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204
Statistical Approach to Shape from Shading: Reconstruction of 3D Face Surfaces from Single 2D Images
 Neural Computation
, 1997
"... The human visual system is proficient in perceiving threedimensional shape from the shading patterns in a twodimensional image. How it does this is not well understood and continues to be a question of fundamental and practical interest. In this paper we present a new quantitative approach to shap ..."
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Cited by 115 (0 self)
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The human visual system is proficient in perceiving threedimensional shape from the shading patterns in a twodimensional image. How it does this is not well understood and continues to be a question of fundamental and practical interest. In this paper we present a new quantitative approach to shapefromshading that may provide some answers. We suggest that the brain, through evolution or prior experience, has discovered that objects can be classified into lowerdimensional objectclasses as to their shape. Extraction of shape from shading is then equivalent to the much simpler problem of parameter estimation in a low dimensional space. We carry out this proposal for an important class of 3D objects; human heads. From an ensemble of several hundred laserscanned 3D heads, we use principal component analysis to derive a lowdimensional parameterization of head shape space. An algorithm for solving shapefromshading using this representation is presented. It works well even on real im...
Moving coframes. I. A practical algorithm
 Acta Appl. Math
, 1998
"... Abstract. This is the first in a series of papers devoted to the development and applications of a new general theory of moving frames. In this paper, we formulate a practical and easy to implement explicit method to compute moving frames, invariant differential forms, differential invariants and in ..."
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Cited by 115 (28 self)
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Abstract. This is the first in a series of papers devoted to the development and applications of a new general theory of moving frames. In this paper, we formulate a practical and easy to implement explicit method to compute moving frames, invariant differential forms, differential invariants and invariant differential operators, and solve general equivalence problems for both finitedimensional Lie group actions and infinite Lie pseudogroups. A wide variety of applications, ranging from differential equations to differential geometry to computer vision are presented. The theoretical justifications for the moving coframe algorithm will appear in the next paper in this series.
Differential and numerically invariant signature curves applied to object recognition
 Int. J. Computer Vision
, 1998
"... Abstract. In this paper we introduce a new paradigm, the differential invariant signature curve or manifold, for the invariant recognition of visual objects. A general theorem of É. Cartan implies that two curves are related by a group transformation if and only if their signature curves are identic ..."
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Cited by 99 (28 self)
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Abstract. In this paper we introduce a new paradigm, the differential invariant signature curve or manifold, for the invariant recognition of visual objects. A general theorem of É. Cartan implies that two curves are related by a group transformation if and only if their signature curves are identical. The important examples of the Euclidean and equiaffine groups are discussed in detail. Secondly, we show how a new approach to the numerical approximation of differential invariants, based on suitable combination of joint invariants of the underlying group action, allows one to numerically compute differential invariant signatures in a fully groupinvariant manner. Applications to a variety of fundamental issues in vision, including detection of symmetries, visual tracking, and reconstruction of occlusions, are discussed.
A Theory of Specular Surface Geometry
, 1996
"... A theoretical framework is introduced for the perception of specular surface geometry. When an observer moves in threedimensional space, real scene features such as surface markings remain stationary with respect to the surfaces they belong to. In contrast, a virtual feature which is the specular r ..."
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Cited by 90 (2 self)
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A theoretical framework is introduced for the perception of specular surface geometry. When an observer moves in threedimensional space, real scene features such as surface markings remain stationary with respect to the surfaces they belong to. In contrast, a virtual feature which is the specular reflection of a real feature, travels on the surface. Based on the notion of caustics, a feature classification algorithm is developed that distinguishes real and virtual features from their image trajectories that result from observer motion. Next, using support functions of curves, a closedform relation is derived between the image trajectory of a virtual feature and the geometry of the specular surface it travels on. It is shown that, in the 2D case, where camera motion and the surface profile are coplanar, the profile is uniquely recovered by tracking just two unknown virtual features. Finally, these results are generalized to the case of arbitrary 3D surface profiles that are traveled by virtual features when camera motion is not confined to a plane. This generalization includes a number of mathematical results that substantially enhance the present understanding of specular surface geometry. An algorithm is developed that uniquely recovers 3D surface profiles using a single virtual feature tracked from the occluding boundary of the object. All theoretical derivations and proposed algorithms are substantiated by experiments.
From high energy physics to low level vision
 Presented in ONR workshop, UCLA
, 1996
"... Abstract. A geometric framework for image scale space, enhancement, and segmentation is presented. We consider intensity images as surfaces in the (x � I) space. The image is thereby a 2D surface in 3D space for gray level images, and a 2D surface in 5D for color images. The new formulation uni es m ..."
