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Code available at http://github.com/rfabbri/minus; International audience; We present a method for solving two minimal problems for relative camera pose estimation from three views, which are based on three view correspondences of (i) three points and one line and (ii) three points and two lines through two of the points. These problems are too difficult to be efficiently solved by the state of the art Grobner basis methods. Our method is based on a new efficient homotopy continuation (HC) solver, which dramatically speeds up previous HC solving by specializing HC methods to generic cases of our problems. We show in simulated experiments that our solvers are numerically robust and stable under image noise. We show in real experiment that (i) SIFT features provide good enough point-and-line correspondences for three-view reconstruction and (ii) that we can solve difficult cases with too few or too noisy tentative matches where the state of the art structure from motion initialization fails.

A major challenge for image guided tumor ablation is the high treatment variability due to heterogeneous tissue characteristics and thermal sinks. In this work, we present a methodology to analyze the geometry of the treatment zones and treatment zone variability. Our first contribution is an applicator centric co-ordinate system which enables us to compare treatment zones and vendor specifications across patients. Our second contribution is the analysis of the shape of the ablation zone using applicator centric longitudinal 2D cross sections. We present initial results of applying this methodology to analyze the geometry and variability in synthetic examples like ellipsoid, sphere and real microwave ablation zones in lung and liver.

The main question we address is: What is the minimal information required to generate closed, nonintersecting planar boundaries? For this paper, we restrict "shape" to this meaning. More precisely, we examine whether the medial axis, together with dynamics, can serve as a language to design shapes and to effect shape changes. We represent the medial axis together with a direction of flow along the axis as the shock graph and examine the reconstruction of shape along each of the three types of medial axis points, A/sub 1//sup 2/, A/sub 1//sup 3/, A/sub 3/, and the associated six types of shock points. First, we show that the tangent and curvature of the medial axis and the speed and acceleration of the shock with respect to time of propagation are sufficient to determine the boundary tangent and curvature at corresponding points of the boundary. This implies that a rather coarse sampling of the symmetry axis, its tangent, curvature, speed, and acceleration is sufficient to regenerate accurately a local neighborhood of shape at regular axis points (A/sub 1//sup 2/). Second, we examine the reconstruction of shape at branch points (A/sub 1//sup 3/) where three regular branches are joined. We show that the three pairs of geometry (that is, curvature) and dynamics (that is, acceleration) must satisfy certain constraints. Finally, we derive similar results for the end points of shock branches (A/sub 3/ points). These formulas completely specify the local reconstruction of a shape from its shock-graph or medial axis and the conditions required to form a coherent shape from the medial axis.

A key challenge underlying theories of vision is how the spatially restricted, retinotopically represented feature analysis can be integrated to form abstract, coordinate-free object models. A resolution likely depends on the use of intermediate-level representations which can on the one hand be populated by local features and on the other hand be used as atomic units underlying the formation of, and interaction with, object hypotheses. The precise structure of this intermediate representation derives from the varied requirements of a range of visual tasks which motivate a significant role for incorporating a geometry of visual form. The need to integrate input from features capturing surface properties such as texture, shading, motion, color, etc., as well as from features capturing surface discontinuities such as silhouettes, T-junctions, etc., implies a geometry which captures both regional and boundary aspects. Curves, as a geometric model of boundaries, have been extensively used as an intermediate representation in computational, perceptual, and physiological studies, while the use of the medial axis (MA) has been popular mainly in computer vision as a geometric region-based model of the interior of closed boundaries. We extend the traditional model of the MA to represent images, where each MA segment represents a region of the image which we call a visual fragment. We present a unified theory of perceptual grouping and object recognition where through various sequences of transformations of the MA representation, visual fragments are grouped in various configurations to form object hypotheses, and are related to stored models. The mechanisms underlying both the computation and the transformation of the MA is a lateral wave propagation model. Recent psychophysical experiments depicting contrast sensitivity map peaks at the medial axes of stimuli, and experiments on perceptual filling-in, and brightness induction and modulation, are consistent with both the use of an MA representation and a propagation-based scheme. Also, recent neurophysiological recordings in V1 correlate with the MA hypothesis and a horizontal propagation scheme. This evidence supports a geometric computational paradigm for processing sensory data where both dynamic in-plane propagation and feedforward-feedback connections play an integral role.

