The paper presents a fuzzy chamfer distance and its probabilistic formulation for edge-based visual tracking. First, connections of the chamfer distance and the Hausdorff distance with fuzzy objective functions for clustering are shown using a reformulation theorem. A fuzzy chamfer distance (FCD) based on fuzzy objective functions and a probabilistic formulation of the fuzzy chamfer distance (PFCD) based on data association methods are then presented for tracking, which can all be regarded as reformulated fuzzy objective functions and minimized with iterative algorithms. Results on challenging sequences demonstrate the performance of the proposed tracking method.

We propose an image-based shape model for view-invariant human motion recognition. Image-based visual hull explicitly represents the 3D shape of an object, which is computed from a set of silhouettes. We then use the set of silhouettes to implicitly represent the visual hull. Due to the fact that a silhouette is the 2D projection of an object in the 3D world with respect to a certain camera, which is sensitive to the point of view, our multi-silhouette representation for the visual hull entails the correspondence between views. To guarantee the correspondence, we define a canonical multi-camera system and a canonical human body orientation in motions. We then "normalize" all the constructed visual hulls into the canonical multi-camera system, align them to follow the canonical orientation, and finally render them. The rendered views thereby satisfy the requirement of the correspondence. In our visual hull's representation, each silhouette is represented as a fixed number of sampled points on its closed contour, therefore, the 3D shape information is implicitly encoded into the concatenation of multiple 2D contours. Each motion class is then learned by a Hidden Markov Model (HMM) with mixture of Gaussians outputs. Experiments using our algorithm over some data sets give encouraging results.

The paper proposes an edge-based multi-object tracking framework which deals with tracking multiple objects with occlusions using a variational particle filter. Object is modelled by a mixture of a non-parametric contour model and a non-parametric edge model using kernel density estimation. Visual tracking with a mixture model is formulated as a Bayesian incomplete data problem, where measurements in an image are associated with a generative model which is a mixture of mixture models including object models and a clutter model and unobservable associations of measurements to densities in the generative model are regarded as missing data. A likelihood for tracking multiple objects jointly with an exclusion principle is presented, where it is assumed that one measurement can only be generated from one density and one density can generate multiple measurements and it significantly reduces the complexity of enumerating all feasible events. To address the curse of dimensionality in tracking multiple objects jointly, a variational particle filter (VPF) is proposed for multi-object tracking, where the proposal distribution is based on the approximated posterior from variational inference rather than using the prior as the proposal distribution in sampling importance resampling (SIR) particle filter. With the variational particle filter, the number of particles needed for multi-object tracking can be significantly reduced. Experimental results in challenging sequences demonstrate the robust performance of the proposed method.

Dynamic shape is a time series of the outlines of a moving object, which records the temporal variation of the shape of the object during its movement. We believe that the dynamic shape provides clues about the motion performed by the object. In this paper, we borrow tools from system identification to capture the "essence" of the dynamic shape, so that we convert the problems of modelling, learning, and recognizing object motions to the modelling, learning, and comparing of dynamical systems where each motion is represented. Concretely, we use Kenall's definition of shape to represent object contours extracted from each frame, and construct a tangent space with the full Procrustes mean shape as the pole to approximate a linear space for the data set; we then apply these linearized contour representations as training data to learn the dynamical systems, i.e. estimate system parameters; finally supervised pattern classification techniques based on various types of distance measure are adopted for recognition.

The paper presents a robust multi-feature head tracker using importance sampling particle filter with automatic head detection for video surveillance applications. Automatic head detection is based on moving region contour analysis using constraints of head shape and curvature scale space corner detector is introduced to segment contours. Due to the automatic head detection, we propose to exploit detected head shape cue to guide the sampling using importance sampling particle filter and individualized colour and edge features are fused for measurement. Experimental results demonstrate the robustness of the proposed head tracker.