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Cited by 57 (23 self)
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Abstract. A geometric framework for image scale space, enhancement, and segmentation is presented. We consider intensity images as surfaces in the (x � I) space. The image is thereby a 2D surface in 3D space for gray level images, and a 2D surface in 5D for color images. The new formulation uni es many classical schemes and algorithms via a simple scaling of the intensity contrast, and results in new and e cient schemes. Extensions to multi dimensional signals become natural and lead to powerful denoising and scale space algorithms. Here, we demonstrate the proposed framework by applying it to denoise and improve graylevel and color images. 1 Introduction: A philosophical point of view In this paper we adopt an action potential that was recently introduced in physics and use it to produce a natural scale space for images as surfaces. It will lead us to the construction of image enhancement procedures for gray and color images. This model also integrates many existing segmentation and scale space procedures
Joint invariant signatures
 Found. Comput. Math
, 1999
"... Dedicated to the memory of Gian–Carlo Rota Abstract. A new, algorithmic theory of moving frames is applied to classify joint invariants and joint differential invariants of transformation groups. Equivalence and symmetry properties of submanifolds are completely determined by their joint signatures, ..."
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Cited by 46 (25 self)
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Dedicated to the memory of Gian–Carlo Rota Abstract. A new, algorithmic theory of moving frames is applied to classify joint invariants and joint differential invariants of transformation groups. Equivalence and symmetry properties of submanifolds are completely determined by their joint signatures, which are parametrized by a suitable collection of joint invariants and/or joint differential invariants. A variety of fundamental geometric examples are developed in detail. Applications to object recognition problems in computer vision and the design of invariant numerical approximations are indicated.
Invariant Geometric Evolutions of Surfaces and Volumetric Smoothing
, 1997
"... . The study of geometric flows for smoothing, multiscale representation, and analysis of two and threedimensional objects has received much attention in the past few years. In this paper, we first survey the geometric smoothing of curves and surfaces via geometric heattype flows, which are invari ..."
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Cited by 45 (13 self)
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. The study of geometric flows for smoothing, multiscale representation, and analysis of two and threedimensional objects has received much attention in the past few years. In this paper, we first survey the geometric smoothing of curves and surfaces via geometric heattype flows, which are invariant under the groups of Euclidean and affine motions. Second, using the general theory of differential invariants, we determine the general formula for a geometric hypersurface evolution which is invariant under a prescribed symmetry group. As an application, we present the simplest affine invariant flow for (convex) surfaces in threedimensional space, which, like the affineinvariant curve shortening flow, will be of fundamental importance in the processing of threedimensional images. Key words. invariant surface evolutions, partial differential equations, geometric smoothing, symmetry groups AMS subject classifications. 35K22, 53A15, 53A55, 53A20, 35B99 PII. S0036139994266311 1. Intro...
Invariant EulerLagrange equations and the invariant variational bicomplex
, 2003
"... In this paper, we derive an explicit groupinvariant formula for the EulerLagrange equations associated with an invariant variational problem. The method relies on a groupinvariant version of the variational bicomplex induced by a general equivariant moving frame construction, and is of independe ..."
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Cited by 41 (29 self)
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In this paper, we derive an explicit groupinvariant formula for the EulerLagrange equations associated with an invariant variational problem. The method relies on a groupinvariant version of the variational bicomplex induced by a general equivariant moving frame construction, and is of independent interest.
Symmetrycurvature duality
 Computer Vision, Graphics and Image Processing
, 1987
"... Several studies have shown the importance of two very different descriptors for shape: symmetry structure and curvature extrema. The main theorem proved by this paper, i.e. the SymmetryCurvature Duality Theorem, states that there is an important relationship between symmetry and curvature extrema: ..."
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Cited by 39 (3 self)
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Several studies have shown the importance of two very different descriptors for shape: symmetry structure and curvature extrema. The main theorem proved by this paper, i.e. the SymmetryCurvature Duality Theorem, states that there is an important relationship between symmetry and curvature extrema: If we say that curvature extrema are of two opposite types, either maxima or minima, then the theorem states: Any segment of a smooth planar curve, bounded by two consecutive curvature extrema of the same type, has a unique symmetry axis, and the axis terminates at the curvature extremum of the opposite type. The theorem is initially proved using Brady’s SLS as the symmetry analysis. However, the theorem is then generalized for any differential symmetry analysis. In order to prove the theorem, a number of results are established concerning the symmetry structure of Hoffman’s and Richards ’ codons. All results are obtained first by observing that any codon is a string of two, three, or four spirals, and then by reducing the theory of codons to that of spirals. We show that the SLS of a codon is either (1) an SAT, which is a
Generating Differential Invariants
, 2007
"... The equivariant method of moving frames is used to specify systems of generating differential invariants for finitedimensional Lie group actions. ..."
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Cited by 34 (17 self)
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The equivariant method of moving frames is used to specify systems of generating differential invariants for finitedimensional Lie group actions.