Detecting Bilateral Symmetry in Perspective Hugo Cornelius and Gareth Loy Computational Vision & Active Perception Laboratory Royal Institute of Technology (KTH), Stockholm, Sweden {hugoc,gareth}@nada.kth.se Abstract A method is presented for efficiently detecting bilateral symmetry on planar surfaces under perspective projection. The method is able to detect local or global symmetries, locate symmetric surfaces in complex backgrounds, and detect multiple incidences of symmetry. Symmetry is simultaneously evaluated across all locations, scales, orientations and under perspective skew. Feature descriptors robust to local affine distortion are used to match pairs of symmetric features. Feature quadruplets are then formed from these symmetric feature pairs. Each quadruplet hypothesises a locally planar 3D symmetry that can be extracted under perspective distortion. The method is posed independently of a specific feature detector or descriptor. Results are presented demonstrating the efficacy of the method for detecting bilateral symmetry under perspective distortion. Our unoptimised Matlab implementation, running on a standard PC, requires of the order of 2 seconds to process images with 1, feature points. 1. Introduction We live in a 3D world full of symmetries. The most familiar form of symmetry, bilateral or mirror symmetry, is common among many manufactured objects and living organisms. It is this form of symmetry that will be the focus of this paper. Humans are very good at detecting symmetry. Psychophysical evidence points to symmetry detection being a pre-attentive process [2] and also one of the cues directing visual attention [2, 9, 14, 25]. Wagemans [31] views symmetry as one of the most important aspects of early visual analysis, and recent psychophsyical results [3] suggest that the detection of symmetries under perspective distortion is an integral part of 3-D object perception. Symmetry gives balance and form to an object s structure and appearance, but it also ties together visual features that can otherwise appear diffuse. Accordingly, features on symmetric objects can be grouped together on the basis of their common symmetry. Whilst objects exhibiting symmetry will often not appear symmetric in the image plane, their symmetry may still be evident under affine or perspective projection. The contribution of this paper is a new and efficient method for detecting planar bilateral symmetry under affine and perspective projection, and grouping the associated symmetric constellations of features. Modern feature descriptors [2] are used to establish pairs of reflective features, and these are in turn paired to form quadruplets. The potential symmetries of each of these quadruplets are determined, and they are grouped into symmetric constellations about common symmetry foci, identifying both the dominant bilateral symmetries present and a set of symmetric features associated with each foci. The method is independent of the feature detector and descriptor used. Symmetry is simultaneously evaluated across all locations, scales and orientations, and also across a range of affine or perspective distortions limited only by the matching capacity of the features used. The method is able to detect local or global symmetries, locate symmetric surfaces in complex backgrounds, and detect multiple incidences of symmetry. This is all done from a single view and requires no reconstruction of the 3D scene. The remainder of this paper is organised as follows, Section 2 reviews previous work, Section 3 describes the method, Section 4 presents experimental results and discusses the performance of the method, and Section 5 presents our conclusions. 2. Background Symmetry detection has a long history in computer vision dating back to the 197s (e.g. [3, 17, 19, 24, 36]). It has found use in numerous applications, including: facial image analysis [23], vehicle detection [11, 37], reconstruction [7, 32], visual attention [17, 24, 25], indexing of image databases [26], classifying regular patterns [13], completion of occluded shapes [35], object detection [19, 36] and detecting tumours in medical imaging [18]. Existing symmetry detection techniques can be broadly classified into global and local feature-based methods.