In this paper we address the curve completion problem, e.g., the geometric continuation of boundaries of objects which are temporarily interrupted by occlusion. Also known as the gap completion or shape completion problem, this problem is a significant element of perceptual grouping of edge elements and has been approached by using cubic splines or biarcs which minimize total curvature squared (elastica), as motivated by a physical analogy. Our approach is motivated by railroad design methods of the early 1900's which connect two rail segments by “transition curves”, and by the work of Knuth on mathematical typography. We propose that in using an energy minimizing solution completion curves should not penalize curvature as in elastica but curvature variation. The minimization of total curvature variation leads to an Euler Spiral solution, a curve whose curvature varies linearly with arclength. The construction of this curve from a pair of points and tangents at these points is reduced to a nonlinear system of equations involving Fresnel Integrals, whose solution relies on optimization from a suitable initial condition constrained to satisfy given boundary conditions. Since the choice of an appropriate initial curve is critical in this optimization, we analytically derive an optimal solution in the class of biarc curves, which is then used as the initial curve. The resulting interpolations yield intuitive interpolation across gaps and occlusions, and are extensible, in contrast to scale invariant elastica. In addition, Euler Spiral segments can be used in other applications of curve completions, e.g., modeling boundary segments between curvature extrema or skeletal branch geometry.

In this report we explore the local geometry of the medial axes (MA) and shocks (SH), and their structural changes under deformations, by viewing these symmetries as subsets of the symmetry set (SS) and present two results. First, we establish that the local form of the medial axes must generically be one of three cases: endpoints (A/sub 3/)/sup 1/, interior points (A/sub 1//sup 2/), and junctions (A/sub 1//sup 3/). The local form of shocks is a subclassification of these points. Second, we address the (classical) instability of the MA, i.e., abrupt changes in the representation with a slight changes in shape, as when a new branch appears with slight protrusion. The identification of these "transitions" is clearly crucial in robust object recognition. We show that for the medial axis only two such instabilities are possible: (i) when four branches come together (A/sub 1//sup 4/), and (ii) when a new branch grows out of an existing one (A/sub 1/A/sub 3/). Similarly, the six cases of shock instabilities are sub-classifications of these. The identification of these skeletal instabilities allows us to make equivalent structurally distinct skeletons arising from highly similar shapes, thus, regularizing the recognition process.

This paper presents an approach for computing the symmetries (skeletons) of an edge map consisting of a collection of curve segments. This approach is a combination of analytic computations in the style of computational geometry and discrete propagations on a grid in the style of the numerical solutions of PDE's. Specifically, waves from each of the initial curve segments are initialized and propagated as a discrete wavefront along discrete directions. In addition, to avoid error built up due to the discrete nature of propagation, shockwaves are detected and explicitly propagated along a secondary dynamic grid. The propagation of shockwaves, integrated with the propagation of the wavefront along discrete directions, leads to an exact simulation of propagation by the Eikonal equation. The resulting symmetries are simply the collection of shockwaves formed in this process which can be manipulated locally, exactly, and efficiently under local changes in an edge map (gap completion, removal of spurious elements, etc). The ability to express grouping operations in the language of symmetry maps makes it an appropriate intermediate representation between low-level edge maps and high level object hypotheses.

Visual tasks often require a hierarchical representation of shapes and images in scales ranging from coarse to fine. A variety of linear and nonlinear smoothing techniques, such as Gaussian smoothing, anisotropic diffusion, regularization, etc., have been proposed, leading to scalespace representations. We propose ageometricsmoothing method based on local curvature for shapes and images. The deformation by curvature, or the geometric heat equation, is a special case of thereaction?diffusionframework proposed in 41]. For shapes, the approach is analogous to the classical heat equation smoothing, but with a renormalization by arc-length at each infinitesimal step. For images, the smoothing is similar to anisotropic diffusion in that, since the component of diffusion in the direction of the brightness gradient is nil, edge location is left intact. Curvature deformation smoothing for shape has a number of desirable properties: it preserves inclusion order, annihilates extrema and inflection points without creating new ones, decreases total curvature, satisfies the semigroup property allowing for local iterative computations, etc. Curvature deformation smoothing of an image is based on viewing it as a collection of iso-intensity level sets, each of which is smoothed by curvature. The reassembly of these smoothed level sets into a smoothed image follows a number of mathematical properties; it is shown that the extension from smoothing shapes to smoothing images is mathematically sound due to a number of recent results 21]. A generalization of these results 14] justifies the extension of the entireentropy scale spacefor shapes 42] to one for images, where each iso-intensity level curve is deformed by a combination of constant and curvature deformation. The scheme has been implemented and is illustrated for several medical, aerial, and range images.