This paper presents an automatic motion trajectory based surveillance video retrieval system with an efficient and flexible trajectory matching technique based on turning function. Raw trajectories of moving objects are extracted by automatic tracking and described by MPEG-7 motion trajectory descriptor with first order interpolating function. Efficient matching is performed for translation invariant and translation-rotation invariant video retrieval in spatial model and spatio-temporal model. In spatial model, turning function is introduced to represent the trail of motion. In spatio-temporal model, velocity function is used to represent the motion of object. Results of experiment on a prototype video database indicate a promising performance of the system. 1

We address the problem of two-dimensional (2-D) shape representation and matching in presence of self-intersection for large image databases. This may occur when part of an object is hidden behind another part and results in a darker section in the gray level image of the object. The boundary contour of the object must include the boundary of this part which is entirely inside the outline of the object. The Curvature Scale Space (CSS) image of a shape is a multiscale organization of its inflection points as it is smoothed. The CSS-based shape representation method has been selected for MPEG-7 standardization. We study the effects of contour self-intersection on the Curvature Scale Space image. When there is no self-intersection, the CSS image contains several arch shape contours, each related to a concavity or a convexity of the shape. Self intersections create contours with minima as well as maxima in the CSS image. An efficient shape representation method has been introduced in this paper which describes a shape using the maxima as well as the minima of its CSS contours. This is a natural generalization of the conventional method which only includes the maxima of the CSS image contours. The conventional matching algorithm has also been modified to accommodate the new information about the minima. The method has been successfully used in a real world application to find, for an unknown leaf, similar classes from a database of classified leaf images representing different varieties of chrysanthemum. For many classes of leaves, self-intersection is inevitable during the scanning of the image. Therefore the original contributions of this paper is the generalization of the Curvature Scale Space representation to the class of 2-D contours with self-intersection, and its application to the classification of Chrysanthemum leaves.

The paper presents a robust corner tracking method for unconstrained motions. The aim is to track corners in video sequences and identify best correspondences under any transformation, perspective distortion and intensity changes. To extract corners from each video frame, the multi-scale enhanced CSS (curvature scale space) corner detector is applied. In the matching stage, a new two-frame correspondence algorithm using multiple matches is employed. Experiments have been carried out on a wide range of real video sequences depicting translation, scaling, rotation and affine transformation, including non-smooth motions with different lighting and different camera motions. All the results were compared to the results of the popular Tommasini feature tracking algorithm (Tommasini, T. et al., IEEE Conf. on Computer Vision and Pattern Recognition, 1998). Experiments and comparison of the results confirm that the proposed corner tracker is more efficient, and more robust for unconstrained and non-smooth motions. These properties make it specially useful for video database retrieval and multimedia applications.

The paper addresses the problem of corner tracking for multimedia applications in the presence of occlusion. To extract corners from each frame of a video sequence, multi-scale enhanced CSS (curvature scale space) corner detection is applied. In the matching stage, two-frame correspondence using multiple match hypotheses is considered. Due to occlusion, not only the corners of the first frame, but also all the corners that appear for the first time in each frame are tracked through the input sequence. To remember the path of corners when tracking, each corner has a record. The record for each tracked corner includes the numbers of the start and end frames during the tracking of these corners in a sequence. By tracking new corners, the corners of hidden moving objects appear again and are tracked after disocclusion. We apply the proposed method to a number of real video sequences. All experiments confirm that the proposed corner tracker performs more robustly and it is practical even when occlusion happens.

