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Computational Symmetry in Computer Vision and Computer Graphics
17 Jun 2010-
TL;DR: Recognizing the fundamental relevance and group theory of symmetry has the potential to play an important role in computational sciences.
Abstract: In the arts and sciences, as well as in our daily lives, symmetry has made a profound and lasting impact. Likewise, a computational treatment of symmetry and group theory (the ultimate mathematical formalization of symmetry) has the potential to play an important role in computational sciences. Though the term Computational Symmetry was formally defined a decade ago by the first author, referring to algorithmic treatment of symmetries, seeking symmetry from digital data has been attempted for over four decades. Computational symmetry on real world data turns out to be challenging enough that, after decades of effort, a fully automated symmetry-savvy system remains elusive for real world applications. The recent resurging interests in computational symmetry for computer vision and computer graphics applications have shown promising results. Recognizing the fundamental relevance and potential power that computational symmetry affords, we offer this survey to the computer vision and computer graphics communities. This survey provides a succinct summary of the relevant mathematical theory, a historic perspective of some important symmetry-related ideas, a partial yet timely report on the state of the arts symmetry detection algorithms along with its first quantitative benchmark, a diverse set of real world applications, suggestions for future directions and a comprehensive reference list.
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TL;DR: In this paper, the authors offer a new book that enPDFd the perception of the visual world to read, which they call "Let's Read". But they do not discuss how to read it.
Abstract: Let's read! We will often find out this sentence everywhere. When still being a kid, mom used to order us to always read, so did the teacher. Some books are fully read in a week and we need the obligation to support reading. What about now? Do you still love reading? Is reading only for you who have obligation? Absolutely not! We here offer you a new book enPDFd the perception of the visual world to read.
2,250 citations
688 citations
27 Jul 2014
TL;DR: This work proposes a method for automatically guiding patch-based image completion using mid-level structural cues by first estimates planar projection parameters, softly segments the known region into planes, and discovers translational regularity within these planes.
Abstract: We propose a method for automatically guiding patch-based image completion using mid-level structural cues. Our method first estimates planar projection parameters, softly segments the known region into planes, and discovers translational regularity within these planes. This information is then converted into soft constraints for the low-level completion algorithm by defining prior probabilities for patch offsets and transformations. Our method handles multiple planes, and in the absence of any detected planes falls back to a baseline fronto-parallel image completion algorithm. We validate our technique through extensive comparisons with state-of-the-art algorithms on a variety of scenes.
337 citations
TL;DR: This method can accurately recover both the intrinsic low-rank texture and the unknown transformation, and hence both the geometry and appearance of the associated planar region in 3D in the case of planar regions with significant affine or projective deformation.
Abstract: In this paper, we propose a new tool to efficiently extract a class of "low-rank textures" in a 3D scene from user-specified windows in 2D images despite significant corruptions and warping. The low-rank textures capture geometrically meaningful structures in an image, which encompass conventional local features such as edges and corners as well as many kinds of regular, symmetric patterns ubiquitous in urban environments and man-made objects. Our approach to finding these low-rank textures leverages the recent breakthroughs in convex optimization that enable robust recovery of a high-dimensional low-rank matrix despite gross sparse errors. In the case of planar regions with significant affine or projective deformation, our method can accurately recover both the intrinsic low-rank texture and the unknown transformation, and hence both the geometry and appearance of the associated planar region in 3D. Extensive experimental results demonstrate that this new technique works effectively for many regular and near-regular patterns or objects that are approximately low-rank, such as symmetrical patterns, building facades, printed text, and human faces.
284 citations
Cites background from "Computational Symmetry in Computer ..."
...In particular, this problem has been studied extensively for the purpose of rectifying building facades in urban scenes (Liu and Collins 2001; Ma et al. 2004; Liu et al. 2010; Park et al. 2010)....
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References
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TL;DR: A near-real-time computer system that can locate and track a subject's head, and then recognize the person by comparing characteristics of the face to those of known individuals, and that is easy to implement using a neural network architecture.
Abstract: We have developed a near-real-time computer system that can locate and track a subject's head, and then recognize the person by comparing characteristics of the face to those of known individuals. The computational approach taken in this system is motivated by both physiology and information theory, as well as by the practical requirements of near-real-time performance and accuracy. Our approach treats the face recognition problem as an intrinsically two-dimensional (2-D) recognition problem rather than requiring recovery of three-dimensional geometry, taking advantage of the fact that faces are normally upright and thus may be described by a small set of 2-D characteristic views. The system functions by projecting face images onto a feature space that spans the significant variations among known face images. The significant features are known as "eigenfaces," because they are the eigenvectors (principal components) of the set of faces; they do not necessarily correspond to features such as eyes, ears, and noses. The projection operation characterizes an individual face by a weighted sum of the eigenface features, and so to recognize a particular face it is necessary only to compare these weights to those of known individuals. Some particular advantages of our approach are that it provides for the ability to learn and later recognize new faces in an unsupervised manner, and that it is easy to implement using a neural network architecture.
14,562 citations
"Computational Symmetry in Computer ..." refers methods in this paper
...Following a face normalization process analogous to [12, 265], three anatomical feature points, the inner canthus of each eye and the philtrum are the three anchor-points to define an affine deformation, a face-midline centered vertically in each face image, and a set of uniform metric coordinates for all faces (Figure 5....