We present a new implementation of an algorithm aimed at recovering a 3D shape from its 2D gray-level picture. In order to reconstruct the shape of the object, an almost arbitrarily initialized 3D function is propagated on a rectangular grid, so that a level set of this function tracks the height contours of the shape. The method imports techniques from differential geometry, fluid dynamics, and numerical analysis and provides an accurate shape from shading algorithm. The method solves some topological problems and gracefully handles cases of non-smooth surfaces that give rise to shocks in the propagating contours. Real and synthetic images of 3D profiles were submitted to the algorithm and the reconstructed surfaces are presented, demonstrating the effectiveness of the proposed method.

We undertake to develop a general theory of two-dimensional shape by elucidating several principles which any such theory should meet. The principles are organized around two basic intuitions: first, if a boundary were changed only slightly, then, in general, its shape would change only slightly. This leads us to propose an operational theory of shape based on incremental contour deformations. The second intuition is that not all contours are shapes, but rather only those that can enclose “physical” material. A theory of contour deformation is derived from these principles, based on abstract conservation principles and Hamilton-Jacobi theory. These principles are based on the work of Sethian (1985a, c), the Osher-Sethian (1988), level set formulation the classical shock theory of Lax (1971; 1973), as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of Gage-Hamilton-Grayson (1986; 1989). The result is a characterization of the computational elements of shape: deformations, parts, bends, and seeds, which show where to place the components of a shape. The theory unifies many of the diverse aspects of shapes, and leads to a space of shapes (the reaction/diffusion space), which places shapes within a neighborhood of “similar” ones. Such similarity relationships underlie descriptions suitable for recognition.

The instabilities of the medial axis of a shape under deformations have long been recognized as a major obstacle to its use in recognition and other applications. These instabilities, or transitions, occur when the structure of the medial axis graph changes abruptly under deformations of shape. The recent classification of these transitions in 2D for the medial axis and for the shock graph was a key factor in the development of an object recognition system where the classified instabilities were utilized to represent deformation paths. The classification of generic transitions of the 3D medial axis could likewise potentially lead to a similar representation in 3D. In this paper, these transitions are classified by examining the order of contact of spheres with the surface, leading to an enumeration of possible transitions which are then examined on a case-by-case basis. Some cases are ruled out as never occurring in any family of deformations, while others are shown to be nongeneric in a one-parameter family of deformations. Finally, the remaining cases are shown to be viable by developing a specific example for each. Our work is inspired by that of Bogaevsky, who obtained the transitions as part of an investigation of viscosity solutions of Hamilton-Jacobi equations. Our contribution is to give a more down-to-earth approach, bringing this work to the attention of the computer vision community, and to provide explicit constructions for the various transitions using simple surfaces. We believe that the classification of these transitions is vital to the successful regularization of the medial axis in its use in real applications.

This paper addresses a key bottleneck in the use of the 3D medial axis (MA) representation, namely, how the complex MA structure can be regularized so that similar, within-category 3D shapes yield similar 3D MA that are distinct from the non-category shapes. We rely on previous work which (i) constructs a hierarchical MA hypergraph, the medial scaffold (MS), and (ii) the theoretical classification of the instabilities of this structure, or transitions (sudden topological changes due to a small perturbation). The shapes at transition point are degenerate. Our approach is to recognize the transitions which are close-by to a given shape and transform the shape to this transition point, and repeat until no close-by transitions exists. This move towards degeneracy is the basis of simplification of shape. We derive 11 transforms from 7 transitions and follow a greedy scheme in applying the transform. The results show that the simplified MA preserves with-in-category similarity, thus indicating its potential use in various applications including shape analysis, manipulation, and matching.

We define the concept of contact between a plane curve and a circle, and between a surface in 3D and a sphere, and describe all the contacts which generically occur for curves and surfaces, or 1-parameter families of such. We then list the local forms of the symmetry set and medial axis in 2D and 3D and briefly describe the local reconstruction of a boundary surface from its symmetry set or medial axis. Next, we describe the transitions which occur on the medial axis and symmetry set in a 1-parameter family of plane curves and which occur on the medial axis in a 1-parameter family of surfaces in 3D. Finally we make some remarks on the constraints which the medial axis must obey at special points such as Y-junctions. Some applications are given to illustrate these ideas.

