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Wavelet-based Reflection Symmetry Detection viaTextural and Color Histograms

Mohamed Elawady, Christophe Ducottet, Olivier Alata, Cecile Barat,Philippe Colantoni

To cite this version:Mohamed Elawady, Christophe Ducottet, Olivier Alata, Cecile Barat, Philippe Colantoni. Wavelet-based Reflection Symmetry Detection via Textural and Color Histograms. ICCV 2017, IEEE Interna-tional Conference on Computer Vision Workshop Detecting Symmetry in the Wild, Oct 2017, Venice,Italy. �10.1109/ICCVW.2017.202�. �ujm-01637175�

Wavelet-based Reflection Symmetry Detection via Textural and ColorHistograms

Mohamed Elawady, Christophe Ducottet, Olivier Alata, Cecile BaratUniversite de Lyon, UJM-Saint-Etienne, CNRS, IOGS,

Laboratoire Hubert Curien UMR5516, F-42023, Saint-Etienne, Francemohamed.elawady@univ-st-etienne.fr

Philippe ColantoniUniversite de Lyon, UJM-Saint-Etienne, CIEREC EA n0 3068,

Saint-Etienne, France

Abstract

Symmetry is one of the significant visual properties in-side an image plane, to identify the geometrically balancedstructures through real-world objects. Existing symmetrydetection methods rely on descriptors of the local imagefeatures and their neighborhood behavior, resulting incom-plete symmetrical axis candidates to discover the mirrorsimilarities on a global scale. In this paper, we proposea new reflection symmetry detection scheme, based on a re-liable edge-based feature extraction using Log-Gabor fil-ters, plus an efficient voting scheme parameterized by theircorresponding textural and color neighborhood informa-tion. Experimental evaluation on four single-case and threemultiple-case symmetry detection datasets validates the su-perior achievement of the proposed work to find global sym-metries inside an image.

1. Introduction

Reflection symmetry is a geometrical property in naturaland man-made scenes which attracts the human attentionby achieving the state of equilibrium through the appear-ance of the similar visual patterns around [2, 29]. It can bedefined as a balanced region where local patterns are nearlymatching on each side of a symmetry axis. This paper fo-cuses on detecting the global mirror symmetries inside animage plane by highlighting the relations between the edge,textural and color information of major involved objects.

Many detection algorithms of reflection symmetry havebeen introduced in the last decade. Loy and Eklundh [17]proposed the baseline algorithm by analyzing the bilat-eral symmetry from image features’ constellation. Firstly,local feature points (i.e. SIFT) are extracted associated

with local geometrical properties and descriptor vectors.Secondly, the process of pairwise matching is computedbased on the measure of the local symmetry magnitude oftheir descriptors. Finally, Hough-like voting space is con-structed, respect to the accumulation of symmetry candi-dates’ magnitude, parametrized with orientation and dis-placement components, to select the dominant axes insidean image. Park et al. [23] presented a survey for symmetrydetection algorithms, followed by two symmetry detectionchallenges [25, 16], in which Loy and Eklundh [17] havethe best symmetrical results against the participated meth-ods [21, 14, 19, 24]. Later, other feature-based approaches[7, 5] introduced some improvements over the baseline al-gorithm [17]. Edge-based features [20, 9, 28, 3, 8, 4, 10]were more recently proposed instead of the feature points,due to their robust behavior in detecting an accurate sym-metric information inside an image plane.

Log-Gabor filter was introduced by Field [11] in late1980’s, as an alternative to the Gabor filter (formerly usedin [10]), to suppress the effect of DC component throughthe computation of the multi-scale logarithmic function inthe frequency domain. This process extracts more accuratefeature (edge/corner) information from an image, plus en-suring the robustness of these features with respect to il-lumination variations. Feature extraction methods basedon Log-Gabor filter have been successfully used in differ-ent computer vision applications (biometric authentication[13, 1], image retrieval [26], face recognition [15], imageenhancement [27], character segmentation [18], and edgedetection [12]).