Energy-minimising active contour models based on dynamic programming have been proposed by Amini et al. (1988) as a discrete multi-stage decision process. The behaviour of the active contour is generally controlled by its internal and external energies. Internal energy is composed of two parts; the first part, acts to shorten the active contour as it iterates towards the interest object, while the second part is the curvature of the active contour and forces smoothness of active contour during its movement towards interest object. In Amini et al., the three points i, i-1 and i-2 were used to estimate the curvature of the active contour at point i. Also the external energy of active contour at this point was calculated as the distance from its previous point to the nearest edge of underlying image. Then due to both of these problems, locking on to interest object does not occur very accurately especially at some points on the boundary of object where curvature changes very quickly. In this paper a reformulated internal energy is proposed to improve the computation of curvature at point i by making use of the three points i-1, i and i+1. Furthermore, external energy of active contour at any point is defined as its distance to nearest edge of underlying image Consequently our proposed active contour model can lock on to interest objects more accurately using the same snake parameters and initial position. Images with single and multiple objects are selected to evaluate the capability of our proposed method. The results show that locking on to interest objects occurs completely like a membrane or thin plate.

This paper describes a novel approach to multi-view 3D object retrieval. To represent each view of the object, its edge contours are extracted at different levels of scale. Each edge contour, in turn, is segmented by its curvature zero crossing points in a multi-scale fashion. This procedure is carried out using the curvature scale space technique which has been selected for MPEG-7 standardisation. A number of features are then computed for each segment of each edge contour. The image is finally represented by the locations of its segments and the values of their associated features. In response to an input query, geometric hashing is first used to find the best locally matched candidates for the verification stage where the goal is to measure the distance between the input query edge contours and the corresponding model contours after applying a proper transformation. The measurement is then optimised and used as the match value. The method has been successfully tested on a collection of 3D objects consisting of 15 aircraft of different shapes.

The feasibility of representing a three-dimensional (3-D) object with a small number of standard views is studied. The object boundary of each view is considered as a two dimensional (2-D) shape and is represented by the locations of the maxima of its curvature scale space (CSS) image contours. The idea is to identify an unknown object from an image taken from a random view by using the stored descriptions of the standard views. The CSS image has been selected for MPEG-7 standardization. The maxima of CSS image have already been used to represent 2-D shapes in different applications under similarity transforms. Since the new application involves affine transforms, we first examine the effects of general affine transforms on the representation and show that the locations of the maxima of the CSS image do not move dramatically even under large affine transformations. Our system for shape-based retrieval from large image databases is then applied to multiview 3-D object representation and recognition. Our collection of 3-D objects consists of 18 aircraft of different shapes. Three silhouette contours corresponding to random views are separately used as input for each object. Results indicate that robust and efficient 3-D free-form object recognition through multiview representation can be achieved using the CSS representation.

A shape-based method for multi-view 3-D object representation and recogni-tion is introduced and explored in this paper. 3-D objects are recognised by a small number of images taken from different views. The paper addresses the issue of automatic selection of the best and the optimum number of views for each object. The object boundary of each view is considered as a 2-D shape and is represented effectively by less than ten pairs of integer values. These values include the locations of the maxima of its Curvature Scale Space (CSS) image contours. The CSS shape descriptor is expected to be selected for MPEG-7 standardisation. An unknown object is then recognised by a single image taken from an arbitrary viewpoint. The method has been tested on a collection of 3-D objects consisting of 15 aircrafts of different shapes. Each object has been modelled using an op-timised number of silhouette contours obtained from different view points. This number varies from 5 to 25 depending on the complexity of the object and the measure of expected accuracy. A comprehensive analysis of the per-formance of the system has been given in this paper as the number of views varies. Around ten silhouette contours corresponding to random views are sepa-rately used as input for each object. Results indicated that robust and efficient 3-D free-form object recognition through multi-view representation can be achieved using the CSS representation even for large database retrieval ap-plications. 1

The recognition of free-form 3-D objects using multi-scale features recovered from 3-D models, and based on the geometric hashing al-gorithm and global verication is presented. The feature points on the object are detected by smoothing its surface after construction of semigeodesic coordinates at each point (mesh vertex). This technique is the generalisation of the CSS method which is a powerful shape de-scriptor expected to be in the MPEG-7 standard. Smoothing is used to remove noise and to select multi-scale feature points to add to the eÆciency and robustness of the system. The local maxima of Gaussian and mean curvatures are selected as feature points. Furthermore the torsion maxima of the zero-crossing contours of Gaussian and mean curvatures are also selected as feature points. Recognition results are demonstrated for rotated and scaled as well as partially occluded ob-jects. In order to conrm the match, 3D translation, rotation and scaling parameters are used for verication and results indicate that our technique is invariant to those transformations. 1

