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### Depth estimation via non-linear minimization

Although the above-mentioned depth recovery algorithm is non-iterative, the final accuracy of reconstruction highly depends on the accurate estimation of the fundamental matrices. Moreover, it does not embody one of the nice feature of the affine factorization method, that is, use of all of the images at the same time in a uniform manner. This problem can be resolved by estimating depths in an iterative manner[1,5,15].

Let be SVD of the measurement matrix with singular values . If there is no noise in , is rank 4 and hence . With the presence of noise, is not exactly rank 4. If the noise is small, however, is still nearly rank 4 and through are almost zero. So, we define a measure indicating the rank 4 proximity of as

and estimate depths {} which minimize this measure as a criterion.

Let and be n-th column of and respectively. Differentiating both sides of and using relations , and , we obtain

with and , where each is a 3-vector corresponding to i-th frame. Using this equation, we can compute first derivatives of J with respect to . Therefore J can be minimized by standard optimization technique such as a conjugate gradient method. Computing second derivatives is also possible which allows to adopt Newton-like methods. We initialize the minimization process by taking as unity for all i and k which means we start with an affine projection. F+P-1 depth parameters are fixed throughout the iteration process as stated in (12).

Bob Fisher
Wed Apr 21 20:23:11 BST 1999