Our contribution is twofold. Firstly, we extract edge fea-tures upon the application of Log-Gabor filters for symme-try detection instead of using Gabor filters. Secondly, wepropose a similarity measure based on color image infor-mation, to improve the symmetry magnitude estimation in

1

(a) Multiscale EdgeSegment Extraction

(b) Triangulation basedon Symmetry Weights

θ

ρ0

π

(c) Voting Space for Peak Detection

Figure 1: The general framework of the proposed reflec-tion symmetry detection, using multi-scale edge detectionfollowed by symmetry triangulation based on local geomet-rical, spatial and color information.

the voting step. In addition, we evaluate the proposed meth-ods over all public datasets for reflection symmetry (singleand multiple cases), in comparison with state-of-the-art al-gorithms.

Figure 1 shows our proposed algorithm, which detectsglobally the symmetry axes inside an image plane. Wefirstly extract edge features using Log-Gabor filters withdifferent scales and orientations. Afterwards, we use theedge characteristics associated with the textural and colorinformation as symmetrical weights for voting triangula-tion. In the end, we construct a polar-based voting his-togram based on the accumulation of the symmetry contri-bution (local texture and color information), in order to findthe maximum peaks presenting as candidates of the primarysymmetry axes.

The remaining part of the paper is organized as fol-lows. In section 2, we explain Log-Gabor transform and itsfeature-based application on a gray-scale image. In section3, textural and color histograms of window-based featuresare described in details. In section 4, we illustrate the de-tection of symmetry candidates through triangulation andvoting processes. Experimental details and results are pre-sented in section 5. Finally, section 6 contains conclusionand future work.

2. Log-Gabor Edge Detection

Log-Gabor filter consists of logarithmic transformationof a Gabor filter in the Fourier domain, which suppressesthe negative effect of the DC component:

G(η, α; s, o) = Gs(η) Go(α) (1)

Gs(η) = exp(−(log( ηηs ))2

2(log(ση))2) U(η) (2)

Go(α) = exp(−|atan( sin(α−αo)

cos(α−αo) )|2σ2

α

) (3)

where (η, α) are the log-polar coordinates representing ra-dial and angular components over S scales and O orien-tations, associated with the frequency centers (ηs, αo) andtheir bandwidths (ση, σα). Gs(η) is multiplied by low-passButterworth filter U(η) of order 15, and frequency 0.45, toeliminate any extra frequency at Fourier corners.

The modulus of complex wavelet coefficients Is,o(x, y)are computed on an image I (width W and height H)over multiple scales s ∈ {1, . . . , S} and orientations o ∈{ zπO , z = 0, . . . , O − 1} as follows:

I →GS

IGSFT→ IGS (4)

Is,o(x, y) = |FT−1(IGS × G)| (5)

where IGS is the gray-scale version of the image I infrequency domain, and FT (.), FT−1(.) are the Fouriertransform and its inverse. Figure 2a shows Log-Gaborwavelet filter bank with 4 scales and 5 orientations whereschematic contours cover the frequency space in the Fourierdomain. Consequently, the elongation scheme of Log-Gabor wavelets appears in figures 2b, 2c presenting the realand imaginary components in the spatial domain.

Figure 3 presents an example of Log-Gabor responseIs,o on a natural image with a different illuminated fore-ground leaf along side with a blurring background. Ampli-tude map J(x, y) = max

s,oIs,o(x, y) highlights the edge de-

tails of the foreground object in a sharp way, accompaniedby precise angular values in the corresponding orientationmap φ(x, y). Upon a spatial sampling of the input image Iusing non-interleaved cells along a regular grid (stride andcell size are proportional to the maximum image dimensionmax(W,H)). A feature point pi is extracted within eachcellDi containing edges using the wavelet response of Log-Gabor filter Is,o(Di), associated with its maximum waveletresponse Ji = J(pi) along side with the corresponding ori-entation φi and color in HSV color space ψi.

3. Textural and Color HistogramsThe textural and color information around an edge seg-

ment are prominent similarity characteristics for natural im-ages, describing the symmetrical behavior of local edge ori-entation, and the balanced distribution of luminance andchrominance components. Hence, we introduce two his-tograms: Firstly, neighboring textural histogram hi of sizeN :

hi(n) =∑r∈Di

Jr 1Φn(φr), (6)

(a) Fourier (b) Real (c) Imaginary

Figure 2: Multi-resolution log-Gabor arrangement with S = 4 scales (in rows) and O = 5 orientations (in columns). (a)Filters in the Fourier domain. (b,c) Real and imaginary components in the spatial domain.

(a) Input (b) Amplitude (c) Orientation

Figure 3: Log-Gabor response: (a) Input image. (b) Amplitude map J ∈ [0, 1]. (c) Orientation map φ ∈ [−90◦, 90◦].