The curvature scale space (CSS) technique has been used in conjunction with Hermite curves for compression and robust reconstruction of 2-D contours at multiple scales. CSS is a powerful contour shape descriptor which is expected to be an MPEG-7 standard. A parametric representation of the input contour is convolved with Gaussian functions in order to obtain multi-scale descriptions of the contour. Curvature can be computed directly at each point of the smoothed contours. As a result, a set of curvature zero-crossing points can be extracted from each smoothed contour. Hermite curves were used since each Hermite curve is defined by two endpoints and the tangent vectors at those points. No points external to the input contour are required for Hermite curves. Hermite endpoints are defined as consecutive curvature zero-crossing points extracted at multiple scales using the CSS method. Hermite tangent vectors can also be determined using the CSS technique. The only data stored are the endpoints and the tangent vectors needed by the Hermite curves in order to arrive at an approximate reconstruction of the original contour.

We introduce a new method for automatic selection of optimal views in a shape-based method of multi-view 3D object representation and recognition. A 3D object is recognised by an optimum number of images taken from different views. The object boundary of each view is considered as a 2D shape and is represented by the locations of the maxima of its curvature scale space (CSS) image contours. An unknown object is then recognised by a single image taken from an arbitrary viewpoint. The method has been tested on a collection of 3D objects consisting of 15 aircraft of different shapes. Each object has been modelled using an optimised number of silhouette contours obtained from different view points. The results obtained show that robust and efficient 3D free form object recognition through multi-view representation can be achieved using the CSS representation even for large database retrieval applications.

13 - 16 September 1999, A novel technique for multi-scale curvature computation on a smoothed 3-D surface is presented. In our technique, semigeodesic coordinates are constructed at each vertex of themesh which becomes the local origin. A geodesic from the origin is first constructed in an arbitrary direction such as the direction of one of the incident edges. The surface Gaussian and mean curvatures are then estimated. Next, the curvature zero-crossing contours were recovered. Curvature features such as zero-crossing contours and maxima recovered at multiple scales are useful for surface matching and object recognition algorithms, as well as registration of 3-D medical data. The performance of our technique when selecting different directions as an arbitrary direction for the geodesic at each vertex is also evaluated. Our experiments demonstrate that estimation of smoothed surface curvatures are very accurate and not affected by the arbitrary direction of the first geodesic line when constructing semigeodesic coordinates. Our technique is independent of the underlying triangulation and is also more efficient than volumetric diffusion techniques since 2-D rather than 3-D convolutions are employed.

The maxima of curvature scale space (CSS) image have already been used to represent 2-D shapes in different applications under similarity transforms. We also consider affine transforms and examine their effect on the representation. Since the CSS image employs the arc length parametrisation which is not affine invariant, we expect some deviation in the maxima of the CSS image under affine transforms. We first examine the arc length deviation and then show that the locations of the maxima of the CSS image do not change dramatically even under large affine transformations. Applying affine transforms to every object boundary of our database of 1100 images, we construct a large database of 5500 boundary contours. The CSS representation is then used to find similar shapes from this prototype database. The results provide substantial evidence of stability of the the CSS image and its contour maxima under affine transformation. Our system for shape-based retrieval from large image databases is then applied to multi-view 3-D object representation and recognition. Our collection of 3-D objects consists of 10 aircraft of different shapes. Each object is modelled using a small number of silhouette contours obtained from different view points (four contours in one experiment and nine contours in another). Three silhouette contours corresponding to random views are separately used as input for each object. Results indicate that robust and efficient 3-D free-form object recognition through multi-view representation can be achieved using the CSS representation.

