Shift invariance and the neocognitron

-We investigate the ability of the neocognitron to perform shift-invariant pattern recognition. Both an intuitive analysis and a more formal investigation show that the performance of the neocognitron is not intrinsically shift invariant, and that certain model parameters must be chosen appropriately to obtain approximate shift invariance. It is shown how these parameters should be chosen to reach a compromise between invariance and classification sensitivity. Keywords--Shift invariance, Neural classifiers, Pattern recognition, Neocognitron. I. I N T R O D U C T I O N One of the main problems of visual pattern recognition is that the objects to be recognized are generally subjected to various forms of transformation. Thus, a successful system needs to be able to recognize objects despite such transformations. Numerous methods to obtain scale, shift, and rotation invariance have been investigated for this purpose. Conventional methods have used two approaches. One set of methods uses feature spaces which are automatically invariant to some transformations (e.g., moments (Hu, 1962), polar-coordinate Fourier transform (Casasent & Psaltis, 1976)). Another approach (Chin & Dyer, 1986) uses object models and tries to match the observed and stored models by determination of the transformation parameters. Unfortunately, these techniques have only been successful in a limited range of applications; they have not come close to duplicating the human ability to perform transformation-invariant pattern recognition. Thus, invariant pattern recognition has been an ideal target for neural nets, since neural nets will hopefully be able to perform functions similar to those which underlie biological pattern recognition. The neocognitron (Fukushima, 1980) is a neuralnetwork model for visual pattern recognition that has Acknowledgment--Funding for this research was provided by a contract from the Defense Advanced Research Project Agency monitored by the U.S. Army Missile Command (Contract DAAH01-89-C-04180). Requests for reprints should be sent to Dr. D. Casasent, Center for Excellence in Optical Data Processing, Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213. captured considerable attention (Fukushima & Miyake, 1982; Miyake & Fukushima, 1984; Fukushima, 1987; Johnson, Daniell, & Burman, 1988; Menon & Heinemann, 1988). It is loosely based on known properties of the mammalian visual system, performs unsupervised learning, and is claimed to be shiftinvariant. (A supervised version of the neocognitron also exists; however, our interest will be limited to the more popular self-organizing architecture.) Good performance on a task involving the recognition of one of the ten roman numerals with small shifts within a 16 x 16 image has been demonstrated (Fukushima, 1980). However, in a recent study, Menon and Heinemann (1988) found that the neocognitron did not perform satisfactorily when it had to discriminate between three somewhat larger objects with larger shifts in a 128 x 128 image. It was found that shift invariance could only be obtained by creating a model which simply responds to the total energy in the image. This is not acceptable for most applications of pattern recognition, and can certainly be implemented more straightforwardly than with a neocognitron. We show that the reason for the problems encountered by Menon and Heinemann is the lack of intrinsic shift invariance of the neocognitron. In section II a 12 x 12 model-neocognitron is used to show intuitively why the neocognitron fails to be an intrinsically shift-invariant pattern recognizer. The purpose of section II is to obtain an intuitive understanding of why the neocognitron might fail. Therefore, the model parameters used in section II are not necessarily realistic. The intuition gained in studying this artificial model is used in section III, where we show that these problems persist under more general conditions. How-