Φn = [nπ

N,

(n+ 1)π

N[, n = 0, . . . , N − 1

where 1 is the indicator function. hi is l1 normalized andcircular shifted with respect to the orientation of the maxi-mum magnitude φi among the neighborhood cell Di. Sec-ondly, the local HSV histogram gi of size C with sub-sampling rate (Chu : Csa : Cva)

gi(c) =∑r∈D∗

i

1Ψc(ψr), (7)

c = (chu, csa, cva),

chu ∈ {0, . . . , Chu − 1},csa ∈ {0, . . . , Csa − 1},cva ∈ {0, . . . , Cva − 1},

Ψc = [ 2chuπChu , 2(chu+1)π

Chu [× [ csa

Csa ,csa+1Csa [× [ c

va

Cva ,cva+1Cva [

whereD∗i is the neighborhood window around feature pointpi , ψc is a sub-sampled set of HSV space, in terms ofthree components: hue (hu), saturation (sa) and value (va).l1 normalization is applied to gi(.) for bin-wise histogramcomparison.

4. Symmetry Triangulation and Voting

A set of feature pairs (pi, pj) of size P (P−1)2 are elected

such that i 6= j, and P is the number of feature points.Then, we compute the symmetry candidate axis based on atriangulation process with respect to the image origin. Thiscandidate axis is parametrized by angle θi,j (orientation ofthe bisector of the pair segment (pi, pj)), and displacementρi,j (distance of the image origin to this bisector) and has asymmetry weight ωi,j defined as follows:

ωi,j = ω(pi, pj) = m(i, j) t(i, j) q(i, j) (8)

m(i, j) = |τ iR(T⊥ij )τ j | (9)

t(i, j) =

N∑n=1

min(hi(n), hj(n)) (10)

q(i, j) =

C∑c=1

min(gi(c), gj(c)) (11)

where τ i = [cos(φi), sin(φi)]T , R(T⊥ij ) is the reflection

matrix with respect to the perpendicular of the line connect-ing (pi, pj) [9, 10], and hj is the mirror version of hj his-

A1

A2

B3

B4

B1 B2

(a) Input with GT

A1

A2

A2

B1 B2

B3 B4

B3 B4

(b) Voting representation

Figure 4: Symmetry voting process: (a) Input image withmajor (in red) and minor (in green) symmetry axes. (b)Smoothed output of the symmetry histogram H with ρ dis-placement x-axis and θ angle y-axis, highlighting the corre-sponding axes.

togram for element-wise orientation-based comparison. l1normalization is applied to symmetry weights ω.

A symmetry histogram H(ρ, θ) is defined as the sum ofthe symmetry weights of all pairs of feature points such as:

H(ρ, θ) =∑pi,pj

i 6=j

ωi,jδρ−ρi,jδθ−θi,j (12)

where δ is the Kronecker delta.the voting histogram H(ρ, θ) is smoothed using a Gaus-

sian kernel to output a proper density representation (asshown in figure 4), in which the major symmetry peaksare selected by reaching-out clear extreme spots using well-known non-maximal suppression technique [6, 17]. Thespatial boundaries of each symmetry axis is computedthrough the convex hull of the associated voting pairs.

5. Results and DiscussionsThis section describes the details of public symmetry

datasets, evaluation metrics, and experimental settings forperformance comparison of the proposed work. Perfor-mance analysis against the state-of-the-art algorithms is alsoprovided in this section.

5.1. Datasets description

Seven public datasets of reflection symmetry detectionare used from four different databases: (1) PSU datasets(single, multiple): Liu’s vision group proposed symmetrygroundtruth for Flickr images (# images = # symmetries= 157 for single case, # images = 142 and # symmetries= 479 for multiple case) in ECCV20101, CVPR20112 andCVPR20133. (2) AVA dataset (single): Elawady et. al

1http://vision.cse.psu.edu/research/symmetryCompetition/index.shtml2http://vision.cse.psu.edu/research/symmComp/index.shtml3http://vision.cse.psu.edu/research/symComp13/content.html

[10] provided axis groundtruth4 for some professional pho-tographs (# images = # symmetries = 253 for single case)from large-scale aesthetic-based dataset called AVA [22].(3) NY datasets (single, multiple): Cicconet et al. [8] in-troduced a symmetry database (# images = # symmetries= 176 for single case, # images = 63 and # symmetries= 188 for multiple case) in 20165, providing more stablegroundtruth. (4) ICCV2017 training datasets (single, multi-ple): Seungkyu Lee delivered a challenge database associ-ated with reflection groundtruth6 (# images = # symmetries= 100 for single case, # images = 100 and # symmetries =384 for multiple case).