A novel technique for multi-scale smoothing of a free-form 3-D surface is presented. Complete triangulated models of 3-D objects are constructed (through fusion of range images) and are then described at multiple scales. This is achieved by convolving local parametrizations of the surface with 2-D Gaussian filters iteratively. Our method for local parametrization makes use of semigeodesic or goedesic polar coordinates as a natural and efficient way of sampling the local surface shape. The smoothing eliminates surface noise and small surface detail gradually. Our technique for 3-D multi-scale surface smoothing is independent of the underlying triangulation. It is also argued that the proposed technique is preferrable to volumetric smoothing or level set methods since it is applicable to incomplete surface data which occurs during occlusion. 1

This paper describes a new method for image corner detection based on the curvature scale space (CSS) representation. The first step is to extract edges from the original image using a Canny detector. The corner points of an image are defined as points where image edges have their maxima of absolute curvature. The corner points are detected at a high scale of the CSS image and the locations are tracked through multiple lower scales to improve localization. The CSS corner detector is very robust to noise and performed better than three other detectors it was compared to.

This paper describes a novel method for image corner detection based on the curvature scale-space (CSS) representation. The first step is to extract edges from the original image using a Canny detector (1986). The corner points of an image are defined as points where image edges have their maxima of absolute curvature. The corner points are detected at a high scale of the CSS and tracked through multiple lower scales to improve localization. This method is very robust to noise, and we believe that it performs better than the existing corner detectors An improvement to Canny edge detector's response to 45/spl deg/ and 135/spl deg/ edges is also proposed. Furthermore, the CSS detector can provide additional point features (curvature zero-crossings of image edge contours) in addition to the traditional corners.

The user of an image database often wishes to retrieve all images similar to the one (s)he already has. Using some features like texture, color and shape, we can associate a feature vector to every image in the database. A fast indexing method then can be used to retrieve similar images based on their associated vectors. We use the maxima of curvature zero crossing contours of Curvature Scale Space (CSS) image as a feature vector to represent the shapes of object boundary contours. The matching algorithm which compares two sets of maxima and assigns a matching value as a measure of similarity is presented in this paper. The method is robust with respect to noise, scale and orientation changes of objects. It is also capable to retrieve objects which are similar to the mirror-image of the input boundary. We introduce the aspect ratio of the CSS image as a new parameter which can be used for indexing in conjunction with other parameters like eccentricity and circularity. The method has been tested and evaluated on a prototype database of 450 images of marine animals with a vast variety of shapes with very good results. Since shape similarity is a subjective issue, in order to evaluate the methilarity retrieval based on shape on a randomly selected small database. We then compared the results of this experiment to the outputs of our system to the same queries and on the same database. The comparison indicated a promising performance of the system.

A complete, fast and practical isolated object recognition system has been developed which is very robust with respect to scale, position and orientation changes of the objects as well as noise and local deformations of shape (due to perspective projection, segmentation errors and non-rigid material used in some objects). The system has been tested on a wide variety of three-dimensional objects with different shapes and material and surface properties. A light-box setup is used to obtain silhouette images which are segmented to obtain the physical boundaries of the objects which are classified as either convex or concave. Convex curves are recognized using their four high-scale curvature extrema points. Curvature scale space (CSS) representations are computed for concave curves. The CSS representation is a multi-scale organization of the natural, invariant features of a curve (curvature zero-crossings or extrema) and useful for very reliable recognition of the correct model since it places no constraints on the shape of objects. A three-stage, coarse-to-fine matching algorithm prunes the search space in stage one by applying the CSS aspect ratio test. The maxima of contours in CSS representations of the surviving models are used for fast CSS matching in stage two. Finally, stage three verifies the best match and resolves any ambiguities by determining the distance between the image and model curves. Transformation parameter optimization is then used to find the best fit of the input object to the correct model.>

