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The tasks of visual object recognition and classification are natural and effortless for biological visual systems, but exceedingly difficult to replicate in computer vision systems. This difficulty arises from the large variability in images of different objects within a class, and variability in viewing conditions. In this paper we describe a fragment-based method for object classification. In this approach objects within a class are represented in terms of common image fragments, that are used as building blocks for representing a large variety of different objects that belong to a common class, such as a face or a car. Optimal fragments are selected from a training set of images based on a criterion of maximizing the mutual information of the fragments and the class they represent. For the purpose of classification the fragments are also organized into types, where each type is a collection of alternative fragments, such as different hairline or eye regions for face classification. During classification, the algorithm detects fragments of the different types, and then combines the evidence for the detected fragments to reach a final decision. The algorithm verifies the proper arrangement of the fragments and the consistency of the viewing conditions primarily by the conjunction of overlapping fragments. The method is different from previous part-based methods in using class-specific overlapping object fragments of varying complexity, and in verifying the consistent arrangement of the fragments primarily by the conjunction of overlapping detected fragments. Experimental results on the detection of face and car views show that the fragment-based approach can generalize well to completely novel image views within a class while maintaining low mis-classification error rates. We briefly discuss relationships between the proposed method and properties of parts of the primate visual system involved in object perception.