5.2. Evaluation metrics

Assuming a symmetry line defined by two endpoints(a = [ax, by]T , b = [bx, by]T ), quantitative comparisonsare fundamentally performed by considering a detectedsymmetry candidate SC = [aSC , bSC ]T as a true posi-tive (TP) respect to the corresponding groundtruth GT =[aGT , bGT ]T if satisfying the following two conditions:

T (atan(

abs(

∣∣∣∣vSCx vGTxvSCy vGTy

∣∣∣∣)< vSC , vGT >

)) < γ , (13)

√(tSCx − tGTx )2 + (tSCy − tGTy )2 < ζ , (14)

vSC = (aSC − bSC), vGT = (aGT − bGT ) ,

tSC =(aSC + bSC)

2, tGT =

(aGT + bGT )

2,

T (Γ) =

{π − Γ, if Γ > π

2

Γ, otherwise

The conditions represent angular and distance con-straints between detected and groundtruth axes. These con-straints are upper-bounded by the corresponding thresholdsγ and ζ, which are defined in table 1. Furthermore, theprecision PR and recall RC rates are defined by selectingthe symmetry peaks according to the candidates’ amplitudenormalized by the highest detection score, to show the per-formance curve for each algorithm:

PR =TP

TP + FP, RC =

TP

TP + FN(15)

where FP is a false positive (non-matched detection), andFN is a false negative (non-matched groundtruth). In addi-tion, we used the maximum F1 score identifying a uniquecomparison measure, to show the overall accuracy of eachsymmetry algorithm:

4http://github.com/mawady/AvaSym5http://symmetry.cs.nyu.edu/6https://sites.google.com/view/symcomp17/challenges/2d-symmetry

max{F1} = max{2PR×RCPR+RC

} (16)

Table 1: Threshold values of evaluation metrics across dif-ferent reflection symmetry competitions.

Competitions γ ζ

CVPR2011 [25] 10◦ 20%× len(GT )CVPR2013 [16] 10◦ 20%×min{len(MT ), len(GT )}

ICCV2017 3◦ 2.5%×min{W,H}

5.3. Experimental settings

We compare the proposed methods (Lg: without colorinformation and LgC: with color information) against threestate-of-the-art approaches: Loy and Eklundh (Loy2006)[17], Cicconet et al. (Cic2014) [9], and Elawady et al.(Ela2016) [10]). Their source codes are used with defaultparameter values for performance comparison. In Log-Gabor edge detection, we set the number of scales S andnumber of orientations O to 12 and 32. We also set the ra-dial bandwidth ση to 0.55 and the angular bandwidth σαto 0.2. In textural and color histogram calculations, we de-fine the number of bins for textural N and color C to 32and 32 (sampling rate Chu : Csa : Cva = 8 : 2 : 2) respec-tively. We choose those parameters on the basis of extensiveexperiments. In case of gray-scale images, contrast valuesare used instead of color information in HSV color space.In symmetry voting, we declare the 2D histogram space ofρ =

√W 2 +H2 displacement bins and θ = 360 orienta-

tion bins for extrema selection.

5.4. Performance analysis

Table 2: Using evaluation metrics CVPR2013 [16] (Tab. 1),comparison of the true positive rates based on top detec-tion for the proposed methods against the state-of-art algo-rithms. Symmetry datasets are presented as: single-case(first 4 rows) and multiple-case (last 3 rows), highlight-ing (between parenthesis) the number of images for eachdataset. Top 1st and 2nd values are in Bold and underlinedrespectively.

Datasets Loy[17] Cic[9] Ela[10] Lg LgC

PSU(157) 81 90 97 109 116AVA(253) 174 124 170 191 197NY(176) 98 92 109 125 132

ICCV17(100) 52 53 52 70 69PSUm(142) 69 68 67 74 79NYm(63) 32 36 36 39 40

ICCV17m(100) 54 39 53 53 56

Table 3: Using evaluation metrics ICCV17 (Tab. 1), com-parison of the true positive rates based on top detection insingle-case and all detections in multiple-case for the pro-posed methods against the state-of-art algorithms. Symme-try datasets are presented as: single-case (first 4 rows) andmultiple-case (last 3 rows), highlighting (between parenthe-sis) the number of groundtruth for each dataset. Top 1st and2nd values are in Bold and underlined respectively.

