Recovering epipolar geometry from images of smooth surfaces

We present four methods for recovering the epipolar geometry from images of smooth surfaces. In the existing methods for recovering epipolar geometry corresponding feature points are used that cannot be found in such images. The first method is based on finding corresponding characteristic points created by illumination (ICPM-illumination characteristic points method). The second method is based on correspondent tangency points created by tangents from epipoles to outline of smooth bodies (OTPM-outline tangent points method). These two methods are exact and give correct results for real images, because positions of the corresponding illumination characteristic points and corresponding outline are known with small errors. But the second method is limited either to special type of scenes or to restricted camera motion. We also consider two more methods which are termed CCPM (curve characteristic points method, curves, denoted by word "green", are used for this method on Figures) and CTPM (curve tangent points method, curves, denoted by word "red" are used for this method on Figures), for searching epipolar geometry for images of smooth bodies based on a set of level curves (isophoto curves) with a constant illumination intensity. The CCPM method is based on searching correspondent points on isophoto curves with the help of correlation of curvatures between these lines. The CTPM method is based on property of the tangential to isophoto curve epipolar line to map into the tangential to correspondent isophoto curves epipolar line. The standard method termed SM (standard method, curves, denoted by word "blue" are used for this method on Figures) and based on knowledge of pairs of the almost exact correspondent points, has been used for testing of these two methods. The main technical contributions of our CCPM method are following. The first of them consists of bounding the search space for epipole locations. On the face of it, this space is infinite and unbounded. We suggest a method to partition the infinite plane into a finite number of regions. This partition is based on the desired accuracy and maintains properties that yield an efficient search over the infinite plane. The second is an efficient method for finding correspondence between points of two closed isophoto curves and finding homography, mapping between these two isophoto curves. Then this homography is corrected for all possible epipole positions with the help of evaluation function. A finite subset of solution is chosen from the full set given by all possible epipole positions. This subset includes fundamental matrices giving local minimums of evaluating function close to global minimum. Epipoles of this subset lie almost on straight line directed parallel to parallax shift. CTPM method was used to find the best solution from this subset. Our method is applicable to any pair of images of smooth objects taken under perspective projection models, as long as assumption of the constant brightness is taken for granted. The methods have been implemented and tested on pairs of real images. Unfortunately, the last two methods give us only a finite subset of solution that usually includes good solution, but does not allow us to find this good solution among this subset. Exception is the case of epipoles in infinity. The main reason for such result is inaccuracy of assumption of constant brightness for smooth bodies. But outline and illumination characteristic points are not influenced by this inaccuracy. So, the first pair of methods gives exact results.