Template matching by normalized correlations is a common technique for determine the existence and compute the location of a shape within an image. In many cases the run time of computer vision applications is dominated by repeated computation of template matching, applied to locate multiple templates in varying scale and orientation. A straightforward implementation of template matching for an image size n and a template size k requires order of kn operations. There are fast algorithms that require order of n log n operations. We describe a new approximation scheme that requires order n operations. It is based on the idea of "Integral-Images", recently introduced by Viola and Jones.

Recently proposed fast template matching techniques employ rejection schemes derived from lower bounds on the match measure. This paper generalizes that idea and shows that in addition to lower bounds, upper bounds on the match measure can be used to accelerate the search. An algorithm is proposed that utilizes both lower and upper bounds to detect the k best matches in an image. The performance of this dual-bound algorithm is guaranteed; it always detects the k best matches. Theoretical analysis and experimental results show that its runtime compares favorably with previously proposed real-time exact template-matching schemes.

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Approximating a matrix by a small subset of its columns is a known problem in numerical linear algebra. Algorithms that address this problem have been used in areas which include, among others, sparse approximation, unsupervised feature selection, data mining, and knowledge representation. Such algorithms were investigated since the 1960's, with recent results that use randomization. The problem is believed to be NP-Hard, and to the best of our knowledge there are no previously published algorithms aimed at computing optimal solutions. We show how to model the problem as a graph search, and propose a heuristic based on eigenvalues of related matrices. Applying the A* search strategy with this heuristic is guaranteed to find the optimal solution. Experimental results on common datasets show that the proposed algorithm can effectively select columns from moderate size matrices, typically improving by orders of magnitude the run time of exhaustive search. We also show how to combine the proposed algorithm with other non-optimal (but much faster) algorithms in a ``two stage'' framework, which is guaranteed to improve the accuracy of the other algorithms.

The following are two classical approaches to dimensionality reduction: 1. Approximating the data with a small number of features that exist in the data (feature selection). 2. Approximating the data with a small number of arbitrary features (feature extraction). We study a generalization that approximates the data with both selected and extracted features. We show that an optimal solution to this hybrid problem involves a combinatorial search, and cannot be trivially obtained even if one can solve optimally the separate problems of selection and extraction. Our approach that gives optimal and approximate solutions uses a “best first” heuristic search. The algorithm comes with both an a priori and an a posteriori optimality guarantee similar to those that can be obtained for the classical weighted A* algorithm. Experimental results show the effectiveness of the proposed approach.