Global approaches treat the entire image as a signal from which symmetric properties are inferred, often via frequency analysis. Examples include the work of Marola [19] who proposed a method for detecting symmetry in images where the axis of symmetry intersects or passes near the centroid, Keller and Shkolnisky [1] who took an algebraic approach and employed Fourier analysis to detect symmetry, and Sun [27] who showed that the orientation of the dominant bilateral symmetry axis could be computed from the histogram of gradient orientations. The main drawbacks of these global approaches are their inability to detect multiple incidences of symmetry and the adverse effect of background structure on the result. Local feature-based methods use local features, edges, contours or boundary points, to reduce the problem to one of grouping symmetric sets of points or lines. The standard way to accumulate symmetry hypotheses from feature information is to use the Hough transform. Cham and Cipolla [1] detected skew symmetry in 2D edge maps using a modified Hough transform. Yiwa and Cheong [34] also detected skew symmetry in 2D edge maps using a Hough based approach. They used a 3D Hough space and de-skewed the detected symmetric regions. Yip [33] demonstrated a Hough transform based approach for detecting symmetry in artificial images. In 23 Tuytelaars et al. [29] presented impressive results detecting regular repetitions of planar patterns under perspective skew using a geometric framework. The approach detected all planar homologies 1 and could thus find reflections about a point, periodicities, and mirror symmetries. The method built clusters of matching points using a cascaded Hough transform, and typically took around 1 minutes to process an image. In contrast the new method proposed herein provides a simpler and more efficient approach. We utilise local feature information to establish robust symmetric matches on a feature level, and merge these into quadruplets which then vote for single symmetry hypotheses. Lazebnik et al. [12] also noted that affine invariant clusters of features could be matched within an image to detect symmetries under affine projection. The method used rotationally invariant descriptors (providing no orientation information) which restricted this, like [29], to a cluster-based approach. In comparison, our method uses orientation information to provide a more robust assessment of symmetry both between features pairs, and within groups of features. By assembling quadruplets from pairs of symmetric of features, we are able to detect symmetry not only under affine, but also perspective projection. Recently a new method [16] was proposed for grouping 1 A plane projective transformation is a planar homology if it has a line of fixed points (called the axis), together with a fixed point (called the vertex) not on the line [6]. symmetric constellations of features and detecting symmetry in the image plane. Pairs of symmetrically matching features were computed and each pair hypothesised either a bilateral symmetry axis or a centre of rotational symmetry. The pairs were grouped into symmetric constellations of features about common symmetry foci, identifying the dominant symmetries present in the image plane. Here we take the concept of matching reflective pairs of features proposed by [16] and extend this to detect planar bilateral symmetries under affine or perspective skew. Detecting symmetry under perspective skew requires using feature quadruplets rather than just pairs, and estimating vanishing points to assess local symmetries. The directions of the features are used to verify that the pairs of feature pairs in a quadruplet have a common axis of symmetry. This paper presents a generic approach to detecting planar bilateral symmetry in images, and grouping features associated with each instance of symmetry. 3. Detecting Planar Symmetry in Perspective In general, objects and thus symmetric surfaces are observed in perspective. Symmetry in the image plane is a subset of the perspective case, and affine, or weak perspective, provides a close approximation to perspective when the distance from the camera is much greater than the change of depth across the object. The solution presented here for detecting symmetry in perspective will also detect symmetry in the image plane and affine symmetries. However, if it is known a priori that the symmetry will be observed either in the image plane, or under approximately affine viewing conditions, then the method can be constrained to take account of this additional information yielding a more computationally efficient method. See Section 3.3 for the affine case and [16] for symmetry in the image plane. Our method is based on matching symmetric pairs of features that are then grouped into pairs-of-pairs to form feature quadruplets. Each quadruplet specifies a particular axis of symmetry, and we group quadruplets hypothesising common symmetry axes in the image. For a quadruplet to correctly specify a symmetry axis the 3D locations of the features must be close to coplanar, and both reflective pairs must share the same axis of symmetry. A number of image-based constraints are placed on the quadruplets to reject cases that cannot satisfy these conditions. These image-based constraints are necessary, but not sufficient, so we rely on the consistency of correct symmetry hypotheses dominating the noise of incorrect hypotheses, to reveal the presence of true 3D symmetries in the image. The remainder of this section discusses the details of this procedure for detecting planar bilateral symmetry under perspective skew.