Datasets Loy[17] Cic[9] Ela[10] Lg LgC

PSU(157) 41 27 46 52 51AVA(253) 63 36 95 86 87NY(176) 34 35 43 50 53

ICCV17(100) 17 9 19 22 22PSUm(479) 130 65 129 154 157NYm(188) 48 45 69 57 60

ICCV17m(384) 95 38 86 111 111

In our experimental evaluation, the algorithms are exe-cuted to detect and compare the global symmetries insidesynthetic and real-world images. Tables 2 & 3 show the dif-ferent versions of true positive rates for the proposed meth-ods (Lg and LgC) against Loy and Eklundh (Loy) [17], Ci-cconet et al. (Cic) [9], and Elawady et al. (Ela) [10]. LgCperforms the best result among most cases in single andmultiple symmetry, due to the importance of color infor-mation for the voting computations in colorful images. Atthe same time, Lg has the top 2nd result, and sometimes thetop 1st results in gray-scale or low-saturated images, thanksfor the use of the bank of Log-Gabor filters in the featureextraction step. Ela [10] ranked as the top 3rd result, due tothe utilization of small grids to compute window-based fea-tures. Thanks for the advantage of SIFT features, Loy [17]is still strong competent to be ranked as the top 4th result ingeneral. Cic [9] has the lowest performance.

Figure 5 presents performance results in terms of pre-cision and recall curves for single-case and multiple-case symmetry datasets, plus values of the maximum F1

score to measure the performance of the proposed algo-rithms (Lg2017 and LgHSV2017) against Loy and Ek-lundh (Loy2006) [17], Cicconet et al. (Cic2014) [9], andElawady et al. (Ela2016) [10]. In single-case symmetry,our method Lg2017 outperforms the other concurrent al-gorithms (Loy2006, Cic2014, and Ela2016) in the contextof using only gray-scale version of involved images. Fur-thermore, color version of our method LgHSV2017 exploitsslightly improvement over gray-scale one Lg2017. On theother hand, Only LgHSV2017 has better precision perfor-mance among others in PSUm and NYm datasets, due tomany local groundtruth existing inside multiple-case sym-metry. Loy2006 has better precision performance at someintermediate recall rates, for the purpose of detecting minor

(a) PSU (b) AVA (c) NY

(d) ICCV’17 (e) PSUm (f) NYm

(g) ICCV’17m

Figure 5: Using evaluation metrics CVPR2013 [16] (Tab. 1), Precision-Recall curves on: (1) four single-case symmetry(a,b,c,d) datasets, and (2) three multiple-case symmetry (e,f,g) datasets to show the overall superior performance of ourproposed methods (Lg2017 and LgHSV2017) against the three prior algorithms (Loy2006 [17], Cic2014 [9], and Ela2016[10]). The maximum F1 scores are qualitatively presented as square symbols along the curves, and quantitatively indicatedbetween parentheses inside the top-right legends. (x-axis: Recall, y-axis: Precision). Best seen on screen (zoom-in fordetails).

symmetries among some image objects.

As a summary of the previous quantitative evaluations,figure 6 compares qualitatively top performing algorithmsshowing different examples of reflection symmetry detec-tion. Despite the single-case images (1st and 2nd columns)have strong shadowing effect in foreground objects, thecolor version of the proposed method LgC easily finds thecorrect symmetry axes as a first candidate. On the other

hand, the non-color proposed method Lg satisfies the single-symmetry groundtruth in the two examples as first and fifthdetections respectively. In contrast, Ela [10] mismatchesthe provided groundtruth with all detection and Loy [17]only finds the local symmetry axes in the object parts hav-ing same contrast values. In multiple-case images (3rd,4th and 5th columns), LgC clearly detects the global andmost of local symmetries. In the opposite side, the other

(a) 16550 - GT (b) 832486 - GT (c) ref rm 13 - GT (d) ref rm 65 - GT (e) I034 - GT

