# A Maximum Likelihood Stereo Algorithm

Published

Journal Article

A stereo algorithm is presented that optimizes a maximum likelihood cost function. The maximum likelihood cost function assumes that corresponding features in the left and right images are normally distributed about a common true value and consists of a weighted squared error term if two features are matched or a (fixed) cost if a feature is determined to be occluded. The stereo algorithm finds the set of correspondences that maximize the cost function subject to ordering and uniqueness constraints. The stereo algorithm is independent of the matching primitives. However, for the experiments described in this paper, matching is performed on the individual pixel intensities. Contrary to popular belief, the pixel-based stereo appears to be robust for a variety of images. It also has the advantages of (i) providing a dense disparity map, (ii) requiring no feature extraction, and (iii) avoiding the adaptive windowing problem of area-based correlation methods. Because feature extraction and windowing are unnecessary, a very fast implementation is possible. Experimental results reveal that good stereo correspondences can be found using only ordering and uniqueness constraints, i.e., without local smoothness constraints. However, it is shown that the original maximum likelihood stereo algorithm exhibits multiple global minima. The dynamic programming algorithm is guaranteed to find one, but not necessarily the same one for each epipolar scanline, causing erroneous correspondences which are visible as small local differences between neighboring scanlines. Traditionally, regularization, which modifies the original cost function, has been applied to the problem of multiple global minima. We developed several variants of the algorithm that avoid classical regularization while imposing several global cohesiveness constraints. We believe this is a novel approach that has the advantage of guaranteeing that solutions minimize the original cost function and preserve discontinuities. The constraints are based on minimizing the total number of horizontal and/or vertical discontinuities along and/or between adjacent epipolar lines, and local smoothing is avoided. Experiments reveal that minimizing the sum of the horizontal and vertical discontinuities provides the most accurate results. A high percentage of correct matches and very little smearing of depth discontinuities are obtained. An alternative to imposing cohesiveness constraints to reduce the correspondence ambiguities is to use more than two cameras. We therefore extend the two camera maximum likelihood to N cameras. The N-camera stereo algorithm determines the "best" set of correspondences between a given pair of cameras, referred to as the principal cameras. Knowledge of the relative positions of the cameras allows the 3D point hypothesized by an assumed correspondence of two features in the principal pair to be projected onto the image plane of the remaining N - 2 cameras. These N - 2 points are then used to verify proposed matches. Not only does the algorithm explicitly model occlusion between features of the principal pair, but the possibility of occlusions in the N - 2 additional views is also modeled. Previous work did not model this occlusion process, the benefits and importance of which are experimentally verified. Like other multiframe stereo algorithms, the computational and memory costs of this approach increase linearly with each additional view. Experimental results are shown for two outdoor scenes. It is clearly demonstrated that the number of correspondence errors is significantly reduced as the number of views/cameras is increased. © 1996 Academic Press, Inc.

### Full Text

### Duke Authors

### Cited Authors

- Cox, IJ; Hingorani, SL; Rao, SB; Maggs, BM

### Published Date

- January 1, 1996

### Published In

### Volume / Issue

- 63 / 3

### Start / End Page

- 542 - 567

### International Standard Serial Number (ISSN)

- 1077-3142

### Digital Object Identifier (DOI)

- 10.1006/cviu.1996.0040

### Citation Source

- Scopus