3.1. Defining Feature Points A set of feature points p i is determined using any method (e.g. [22]) that offers robustness to affine deformation, and detects distinctive points with good repeatability. Next a feature descriptor k i is generated for each feature point, encoding the local appearance of the feature after it has been normalised with respect to rotation, scale and affine distortion. Any feature descriptor suitable for matching can be used as long as it provides a direction for each feature point. See [21] for a review of leading techniques. The experiments in this paper use the SIFT descriptor [15], which gives k i as a 128 element vector. Our method utilises the orientation of features when assessing symmetry. Affine invariant features typically define elliptical feature regions that are warped to circular regions before the feature descriptor is computed. The feature descriptor computes a feature s orientation after this warping. To obtain the true directions of the features in the image we must reverse the effect of this warping as shown in Fig. 1. Figure 1. An affine-invariant feature region is warped to a circle before its descriptor is computed. The feature orientation must be unwarped to give the true orientation of the feature in the image. A set of mirrored feature descriptors m i is also generated. Here m i is the feature descriptor generated from a mirrored copy of the image patch associated with feature k i. The choice of mirroring axis is arbitrary owing to the orientation normalisation in the generation of the descriptor. The mirrored feature descriptors m i can be directly computed from the reflected local image patch or depending on the construction of the feature descriptor being used determined directly from k i [16]. Matches are sort between the features k i and the mirrored features m j to form a set of (p i, p j ) pairs of potentially symmetric features. 3.2. Symmetric Quadruplets After forming reflective pairs the algorithm searches for symmetric quadruplets. These quadruplets are formed by grouping two pairs of reflective features, and each quadruplet hypothesises an axis of symmetry. Fig. 2 shows a schematic of a quadruplet of features made up of two reflective feature pairs (p 1, p 2 ) and (q 1, q 2 ) lying in the same plane and sharing the same symmetry axis. Such a quadruplet of non-colinear reflective pairs specifies a precise symmetry hypothesis that can be determined via a simple geometric construction. That is, find the intersection points c 1 and c 2 of the lines linking the non-paired members of the quadruplet, and the line linking these points will be the symmetry axis. An alternative is to use the vanishing point v of the lines linking the reflective pairs, and compute the 3D midpoints of these lines v 1 and v 2 using projective geometry. The image co-ordinates of the 3D midpoint v 1 are given by v 1 = d2 p 1 + d1 p 2 (1) d 1 + d 2 d 1 + d 2 where d i is the distance from v to p i in the image plane. The 3D midpoint of the other pair v 2 is determined in a similar manner, and together they define the axis of symmetry. For a quadruplet to be useful for discerning a true instance of symmetry the 3D locations of its four features must be close to coplanar and share a common symmetry axis. For a pair of feature pairs, we cannot know a priori if this is the case, however, we can enforce a number of conditions on each quadruplet necessary for this to hold. The conditions we require a quadruplet to satisfy are: 1. the quadruplet must be convex, 2. lines joining reflective pairs must not intersect, 3. the directions of the features must be consistent with the axis of symmetry estimated from the quadruplet, and 4. the relative difference in scale within each pair must not exceed k. The first three conditions are necessary for a quadruplet to represent a local 3D planar symmetry. The third condition is satisfied if the features within each pair are symmetrically oriented towards or away from each other, such that they either (a) define lines that intersect on or close to the axis of symmetry (as illustrated in Fig. 2), or (b) are close to parallel to the line joining the pair. The final condition restricts the scale change across the symmetric surface. Mikolajczyk et al. [22] report a sharp drop off of feature detector performance in structured scenes where features have changed scale by a factor of two or greater. As our method relies on accurate and robust feature performance, we can discount quadruplets that hypothesise a scale change of a factor two or more within a reflective pair. The relative scale of a pair is directly proportional to the ratio of the distances of each feature to the vanishing point, i.e. where s i is the scale of the i th feature and d i is the distance of p i from the vanishing point v in Fig. 2. For each reflective feature pair, we form n quadruplets by pairing it with the n closest neighbouring pairs in the image that are at least m pixels a part (for our experiments we chose m as 5 pixels and n = 15). Forming quadruplets from pairs in close proximity increases the likelihood s 1 s 2 = d1 d 2