(f) 16550 - LgC (g) 832486 - LgC (h) ref rm 13 - LgC (i) ref rm 65 - LgC (j) I034 - LgC

(k) 16550 - Lg (l) 832486 - Lg (m) ref rm 13 - Lg (n) ref rm 65 - Lg (o) I034 - Lg

(p) 16550 - Ela (q) 832486 - Ela (r) ref rm 13 - Ela (s) ref rm 65 - Ela (t) I034 - Ela

(u) 16550 - Loy (v) 832486 - Loy (w) ref rm 13 - Loy (x) ref rm 65 - Loy (y) I034 - Loy

Figure 6: Some single-case and multiple-case results from AVA [10], PSUm [25, 16], and NYm [8] datasets with groundtruth(blue) in 1st row (a-e). our methods in 2nd and 3rd rows (f-o) produces better results among El2016 [10] in 4rd row (p-t)and Loy2006 [17] in 5th row (u-y). For each algorithm, the top five symmetry results is presented in such order: red, yellow,green, blue, and magenta. Each symmetry axis is shown in a straight line with squared endpoints. Best seen on screen.

methods struggle determining only more than one correctgroundtruth.

6. ConclusionIn this paper, we detect global symmetry axes inside an

image using the edge characteristics of Log-Gabor waveletresponse. For the purpose of improving our results, we ad-ditionally use textural and color histograms as local sym-metrical measure around edge features. We show that theproposed methods provide a great improvement over singleand multiple symmetry cases in all public datasets. Futurework will focus on improving the voting representations,respect to the number of symmetry candidate pairs, result-ing a precise selection of symmetrical axis peaks and theircorresponding voting features. For art and aesthetic rank-ing applications, a unified balance measure is required todescribe the global symmetry degree inside an image.

AcknowledgementsWe would like to thank Peter Kovesi for making the

source code of Log-Gabor wavelet filters available on-line.We also thank Marcelo Cicconet and Gareth Loy to providethe MATLAB implementations of the reflection symmetrydetection.

References[1] N. U. Ahmed, S. Cvetkovic, E. H. Siddiqi, A. Nikiforov, and

I. Nikiforov. Combining iris and periocular biometric formatching visible spectrum eye images. Pattern RecognitionLetters, 91:11–16, 2017. 1

[2] R. Arnheim. Art and visual perception. Stockholms Univer-sitet, Institutionen for Konstvetenskap, 2001. 1

[3] I. Atadjanov and S. Lee. Bilateral symmetry detectionbased on scale invariant structure feature. In Image Process-ing (ICIP), 2015 IEEE International Conference on, pages3447–3451. IEEE, 2015. 1

[4] I. R. Atadjanov and S. Lee. Reflection symmetry detectionvia appearance of structure descriptor. In European Confer-ence on Computer Vision, pages 3–18. Springer, 2016. 1

[5] D. Cai, P. Li, F. Su, and Z. Zhao. An adaptive symmetrydetection algorithm based on local features. In Visual Com-munications and Image Processing Conference, 2014 IEEE,pages 478–481. IEEE, 2014. 1

[6] J. Canny. A computational approach to edge detection. IEEETransactions on pattern analysis and machine intelligence,(6):679–698, 1986. 4

[7] M. Cho and K. M. Lee. Bilateral symmetry detection viasymmetry-growing. In BMVC, pages 1–11. Citeseer, 2009.1

[8] M. Cicconet, V. Birodkar, M. Lund, M. Werman, andD. Geiger. A convolutional approach to reflection symme-try. Pattern Recognition Letters, 2017. 1, 4, 7

[9] M. Cicconet, D. Geiger, K. C. Gunsalus, and M. Werman.Mirror symmetry histograms for capturing geometric prop-

erties in images. In Computer Vision and Pattern Recogni-tion (CVPR), 2014 IEEE Conference on, pages 2981–2986.IEEE, 2014. 1, 3, 5, 6

[10] M. Elawady, C. Barat, C. Ducottet, and P. Colantoni. Globalbilateral symmetry detection using multiscale mirror his-tograms. In International Conference on Advanced Conceptsfor Intelligent Vision Systems, pages 14–24. Springer, 2016.1, 3, 4, 5, 6, 7

[11] D. J. Field. Relations between the statistics of natural im-ages and the response properties of cortical cells. Josa a,4(12):2379–2394, 1987. 1