We present a framework for extracting image contours based on geometric and structural consistency among edge element locations and orientations. The paper presents two contributions. First, we observe that while the traditional edge orientation operators are based on first-order derivatives, orientation as tangent of a localized curve requires third-order derivatives. We derive a numerically stable third-order edge operator and show that it outperforms current techniques. Second, we consider all discrete n-tuples of edges in a local neighborhood (7times7) and retain those that are geometrically consistent with a third-order local curve model. This results in a number of ordered discrete combinations of edges, each represented by a bundle of curves. The resulting curve bundle map is a representation of all possible local groupings from which longer contour fragments are constructed. We validate our results and show that our framework outperforms traditional approaches to contour extraction.

The relationship between the orientation and curvature of projected curves and the orientation and curvature of the underlying space curve has been previously established. This has allowed a disambiguation of correspondences in two views and a transfer of these properties to a third view for confirmation. We propose that a higher-order intrinsic differential geometry attribute, namely, curvature derivative, is necessary to account for the range of variation of space curves and their projections. We derive relationships between curvature derivative in a projected view, and curvature derivative and torsion of the underlying space curve. Regardless of the point, tangent, and curvature, any pair of curvature derivatives are possible correspondences, but most would lead to very high torsion and curvature derivatives. We propose that the minimization of third order derivatives of the reconstruction, which combines torsion and curvature derivative of the space curve, regularizes the process of finding the correct correspondences.

This paper proposes a novel hypergraph skeletal representation for 3D shape based on a formal derivation of the generic structure of its medial axis. By classifying each skeletal point by its order of contact, we show that, genetically, the medial axis consists of five types of points, which are then organized into sheets, curves, and points: 1) sheets (manifolds with boundary) which are the locus of bitangent spheres with regular tangency A/sub 1//sup 2/ (A/sub k//sup n/ notation means n distinct k-fold tangencies of the sphere of contact, as explained in the text); two types of curves, 2) the intersection curve of three sheets and the locus of centers of tritangent spheres, A/sub 1//sup 3/, and 3) the boundary of sheets, which are the locus of centers of spheres whose radius equals the larger principal curvature, i.e., higher order contact A/sub 3/ points; and two types of points, 4) centers of quad-tangent spheres, A/sub 1//sup 4/, and 5) centers of spheres with one regular tangency and one higher order tangency, A/sub 1/A/sub 3/. The geometry of the 3D medial axis thus consists of sheets (A/sub 1//sup 2/) bounded by one type of curve (A/sub 3/) on their free end, which corresponds to ridges on the surface, and attached to two other sheets at another type of curve (A/sub 1//sup 3/), which support a generalized cylinder description. The A/sub 3/ curves can only end in A/sub 1/ A/sub 3/ points where they must meet an A/sub 1//sup 3/ curve. The A/sub 1//sup 3/ curves meet together in fours at an A/sub 1//sup 4/ point. This formal result leads to a compact representation for 3D shape, referred to as the medial axis hypergraph representation consisting of nodes (A/sub 1//sup 4/ and A/sub 1/ A/sub 3/ points), links between pairs of nodes (A/sub 1//sup 3/ and A/sub 3/ curves) and hyperlinks between groups of links (A/sub 1//sup 2/ sheets). The description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes, in analogy to 2D. We derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph together with the radius function. Thus, this information completely characterizes 3D shape and lays the theoretical foundation for its use in recognition, morphing, design, and manipulation of shapes.

This paper presents a novel approach to surface representation based on its differential deformations. The evolution of an arbitrary curve by curvature deforms it to a round point while in the process simplifying it. Similarly, we seek a process that deforms an arbitrary surface into a sphere without developing self-intersections, in the process creating a sequence of increasingly simpler surfaces. No previously studied curvature dependent flow satisfies this requirement: mean curvature flow leads to a splitting of the surface, while Gaussian curvature flow leads to instabilities. Thus, in search for such a process, we impose constraints (motivated by visual representation) to narrow down the space of candidate flows. Our main result is to establish a direction for the movement of points to avoid self-intersections: (1) convex elliptic points should move in, while concave elliptic points move out; and (2) hyperbolic and parabolic points should not move at all. Accordingly, we propose /spl part//spl psi///spl part/t=sign(H)/spl radic/(G. >

The extraction of figure symmetry from image contours faces a number of fundamental difficulties: object symmetries are distorted due to (i) gaps in the bounding contour of a shape due to figure-ground blending, weak contrast edges, highlights, noise, etc.; (ii) an introduction of parts and occluders, and (iii) spurious edge elements due to surface markings, texture, etc. A framework for extracting such symmetries from real images is proposed based on the propagation of contour orientation information and the detection of four types of singularities (shocks) arising from the collision of propagating elements. In this paper, we show that an additional labeling of shocks based on whether the colliding wavefronts carry true orientation information (regular vs. rarefaction waves) allows a division of shocks into three sets: regular shocks are the partial shocks of partial contours as they remain invariant to the completion of the contour; semi-degenerate and degenerate shocks depict potential parts and gaps. Finally, shocks altered due to spurious edges, occlusion, and gaps are recovered via a simulation of inter-penetrating waves generated at select shock groups which with the aid of the above shock labels leads to second and further generations of shocks.