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TL;DR: A face recognition algorithm which is insensitive to large variation in lighting direction and facial expression is developed, based on Fisher's linear discriminant and produces well separated classes in a low-dimensional subspace, even under severe variations in lighting and facial expressions.
Abstract: We develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression. Taking a pattern classification approach, we consider each pixel in an image as a coordinate in a high-dimensional space. We take advantage of the observation that the images of a particular face, under varying illumination but fixed pose, lie in a 3D linear subspace of the high dimensional image space-if the face is a Lambertian surface without shadowing. However, since faces are not truly Lambertian surfaces and do indeed produce self-shadowing, images will deviate from this linear subspace. Rather than explicitly modeling this deviation, we linearly project the image into a subspace in a manner which discounts those regions of the face with large deviation. Our projection method is based on Fisher's linear discriminant and produces well separated classes in a low-dimensional subspace, even under severe variation in lighting and facial expressions. The eigenface technique, another method based on linearly projecting the image space to a low dimensional subspace, has similar computational requirements. Yet, extensive experimental results demonstrate that the proposed "Fisherface" method has error rates that are lower than those of the eigenface technique for tests on the Harvard and Yale face databases.
11,674 citations
"Computational Symmetry in Computer ..." refers methods in this paper
...Following a face normalization process analogous to [12, 265], three anatomical feature points, the inner canthus of each eye and the philtrum are the three anchor-points to define an affine deformation, a face-midline centered vertically in each face image, and a set of uniform metric coordinates for all faces (Figure 5....
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TL;DR: It is shown that recognition of geometrical patterns and alphabetical characters independently of position, size and orientation can be accomplished and it is indicated that generalization is possible to include invariance with parallel projection.
Abstract: In this paper a theory of two-dimensional moment invariants for planar geometric figures is presented. A fundamental theorem is established to relate such moment invariants to the well-known algebraic invariants. Complete systems of moment invariants under translation, similitude and orthogonal transformations are derived. Some moment invariants under general two-dimensional linear transformations are also included. Both theoretical formulation and practical models of visual pattern recognition based upon these moment invariants are discussed. A simple simulation program together with its performance are also presented. It is shown that recognition of geometrical patterns and alphabetical characters independently of position, size and orientation can be accomplished. It is also indicated that generalization is possible to include invariance with parallel projection.
7,963 citations
"Computational Symmetry in Computer ..." refers background in this paper
...In [103], moment invariants are developed for pattern recognition....
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TL;DR: The dorsocaudal medial entorhinal cortex (dMEC) contains a directionally oriented, topographically organized neural map of the spatial environment, whose key unit is the ‘grid cell’, which is activated whenever the animal's position coincides with any vertex of a regular grid of equilateral triangles spanning the surface of the environment.
Abstract: The ability to find one's way depends on neural algorithms that integrate information about place, distance and direction, but the implementation of these operations in cortical microcircuits is poorly understood. Here we show that the dorsocaudal medial entorhinal cortex (dMEC) contains a directionally oriented, topographically organized neural map of the spatial environment. Its key unit is the 'grid cell', which is activated whenever the animal's position coincides with any vertex of a regular grid of equilateral triangles spanning the surface of the environment. Grids of neighbouring cells share a common orientation and spacing, but their vertex locations (their phases) differ. The spacing and size of individual fields increase from dorsal to ventral dMEC. The map is anchored to external landmarks, but persists in their absence, suggesting that grid cells may be part of a generalized, path-integration-based map of the spatial environment.
3,445 citations
"Computational Symmetry in Computer ..." refers background in this paper
...Surprisingly strong hexagonal symmetry is observed in the the firing fields of grid cells found in the rat dorsolateral medial entorhinal cortex [88] (top of Figure 4....
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01 Jan 2002
TL;DR: The wide-baseline stereo problem, i.e. the problem of establishing correspondences between a pair of images taken from different viewpoints, is studied and an efficient and practically fast detection algorithm is presented for an affinely-invariant stable subset of extremal regions, the maximally stable extremal region (MSER).
Abstract: The wide-baseline stereo problem, i.e. the problem of establishing correspondences between a pair of images taken from different viewpoints is studied. A new set of image elements that are put into correspondence, the so called extremal regions , is introduced. Extremal regions possess highly desirable properties: the set is closed under (1) continuous (and thus projective) transformation of image coordinates and (2) monotonic transformation of image intensities. An efficient (near linear complexity) and practically fast detection algorithm (near frame rate) is presented for an affinely invariant stable subset of extremal regions, the maximally stable extremal regions (MSER). A new robust similarity measure for establishing tentative correspondences is proposed. The robustness ensures that invariants from multiple measurement regions (regions obtained by invariant constructions from extremal regions), some that are significantly larger (and hence discriminative) than the MSERs, may be used to establish tentative correspondences. The high utility of MSERs, multiple measurement regions and the robust metric is demonstrated in wide-baseline experiments on image pairs from both indoor and outdoor scenes. Significant change of scale (3.5×), illumination conditions, out-of-plane rotation, occlusion, locally anisotropic scale change and 3D translation of the viewpoint are all present in the test problems. Good estimates of epipolar geometry (average distance from corresponding points to the epipolar line below 0.09 of the inter-pixel distance) are obtained.
3,400 citations
"Computational Symmetry in Computer ..." refers methods in this paper
...The leftmost column shows the first iteration of the algorithm after initialization from the maximally stable extremal region interest point detector [184]....
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