Objective Functions for Feature Discrimination

We propose and evaluate a class of objective funct ions tha t rank hypotheses for feature labels. Our approach takes into account the representation cost and qual i ty of the shapes themselves, and balances the geometric requirements against the photometr ic evidence Th is balance is essential for any system using underconstrained or generic feature models. We introduce examples of specific models al lowing the actual computa t ion of the terms in the object ive func t ion , and show how this framework leads natura l ly to control parameters that have a clear semantic meaning. We i l lustrate the properties of our object ive functions on synthetic and real images. 1 I n t r o d u c t i o n A l l approaches to the problem of extract ing features f rom images can in pr incip le be phrased in terms of decision theory; however, the concepts of decision theory are very hard to put in to practice because of the di f f i cul ty of evaluat ing the required probabi l i ty measures. Therefore, most pract ical approaches to model-based vision for both specific models, e.g., [B inford, 1982, Bolles and Horaud, 1986, Brooks, 1981, Shneier et a/., 1986], and generic models, e.g., [Fischler et r/./., 1981, Oh ta et a/., 1979, McKeown and Denlinger, 1984, Huertas and Nevatia, 1988], rely on heuristic measures to select among compet ing scene parses. These methods, al though they may be effective in the context for which they were designed, are extremely hard to extend and require the use of many parameters whose significance is not clearly understood. On the other hand, approaches such as those of Feldman and Yakimovsky [1974], Georgeff and Wallace [1984], and Rissanen [1983, 1987] provide a sound theoret ical basis for the decision problem but offer few pract ica l computat ional methods for dealing w i t h complex scenes in real images. In this paper, we focus on an objective funct ion approach to the task of rank ing scene-labeling hypotheses. *This research was supported in part by the Defense Advanced Research Projects Agency under Contract Nos. MDA903-86-C-0084 and DACA76-85-C-0004. For brevity, we omi t discussion of the related problem of hypothesis-generation, and refer the reader to [Fua and Hanson, 1989]. We define a class of objective funct ions based upon theoretical arguments simi lar to those of Georgeff, Wallace and Rissanen, and show tha t the required probabi l i ty estimates can actual ly be computed in the context of a few natura l assumptions. Our formulat ion has many desirable features, but is not by itself a complete solut ion to the feature extract ion problem. To be effective it must be coupled w i th a robust hypothesis generation mechanism and an efficient opt imizat ion procedure. Furthermore, one would like to have models for geometric qual i ty analysis much more complex than those presented here. It should come as no surprise that discovering good models and hypothesisgeneration strategies are the most di f f icul t tasks in the development of a system a t tempt ing to perform shape perception. The strength of our approach is that it provides a unified framework tha t clearly exposes the cr i t i cal components and characteristics of model-based vision systems. 2 D e r i v a t i o n o f t h e O b j e c t i v e F u n c t i o n The goal of feature extract ion is to parse a scene in terms of objects conforming to part icular models. To discr iminate among compet ing parses, an objective funct ion must be able to measure the goodness of f it to feature models that include such characteristics as area photometry, edge photometry, shape, and semantic relat ionships. In this section, we define a basic class of models, discuss the parameters we expect to control our objective functions, derive the theoretical forms of the object ive functions themselves, and provide an interpreta t ion of the result ing funct ions in terms of in format ion theory. 2 .1 O b j e c t M o d e l i n g For the purposes of this work, we define a model to be a geometric description of an object in the wor ld characterized by its geometric constraints and its photometric signature; we define the evidence for such objects in digi ta l images to be a collection of delineated areas corresponding to major object parts, together w i t h associated quanti t ies direct ly derivable f r om the pixel values in such areas. 1596 Vision and Robotics Here F is what we call the encoding-effectiveness of the set of models The first term in F is the number of bits needed to describe the evidence in the absence of the model, while the second term gives the number of bits needed to describe the evidence in terms of the model. The term effectiveness is thus motivated by the fact that F represents the number of bits saved by representing the evidence using the model, and the fact that F increases as the f i t improves. G is the number of bits needed to encode the evidencefree model representation informat ion, and quantifies the elegance of the chosen set of model instances as well as their dependencies.