Subpixel methods that locate curves and their singularities, and that accurately measure geometric quantities, such as orientation and curvature, are of significant importance in computer vision and graphics. Such methods often use local surface fits or structural models for a local neighborhood of the curve to obtain the interpolated curve. Whereas their performance is good in smooth regions of the curve, it is typically poor in the vicinity of singularities. Similarly, the computation of geometric quantities is often regularized to deal with noise present in discrete data. However, in the process, discontinuities are blurred over, leading to poor estimates at them and in their vicinity. In this paper we propose a geometric interpolation technique to overcome these limitations by locating curves and obtaining geometric estimates while (1) not blurring across discontinuities and (2) explicitly and accurately placing them. The essential idea is to avoid the propagation of information across singularities. This is accomplished by a one-sided smoothing technique, where information is propagated from the direction of the side with the “smoother” neighborhood. When both sides are nonsmooth, the two existing discontinuities are relieved by placing a single discontinuity, or shock. The placement of shocks is guided by geometric continuity constraints, resulting in subpixel interpolation with accurate geometric estimates. Since the technique was originally motivated by curve evolution applications, we demonstrate its usefulness in capturing not only smooth evolving curves, but also ones with orientation discontinuities. In particular, the technique is shown to be far better than traditional methods when multiple or entire curves are present in a very small neighborhood.

The instabilities of the medial axis of a shape under deformations have long been recognized as a major obstacle to its use in recognition and other applications. These instabilities, or transitions, occur when the structure of the medial axis graph changes abruptly under deformations of shape. The recent classification of these transitions in 2D for the medial axis and for the shock graph, was a key factor both in the development of an object recognition system and an approach to perceptual organization. This paper classifies generic transitions of the 3D medial axis, by examining the order of contact of spheres with the surface, leading to an enumeration of possible transitions, which are then examined on a case by case basis. Some cases are ruled out as never occurring in any family of deformations, while others are shown to be non-generic in a one-parameter family of deformations. Finally, the remaining cases are shown to be viable by developing a specific example for each. We relate these transitions to a classification by Bogaevsky of singularities of the viscosity solutions of the Hamilton-Jacobi equation. We believe that the classification of these transitions is vital to the successful regularization of the medial axis and its use in real applications.

A key challenge underlying theories of vision is how the spatially restricted, retinotopically represented feature computations can be integrated to form abstract, coordinate-free object models. A resolution likely depends on the use of intermediate-level representations which can on the one hand be populated by local features and on the other hand be used as atomic units underlying the formation of, and interaction with, object hypotheses. The precise structure of this intermediate representation derives from the varied requirements of a range of visual tasks, which motivate a significant role for incorporating a geometry of visual form. The need to integrate input from features capturing surface properties such as texture, shading, motion, color, etc., as well as from features capturing surface discontinuities such as silhouettes, T-junctions, etc., implies a geometry which captures both regional and boundary aspects. Curves, as a geometric model of boundaries, have been extensively and explicitly used as an intermediate representation in computational, perceptual, and physiological studies. However, the medial axis which has been popular in computer vision as a geometric region-based model of the interior of closed boundaries, has not been explicitly used as an intermediate representation. We present a unified theory of perceptual grouping and object recognition where the intermediate representation is a visual fragment which itself is based on the medial axis. Through various sequences of transformations of the medial axis representation, visual fragments are grouped in various configurations to form object hypotheses, and are related to stored models. The mechanisms underlying both the computation and the transformation of the medial axis is a lateral wave propagation model. Recent psychophysical experiments depicting contrast sensitivity map peaks at the medial axes of stimuli, and experiments on perceptual filling-in, and brightness induction and modulation, are consistent with both the use of a medial axis representation and a propagation-based scheme. Also, recent neurophysiological recordings in V1 correlate with the medial axis hypothesis and a horizontal propagation scheme. This evidence supports a geometric computational paradigm for processing sensory data where both dynamic in-plane propagation and feedforward-feedback connections play an integral role.