symmetry plane v symmetry axis q 1 c 2! p 1 v 2 c 1 v 1 q 2 p 2 feature plane Figure 2. A schematic of a quadruplet of features made up of two reflective feature pairs (p 1, p 2) and (q 1, q 2) sharing the same symmetry axis, showing the vanishing point v and the geometric constructions that give the symmetry axis. that these pairs will be associated with the same instance of symmetry. However, pairs that are too close together will not give a reliable symmetry estimate. We test if the quadruplets satisfy the above conditions, and those that do not are discarded. Using the procedure described above means that we get up to n quadruplets per symmetric feature pair. Incorrectly matched feature pairs that do not lie on a symmetric surface will in general give no, or very few quadruplets, and the total number of incorrect quadruplets in an image will usually be small. The axes of symmetry estimated from the quadruplets can then be represented using the standard rθ polar coordinate parameterisation for a line r = x cos θ + y sin θ where (x, y) are the image centred co-ordinates of any point on the line (such as one of the 3D mid-points v i ) and θ is the angle perpendicular to the direction of the axis of symmetry. The linear Hough transform is applied to determine the dominant symmetry axes. Each quadruplet casts a single vote in rθ Hough space. The resulting Hough space is blurred with a Gaussian and the maxima extracted as potential symmetry axes. The quadruplets voting in the neighbourhood of a maximum are grouped together. Strictly speaking, for these groups of symmetric feature pairs sharing the same axis of symmetry to be consistent with an instance of planar bilateral symmetry the lines joining all pairs must have a common vanishing point. However, in practice noise in the feature locations and a desire to tolerate objects whose symmetry is not absolutely perfect, means that the vanishing points will be dispersed. Feature pairs close to each other will always be close to parallel, and when the vanishing point approaches infinity, this will be true for all pairs of pairs. If quadruplets with feature pairs that are close to parallel are used to estimate the vanishing point, small changes in the feature locations will lead to large changes of the position of the vanishing point. However, the position of the vanishing point will only change along the directions of the lines joining the features in the pairs, and these lines should still point towards one common vanishing point. We can therefore measure the consistency of the feature pairs by comparing the directions of the lines joining the features in the pairs. We choose a point on the axis of symmetry, and for each feature pair, we measure its distance to this point. We also calculate the tangent of the angle between the line connecting the features in a pair and the axis of symmetry. If the lines between the pairs have a common vanishing point, the tangent of these angles should vary linearly with the distance to the chosen point on the symmetry axis. This linear relationship can be robustly estimated using RANSAC [5]. This procedure greatly refines our grouping, giving our final estimate of the symmetry axis and its associated features. 3.3. Affine Assumption Pairs of symmetric features sharing a common symmetry plane will be joined by lines that are parallel in 3D. Under affine projection these lines will also appear parallel in the image. This greatly simplifies the symmetry detection process. Feature pairs with a common axis of symmetry are parallel, and we know that the symmetry axis intersects the midpoints of the lines joining the pairs. The direction of the axis of symmetry can be found using the directions of the features, since the intersection of lines through the feature positions and parallel to the feature directions has to be on the axis of symmetry. The affine detection process is as follows, first we determine the angle ψ of the line joining each pair. The pairs are then grouped according to their ψ angle, and for each feature pair, an axis of symmetry is estimated using the midpoint between the features and the intersection of the directions of the features. For groups of feature pairs with common ψ, dominant symmetry axes are found via the Hough transform, with each feature pair in the group voting for one orientation and location. The resulting feature constellations will always have consistent direction and thus consistent vanishing points (at infinity). 3.4. Using Feature Scale In theory the scale of reflectively matched pairs of features could also be used. Using the scale of a pair of matched