[12] X. Gao, F. Sattar, and R. Venkateswarlu. Multiscale cornerdetection of gray level images based on log-gabor wavelettransform. IEEE Transactions on Circuits and Systems forVideo Technology, 17(7):868–875, 2007. 1

[13] H. Kaur and P. Khanna. Cancelable features using log-gaborfilters for biometric authentication. Multimedia Tools andApplications, 76(4):4673–4694, 2017. 1

[14] S. Kondra, A. Petrosino, and S. Iodice. Multi-scale ker-nel operators for reflection and rotation symmetry: furtherachievements. In Computer Vision and Pattern Recogni-tion Workshops (CVPRW), 2013 IEEE Conference on, pages217–222. IEEE, 2013. 1

[15] J. Li, N. Sang, and C. Gao. Log-gabor weber descrip-tor for face recognition. Journal of Electronic Imaging,24(5):053014–053014, 2015. 1

[16] J. Liu, G. Slota, G. Zheng, Z. Wu, M. Park, S. Lee,I. Rauschert, and Y. Liu. Symmetry detection from realworldimages competition 2013: Summary and results. In Com-puter Vision and Pattern Recognition Workshops (CVPRW),2013 IEEE Conference on, pages 200–205. IEEE, 2013. 1,5, 6, 7

[17] G. Loy and J.-O. Eklundh. Detecting symmetry and sym-metric constellations of features. In Computer Vision–ECCV2006, pages 508–521. Springer, 2006. 1, 4, 5, 6, 7

[18] C. Mancas-Thillou and B. Gosselin. Charactersegmentation-by-recognition using log-gabor filters. InPattern Recognition, 2006. ICPR 2006. 18th InternationalConference on, volume 2, pages 901–904. IEEE, 2006. 1

[19] E. Michaelsen, D. Muench, and M. Arens. Recognition ofsymmetry structure by use of gestalt algebra. In ComputerVision and Pattern Recognition Workshops (CVPRW), 2013IEEE Conference on, pages 206–210. IEEE, 2013. 1

[20] Y. Ming, H. Li, and X. He. Symmetry detection via contourgrouping. In Image Processing (ICIP), 2013 20th IEEE In-ternational Conference on, pages 4259–4263. IEEE, 2013.1

[21] Q. Mo and B. Draper. Detecting bilateral symmetry withfeature mirroring. In CVPR 2011 Workshop on SymmetryDetection from Real World Images, 2011. 1

[22] N. Murray, L. Marchesotti, and F. Perronnin. Ava: A large-scale database for aesthetic visual analysis. In Computer Vi-sion and Pattern Recognition (CVPR), 2012 IEEE Confer-ence on, pages 2408–2415. IEEE, 2012. 4

[23] M. Park, S. Lee, P.-C. Chen, S. Kashyap, A. A. Butt, andY. Liu. Performance evaluation of state-of-the-art discrete

symmetry detection algorithms. In Computer Vision and Pat-tern Recognition, 2008. CVPR 2008. IEEE Conference on,pages 1–8. IEEE, 2008. 1

[24] V. Patraucean, R. G. von Gioi, and M. Ovsjanikov. Detec-tion of mirror-symmetric image patches. In Computer Visionand Pattern Recognition Workshops (CVPRW), 2013 IEEEConference on, pages 211–216. IEEE, 2013. 1

[25] I. Rauschert, K. Brocklehurst, S. Kashyap, J. Liu, and Y. Liu.First symmetry detection competition: Summary and results.Technical report, Technical Report CSE11-012, Departmentof Computer Science and Engineering, The PennsylvaniaState University, 2011. 1, 5, 7

[26] E. Walia and V. Verma. Boosting local texture descriptorswith log-gabor filters response for improved image retrieval.International Journal of Multimedia Information Retrieval,5(3):173–184, 2016. 1

[27] W. Wang, J. Li, F. Huang, and H. Feng. Design and imple-mentation of log-gabor filter in fingerprint image enhance-ment. Pattern Recognition Letters, 29(3):301–308, 2008. 1

[28] Z. Wang, Z. Tang, and X. Zhang. Reflection symmetry de-tection using locally affine invariant edge correspondence.Image Processing, IEEE Transactions on, 24(4):1297–1301,2015. 1

[29] R. D. Zakia and D. Page. Photographic Composition: AVisual Guide. Taylor & Francis, 2010. 1