This paper presents an approach for computing the symmetries (skeletons) of an edge map consisting of a collection of curve segments. This approach is a combination of analytic computations in the style of computational geometry and discrete propagations on a grid in the style of the numerical solutions of PDE's. Specifically, waves from each of the initial curve segments are initialized and propagated as a discrete wavefront along discrete directions. In addition, to avoid error built up due to the discrete nature of propagation, shockwaves are detected and explicitly propagated along a secondary dynamic grid. The propagation of shockwaves, integrated with the propagation of the wavefront along discrete directions, leads to an exact simulation of propagation by the Eikonal equation. The resulting symmetries are simply the collection of shockwaves formed in this process which can be manipulated locally, exactly, and efficiently under local changes in an edge map (gap completion, removal of spurious elements, etc.). The ability to express grouping operations in the language of symmetry maps makes it an appropriate intermediate representation between low-level edge maps and high level object hypotheses.

We confront the theoretical and practical difficulties of computing a representation for two-dimensional shape, based on shocks or singularities that arise as the shape's boundary is deformed. First, we develop subpixel local detectors for finding and classifying shocks. Second, to show that shock patterns are not arbitrary but obey the rules of a grammar, and in addition satisfy specific topological and geometric constraints. Shock hypotheses that violate the grammar or are topologically or geometrically invalid are pruned to enforce global consistency. Survivors are organized into a hierarchical graph of shock groups computed in the reaction-diffusion space, where diffusion plays a role of regularization to determine the significance of each shock group. The shock groups can be functionally related to the object's parts, protrusions and bends, and the representation is suited to recognition: several examples illustrate its stability with rotations, scale changes, occlusion and movement of parts, even at very low resolutions.

A general theory of shape is developed from basic principles. These principles are organized around two intuitions: first, if a boundary changes only slightly, then its shape should change only slightly. This leads to a proposal of an operational theory of shape based on incremental contour deformations. The second intuition is that not all contours are shapes, but rather only those that can enclose “physical” material. A novel theory of contour deformation is derived from these principles, based on abstract conservation principles and the Hamilton-Jacobi theory. The result is a characterization of the computational elements of shape: deformations, parts, bends, and seeds, which show where to place the components of a shape. The theory unifies many of the diverse aspects of shapes, and leads to a space of shapes (the reaction/diffusion space), which places shapes within a neighbourhood of “similar” ones. Such similarity relationships underlie descriptions suitable for recognition.

There are two classical approaches to approximating the shape of objects. The first is based on diffusion and often leads tothe (Gaussian) smoothing of contour information. The resulting scale-space may often be viewed as generated by parabolicoperators which progressively and globally smooth shapes. The second approach is based on morphological morphologyoperations which represent the interior of shapes as sets, e.g., a collection of disks. The resulting morphological spacecan be viewed as being defined via a hyperbolic operator whose weak or viscosity solutions progressively smooth shapesin a local manner. We are developing a general theory of shape which unifies these two different approaches in theentropy scale space. The theory is organized around two basic intuitions: first, if a boundary were changed only slightly,then, in general, its shape would change only slightly. This leads us to propose an operational theory of shape basedon incremental contour deformations. The second intuition is that not all contours are shapes, but rather only thosethat can enclose "physical" material. A novel theory of contour deformation is derived from these intuitions, based onabstract conservation principles and the Hamilton-Jacobi theory. The result is a characterization of the computationalelements of shape: protrusions, parts, bends, and seeds (which show where to place the components of a shape); andleads to a space of shapes (the reaction-diffusion space) which places shapes within a neighborhood of "similar" ones.Previously, these elements of shape have been used for description. We now show how they can be used to generateanother space for shapes, the entropy scale space, which is obtained from the reaction-diffusion space by running the"reaction" portion of the equations "backwards" in time. As a result distinct components of a shape can be removed byintroducing a minimal disturbance to the remainder of the shape. For example, imagine an image of don Quixote holdinga lance. Within the entropy scale space the lance can be removed without disturbing the shape of his body. In contrast,if Gaussian smoothing techniques were used, the extent of the lance would effect the amount of distortion introducedto the rest of his body. As such, the entropy scale space is a combination of smoothing due to shocks as "black holes"of information and the subsequent rarefaction wave reconstruction, and the anisotropic diffusion process spreading ofcontour information. Our technique is numerically stable, and several examples will be shown.