features, the 3D midpoint between the features can be estimated directly, without having to form a quadruplet. However, in practice inaccuracies in the scale information obtained by current affine-invariant feature techniques make this approach unfeasible. 4. Performance The method was implemented in Matlab, and feature points were detected and descriptors computed using code made available by the authors of these methods [15, 21, 22]. We tested our algorithm using several different feature detection methods: hessian-affine and Harris affine detectors [2], an intensity extrema-based detector (IBR), and an edge based region detector (EBR) [28]. The SIFT descriptor was used in all experiments. Fig. 3 shows the results of our algorithm on natural images. The dominant symmetry is indicated together with other symmetries with at least half as much support as the dominant axis. The results demonstrate the method s ability to detect multiple instances of symmetry and detect symmetry planes in the presence of clutter and partial occlusion. To quantitatively evaluate the algorithm s ability to detect symmetry under different amounts of perspective skew and in the presence of noise we generated a set of 1,3 symmetric images with known ground truth. We gathered 3 natural images, not containing any obvious symmetries. Each image was concatenated with a mirrored version of itself forming a perfectly symmetric image. These symmetric images were then skewed using a homography with the following effect: The image was rotated a random amount, and then tilted around an axis in the image plane passing through the image centre. The tilted image was viewed through a pinhole camera positioned a distance d away, and pointing towards, the image centre. The parameter d was set equal to the longest side of the image. Gray columns or rows were added to the right side or bottom of the image giving approximately 6% non-symmetric background in each image. Finally Gaussian noise was added to the image. For each of the 3 images gathered 45 test images were generated, showing symmetry with perspective tilt from to 8 degrees and Gaussian noise with standard deviation σ of, 25,, 75 and 1 (the graylevels of the images ranged between and 255). The harris-affine feature detector was used throughout the experiment, and the images were scaled so that just over 1, features were detected in every image. Fig. 4 shows examples of two test images and the resulting detected symmetries. The results of the test are shown in Fig. 5. The first diagram shows the percentage of correctly detected symmetry axes for all different tilts and noise levels. The second diagram shows the mean number of feature pairs supporting the correct symmetry hypothesis (when it was detected). (a) (b) Figure 4. Sample images from our test database, (a) tilt 6o no noise, (b) tilt 6o and Gaussian noise σ = 75. Note that there are approximately 1, feature points in each test image. With no noise or tilt we see close to perfect matching between mirrored pairs of features giving a little less than feature pairs supporting the correct hypothesis. As the tilt of the symmetric surface, or the noise level, increases we cannot expect to detect identical features in both parts of the symmetric image, and the number of feature pairs that agree with the correct axis of symmetry decreases. However, the correct axis of symmetry is still found in almost all images for tilts up to 6 degrees and Gaussian noise with standard deviation up to 75. Even though only a small part of the total number of feature pairs formed agrees with the correct axis of symmetry, we are still able to detect it. If the same experimets are run using the affine, or 2D symmetry detector instead of the perspective, the correct symmetry axes are found in the un-tilted images, With 1 degrees tilt, the methods still perform reasonably well, detecting, around 6% of the symmetry axes, but as the tilt increases, the performances of both methods rapidly decrease. This is expected since the perspective effects are quite strong in the images, and the symmetries cannot be be well modeled using the affine assumption. To assess the effect of clutter in the background we repeated a subset of the above experiments using a random selection one of the 3 images as background as shown in Fig. 6. The results from these tests are shown in Fig. 7. As can be seen, the textured backgrounds have little effect on the results.