Space curves are efficient, highly descriptive features of 3-D objects. A new multiscale shape representation (the resampled torsion scale space or TSS image) for space curves suitable for recognition of a noisy curve at any scale or orientation is introduced. A two-phase matching algorithm consisting of TSS matching (using extrema of TSS zero-crossing contours) followed by transformation parameter optimization demonstrates the usefulness of the representation in recognition tasks.>

A complete and practical isolated-object recognition system has been developed which is very robust with respect to scale, position and orientation changes of the objects as well as noise and local deformations of shape due to perspective projection, segmentation errors and non-rigid material used in some objects. The system has been tested on a wide variety of 3-D objects with different shapes and surface properties. A light-box setup is used to obtain silhouette images which are segmented to obtain the physical boundaries of the objects which are classified as either convex or concave. Convex curves are recognized using their four high-scale curvature extrema points. Curvature scale space (CSS) representations are computed for concave curves. The CSS representation is a multi-scale organization of the natural invariant features of a curve. A three-stage coarse-to-fine matching algorithm quickly detects the correct object in each case.>

A shape representation technique suitable for tasks that call for recognition of a noisy curve of arbitrary shape at an arbitrary scale or orientation is presented. The method rests on the describing a curve at varying levels of detail using features that are invariant with respect to transformations that do not change the shape of the curve. Three different ways of computing the representation are described. They result in three different representations: the curvature scale space image, the renormalized curvature scale space image, and the resampled curvature scale space image. The process of describing a curve at increasing levels of abstraction is referred to as the evolution or arc length evolution of that curve. Several evolution and arc length evolution properties of planar curves are discussed.>

It has been shown by A.L. Yuille and T. Poggio (1983) that the scale-space image of a signal determines that signal uniquely up to constant scaling. Here, generalization of the proof given by Yuille and Poggio is presented. It is shown that the curvature scale-space image of a planar curvature determines the curvature uniquely, up to constant scaling and a rigid motion. The results show that a 1-D signal can be reconstructed using only one point from its scale-space image. This is an improvement of the result obtained by Yuille and Poggio.>

The curvature scale-space image of a planar curve is computed by convolving a path-based parametric representation of the curve with a Gaussian function of variance sigma /sup 2/, extracting the zeros of curvature of the convolved curves and combining them in a scale space representation of the curve. For any given curve Gamma , the process of generating the ordered sequence of curves ( Gamma /sub sigma / mod sigma >or=0) is the evolution of Gamma . It is shown that the normalized arc length parameter of a curve is, in general, not the normalized arch length parameter of a convolved version of that curve. A novel method of computing the curvature scale space image reparametrizes each convolved curve by its normalized arc length parameter. Zeros of curvature are then expressed in that new parametrization. The result is the renormalized curvature scale-space image and is more suitable for matching curves similar in shape. Scaling properties of planar curves and the curvature scale space image are also investigated. It is shown that no new curvature zero-crossings are created at the higher scales of the curvature scale space image of a planar curve in C/sub 1/ if the curve remains in C/sub 1/ during evolution. Several results are presented on the preservation of various properties of planar curves under the evolution process.>

The authors address the problem of representing the shape of three-dimensional or space curves. This problem is important since space curves can be used to model the shape of many three dimensional objects effectively and economically. A number of shape representation methods that operate on two-dimensional objects and can be extended to apply to space curves are reviewed briefly and their shortcomings discussed. Next, the concepts of curvature and torsion of a space curve are explained. Arc-length parametrization followed by Gaussian convolution is used to compute curvature and torsion on a space curve at varying levels of detail. Information of both the curvature and torsion of the curve over a continuum of scales are combined to produce the curvature and torsion scale-space images of the curve. These images are essentially invariant under rotation, uniform scaling, and translation of the curve and are used as a representation for it. The application of this technique to a common three-dimensional object is demonstrated. The proposed representation is then evaluated according to several criteria that any shape representation method should ideally satisfy.>

