Visual object recognition requires the matching of an image with a set of models stored in memory. In this paper, we propose an approach to recognition in which a 3-D object is represented by the linear combination of 2-D images of the object. IfJLk{M1,." .Mk} is the set of pictures representing a given object and P is the 2-D image of an object to be recognized, then P is considered to be an instance of M if P= C~=,aiMi for some constants (pi. We show that this approach handles correctly rigid 3-D transformations of objects with sharp as well as smooth boundaries and can also handle nonrigid transformations. The paper is divided into two parts. In the first part, we show that the variety of views depicting the same object under different transformations can often be expressed as the linear combinations of a small number of views. In the second part, we suggest how this linear combination property may be used in the recognition process. Index Terms-Alignment, linear combinations, object recognition, recognition, 3-D object recognition, visual recognition. A. Recognition by Alignment V ISUAL OBJECT recognition requires the matching of an image with a set of models stored in memory. Let M = {Ml, . , M,} be the set of stored models and P be the image to be recognized. In general, the viewed object, depicted by P, may differ from all the previously seen images of the same object. It may be, for instance, the image of a three-dimensional object seen from a novel viewing position. To compensate for these variations, we may allow the models (or the viewed object) to undergo certain compensating transformations during the matching stage. If 7 is the set of allowable transformations, the matching stage requires the selection of a model A4, E M and a transformation T E 7, such that the viewed object P and the transformed model TM, will be as close as possible. The general scheme is called the alignment approach since an alignment transformation is applied to the model (or to the viewed object) prior to, or during, the matching stage. Such an approach is used in [S] In this paper, we suggest a different approach, in which each model is represented by the linear combination of 2-D images of the object. The new approach has several advantages. First, it handles all the rigid 3-D transformations, but it is not restricted to such transformations. Second, there is no need in this scheme to explicitly recover and represent the 3-D structure of objects. Third, the computations involved are often simpler than in previous schemes. The paper is divided into two parts. In the first (Section I), we show that the variety of views depicting the same object under different transformations can often be expressed as the linear combinations of a small number of views. In the second part (Section II), we suggest how this linear combination property may be used in the recognition process. The modeling of objects using linear combinations of images is based on the following observation. For many continuous transformations of interest in recognition, such as 3-D rotation, translation, and scaling, all the possible views of the transforming object can be expressed simply as the linear combination of other views of the same object. The coefficients of these linear combinations often follow in addition to certain functional restrictions. In the next two sections, we show that the set of possible images of an object undergoing rigid 3-D transformations and scaling is embedded in a linear space and spanned by a small number of 2-D images. The images we will consider are 2-D edge maps produced in the image by the (orthographic) projection of the bounding contours and other visible contours on 3-D objects. We will make use of the following definitions. Given an object and a viewing direction, the rim is the set of all the points on the object's surface whose normal is perpendicular to the viewing direction [13]. This set is also called the contour generator [17]. A silhouette is an image generated by the orthographic projection of the rim. In the analysis below, we assume that every point along the silhouette is generated by a single rim 0162~8828/91$01.00 0 1991 IEEE
It makes use of a linear combination of two prototypes in a feature space, whereas in REF a 3-D object is represented by a linear combination of 2-D boundary maps of the object and the knowledge of imaging parameters is not required.
Recognition by Linear Combinations of Models
Computer Science