(a) (d) (b) (e) (c) (f) Figure 3. Symmetry detected in perspective images, showing reflective matches and axes of symmetry detected. The algorithm s performance relies heavily on the matching capability of the feature method used, and it performs best when a large number of features are detected on a symmetric surface. The basic mechanism of matching instances of texture that are locally symmetric under some affine transformation performs well for strongly textured surfaces that are close to planar, such as the carpets in Fig. 3. Less textured surfaces such as the car or the phone generate less matches and are not as strongly detected. This predisposition towards textured surfaces is a result of the feature type chosen, our algorithm could equally well be applied to features from a shape-based descriptor such as [8]. A substantial amount of time consumed when detecting symmetry is used computing features and performing the matching. The analysis of the feature pairs, formation of quadruplets, grouping by symmetry axis and vanishing points, and final assessment of symmetry takes approximately 2 seconds for images with 1, feature points, running unoptimised Matlab code on a 2.8 GHz Pentium 4. In this scenario, using Lowe s pre-compiled SIFT code and a naive Matlab matching routine the time for computing features and performing matching was approximately 1 seconds. The results demonstrate the suitability of this approach for detecting planar bilateral symmetries under affine or per- spective distortion. Further work will consider segmenting symmetric regions, grouping local planar symmetries with a common symmetry plane, and detecting non-coplanar symmetry planes. The groups of features associated with each symmetry axis provide a good starting point for segmenting the symmetric region. Additional feature points could be generated in the vicinity of the symmetric matches, to grow symmetric regions possibly in a similar fashion to Ferrari et al. s object segmentation approach [4]. Segmenting the symmetric regions would also provide a more accurate measure of the extent of the axes of symmetry in the image, and allow a more rigourous verification of the correctness of each proposed symmetry axis. 5. Conclusions This paper has presented a method for detecting bilateral planar symmetry in images under perspective or affine distortion. The method is feature-based and relies on robust, repeatable matches under local affine distortion. It is posed independently of a specific feature detector or descriptor method, and provides a generic means of grouping features that are likely to correspond to an instance of planar symmetry in 3D. Hypotheses are built from quadruplets made of pairs of reflective feature matches and the method is thus able to consider symmetry at all locations, scales, and ori-
1 1 9 9 8 σ= 7 Percentage of correct detections Percentage of correct detections 8 σ = 25 σ = 6 σ = 75 σ = 1 4 3 7 6 4 3 2 2 1 1 σ= σ = 1 2 3 4 Tilt angle 6 7 8 9 1 2 3 (a) 4 Tilt angle 6 7 8 9 (a) 3 σ= σ = 25 σ = σ = 75 σ = 1 4 4 σ= σ = 2 Number of feature pairs Number of feature pairs 3 3 2 2 2 1 1 1 1 1 2 3 4 Tilt angle 6 7 8 9 1 2 3 4 Tilt angle 6 7 8 9 (b) (b) Figure 5. Results of test on 1,3 images for different noise levels σ. (a) Percentage of tests where correct symmetry axis was found, (b) number of pairs found supporting correct axis. Figure 7. Results of test with cluttered background on 54 images for two different noise levels σ. (a) Percentage of tests where correct symmetry axis was found, (b) number of pairs found supporting correct axis. thesised images with varying amounts of noise and perspective distortion. References Figure 6. A sample test image with a cluttered background, tilt 6o and Gaussian noise σ =. entations in the presence of perspective or affine distortion. The orientation of the features are used to verify the symmetry of each quadruplet and improve the robustness of the symmetry axis estimate. If it is known a priori that an affine assumption is appropriate then, using the orientations of the features makes it possible to estimate the axis of symmetry for each feature pair directly, without having to form quadruplets, giving increased computational efficiency under affine conditions. The algorithm has been employed in conjunction with different feature detectors and has been shown to successfully detect symmetry in natural images and its performance has been quantified on over 1, syn- [1] T.-J. Cham and R. Cipolla. Symmetry detection through local skewed symmetries. Image and Vision Computing, 13(5), 1995. 2 [2] S. C. Dakin and A. M. Herbert. The spatial region of integration for visual symmetry detection. Proc of the Royal Society London B. Bio Sci, 265:659 664, 1998. 1 [3] L. S. Davis. Understanding shape, ii: Symmetry. SMC, 7:24 212, 1977. 1 [4] V. Ferrari, T. Tuytelaars, and L. V. Gool. Simultaneous object recognition and segmentation from single or multiple model views. International Journal of Computer Vision, 25. 6 [5] M. Fischler and R. Bolles. Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography. Commun. ACM, 24(6):381 395, June 1981. 4 [6] R. I. Hartley and A. Zisserman. Multiple View Geometry in Computer Vision. Cambridge University Press, ISBN: 52154518, second edition, 24. 2
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