An evolved version I?, of a space curve l7 is obtained by convolving a parametric representation of r with a Gaussian function of variance 0'. The process of generating the ordered sequence of curves {I',\o>O} is referred to as the evolution of r. Evolved space curves arise when computing the Torsion and Curvature Scale Space representation of a space curve. A number of evolution properties of space curves are investigated in this paper. It is shown that the evolution of space curves is invariant under rotation, uniform scaling and translation of those curves. This is an essential propert for any reliable shape representation. It is also shown tiat properties such as connectedness and closedness of a space curve are preserved during evolution of the curve and that the center of mass of a s ace curve remains the same as the curve evolves. AnotRer result is that a space curve remains inside its convex hull during evolution. The two main theorems of the paper examine a s ace curve during its evolution just before and just after tie formation of a cusp point. It is shown that strong constraints on the shape of the curve in the neighborhood of the cusp point exist just before and just after the formation of that point. Final1 it is argued that each one of the results obtained in tiis paper is important and useful for practical applications. I. Introduction A multi-scale representation for one-dimensional functions and signals was first proposed by Stansfield [1980] and later developed by Witkin [1983]. The signal fp) is convolved with a Gaussian function as its variance 0 varies from a small to a large value. The aero-crossings of the second derivative of each convolved signal are extracted and marked in the z-o space. The result is the Scale Space Image of the signal. Mokhtarian and Mackworth [1986] generalized that concept to planar curves. A planar curve r is parametrized by arc length U and represented using its coordinate functions. An evolved version of I' is computed by convolving each of its coordinate functions with a Gaussian function of variance 0' and denoted by I',,. The process of generating the ordered sequence of curven {r,lo>O} is referred to as the evolution of r. The curvature of each Fa can be expressed in terms of the first and second derivatives of convolved versions of functions z( U) and y(u). It is therefore possible to extract the curvature zero-crossings of each I', as 5 varies from a small to a large value and mark them in the U-o space. The result is referred to as the Curvature Scale Space Image of the curve. Mokhtarian [1988b] generalized the above concept further to space curves. The parametrization of a space curve can be expressed as: I' = (z(u),y(u),z(u)). Curvature and torsion of an evolved space curve can be expressed in terms of the first three derivatives of convolved versions of functions z(u), y(~) and n(u). A scale space representation for space curves consists of the Torsion and Curvature Scale Space Images which contain the torsion zero-crossings and the curvature level-crossings maps of the curve respectively. Scale space representations for planar and space curves satisfy several useful criteria such as Eficiency, Invariance, Sensitivity, Uniqueness, Detail and Robustness [Mokhtarian 1988bI. These properties make them specially suitable for recognition of arbitrarily shaped objects. Mackworth and Mokhtarian [1988] have investigated a number of evolution properties of planar curves. This paper generaliies some of those properties and investigates other evolution properties of space curves. Lemma 1 shows that evolution of a space curve is invariant under rotation, uniform scaling and translation of the curve. Lemmas 2 and 3 show that connectedness and closedness of a space curve are preserved during its evolution. Lemma 4 shows that the center of mass of a closed space curve does not move as the curve evolves. Lemma 5 shows that a space curve remains inside its convex hull during evolution.

The problem of finding a description, at varying levels of detail, for planar curves and matching two such descriptions is posed and solved in this paper. A number of necessary criteria are imposed on any candidate solution method. Path-based Gaussian smoothing techniques are applied to the curve to find zeros of curvature at varying levels of detail. The result is the ``generalized scale space'' image of a planar curve which is invariant under rotation, uniform scaling and translation of the curve. These properties make the scale space image suitable for matching. The matching algorithm is a modification of the uniform cost algorithm and finds the lowest cost match of contours in the scale space images. It is argued that this is preferable to matching in a so-called stable scale of the curve because no such scale may exist for a given curve. This technique is applied to register a Landsat satellite image of the Strait of Georgia, B.C. (manually corrected for skew) to a map containing the shorelines of an overlapping area.