The importance of face-image retrieval cannot be overemphasized. This paper presents a method for retrieving face images from a database. The method is based on singular value decomposition (SVD) and potential-field representation.

The authors first show that the dominant singular values of a face-image matrix are rotation-shift-scale invariant (theorems 1, 2, and 3). They claim that this is the first paper that completely proves the invariant feature (page 10). Actually, it is a straightforward result from the definition of SVD. For example, let A = U*S*V’ be the SVD of an image matrix A and G a rotation, then G*A = (G*U)*S*V’ is the SVD of the rotated image G*A since rotation G is orthogonal. Thus, the singular values remain unchanged.

Similarly, the singular values of a scaled image a*A are the singular values of A scaled by a, since a*A = U*(a*S)*V’. For the (circular) shift, if A = (a(1) a(2) ... a(n)) is a column partition of an image matrix A, then the image (circularly) shifted by m columns defined in the paper is (a(m+1) ... a(n) a(1) ... a(m)) (see right column of page 11). Actually, it should be (a(n-m+1) ... a(n) a(1) ... a(n-m)) and it is a permutation, that is, the last m columns are permuted to the front. Then, since a permutation is orthogonal, the singular values remain unchanged after the permutation. In general, pre-multiplying or post-multiplying an orthogonal matrix alters only the singular vectors, not singular values.

Then the authors adopt potential-field representation, which smooths out the edge information in an image so that it can be applied to grayscale images and is less sensitive to illumination.

By integrating the SVD and potential-field representation, the authors present an effective and robust method. This integration is the novelty of the paper. The authors tested their method on two benchmark face-image databases, GTAV and BioID, and reported that the method had higher success rates than five other state-of-the-art methods.

The accuracy of the paper could be improved. For example, it is stated that the singular values are “the square root of the eigenvalues” of the same matrix (page 10), which is incorrect. Also on page 10, the authors state that “k is the number of singular values or eigenvectors to be retained,” mixing singular values with eigenvectors. Nevertheless, eigenvalues and eigenvectors are not involved in their method.