectively. They were obtained by applying suitable hysteresisthresholds to the local maxima in Figs. 6b and 6c. It is seen that more

5.4 Edge LlFig. 7a showsda tabase [18].correspondingand phase coni

Fig. 6. Comparison between the FIR HBT and GFD filters on noisy images. (a) Outdoor scene. (b) Similarity map, GFD filter. (c) Similarity map, FIR HBT filter. (d) Edge

map, GFD filter. (e) Edge map, HBT filter.

TABLE 4Quantitative Performance of Edge Detection with Noise

SNR(dB) GFD HBTCorrect edges Missed edges False edges Correct edges Missed edges False edges

No noise 256 4 0 252 8 010 255 5 2 232 28 05 254 6 28 222 38 30 165 95 256 176 84 62

allyreaseakold.5c,

ima, inical

accurate detection and localization of edges is obtained with theHBT filter.

The noise sensitivity of both GFD and HBT filters are alsocompared by applying them to a synthetic binary image comprisinga white square on a black background and corrupted with whiteGaussian noise. The comparison is made over a range of SNR levelsfor a filter width of W = 2. Using Steps 2 and 3 from the algorithm inSection 4, U2 = 1.36 for SNR 2: 5 dB and U2 = 1.66 for SNR < 5 dBfor the HBT filter. The s values for the GFD filters remain fixed at 0.8.

Since it is known a priori that 3.4 percent of the imagecomprises edge pixels, we impose a threshold such that only pixelswith edge strengths in the top 3.4 percent are admitted as edgepixels. The results of the comparison are shown in Table 4 wherethe total number of true edge pixels is 260. The GFD filter, ingeneral, detects more true edge pixels than the HBT filter.However, HBT detects fewer false edge pixels and tends toperform better than GFD for SNR levels below 5 dB. Theperformance of GFD degrades more rapidly as the noise levelincreases, indicating a lower robustness to noise.

5.4 Edge LocalizationFig. 7a shows a 512 x 512 synthetic image from the USC-SIPIdatabase [18]. Figs. 7b and 7c represent, respectively, thecorresponding edge maps obtained using the proposed methodand phase congruency (PC) [7]. The PC method employs filters at

four scales with six filter orientations at each scale to generate amap (which we term the PC map) from the phase congruencyinformation. The edge maps are obtained by applying nonmaximasuppression to the PC map and similarity map using R;, followedby a threshold of 0.1 on the local maxima.

A quantitative measure of edge detection performance isprovided by Pratt's figure of merit [14],

1 ED 1

F = max(E"ED) ~ 1+ O'cF(i)'

l~

==::=:==~I !I I

Fig. 7. Edge localization comparison of the proposed and PC methods.(a) Synthetic image. (b) Edge map from proposed method. (c) Edge map fromPC method.

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Fig. 8. Edge localization comparison between the proposed and PC methods. (a) Lena image. (b) Similarity map from proposed method. (c) PC map. (d) Edge map from

proposed method. (e) Edge map from PC method.

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where Ej and ED are the number of ideal and detected edge pixels,respectively, d(i) the Euclidean distance between the ith edge pixeldetected and the ideal edge pixel nearest to it, and a a scalingconstant, which we have set to 1. The proposed method givessignificantly better edge localization (F = 0.86) than the PC method(F = 0.63), which detects false edges in the vicinity of sharptransitions in an image and gives poor edge localization. The valueof F for the proposed method is smaller than 1 since, being a stepedge detector, it does not detect the corners in the image (thejunctions where the edges meet).

We also compare the performance of our method with the PCapproach using the Lena image of Fig. 8a. Figs. 8b and 8c show thesimilarity map using Ri and the PC map, both normalized to therange [0,1]. Figs. 8d and 8e show the corresponding edge mapsobtained by first detecting the local edge maxima, followed byapplying Canny's hysteresis thresholding. Compared to PC, theproposed method results in a higher gradient magnitude response(Figs. 8b and 8c) and more accurate detection of both strong andweak edges (Figs. 8d and 8e).

We have presented an edge-similarity measure that is simulta-neously robust to varying image illumination and contrast unliketraditional GM and AN methods, which are sensitive to contrastand illumination, respectively. The subtraction of mean luminancefrom the image neighborhoods ensures that the AN method isinvariant to contrast changes but results in increased sensitivity tonoise. The use of a regularization term -y, estimated using the MADcriterion, achieves a good compromise between the opposingobjectives of reduced noise sensitivity and contrast invariance. Inaddition, the edge filter used has a FIR HBT profile, which offersbetter edge localization and reduced noise sensitivity compared tothe classical GFD filter. The proposed method shows better edgelocalization and detection compared to the PC method and hasbeen successfully applied to both synthetic and natural images.

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