Local Shape from Mirror Reflections

Abstract

We study the problem of recovering the 3D shape of an unknown smooth specular surface from a single image. The surface reflects a calibrated pattern onto the image plane of a calibrated camera. The pattern is such that points are available in the image where position, orientations, and local scale may be measured (e.g. checkerboard). We first explore the differential relationship between the local geometry of the surface around the point of reflection and the local geometry in the image. We then study the inverse problem and give necessary and sufficient conditions for recovering surface position and shape. We prove that surface position and shape up to third order can be derived as a function of local position, orientation and local scale measurements in the image when two orientations are available at the same point (e.g. a corner). Information equivalent to scale and orientation measurements can be also extracted from the reflection of a planar scene patch of arbitrary geometry, provided that the reflections of (at least) 3 distinctive points may be identified. We validate our theoretical results with both numerical simulations and experiments with real surfaces.

Shape from Specular Reflection and Optical Flow

“…A striking consequence of the theorem is that the optical flow is only influenced by surface properties up to second order. This is in close analogy to a result of Savarese and Perona [SCP05] for a related problem. Note that the optical flow computation only makes sense if one can presume that the law of reflection holds.…”

“…Kickingereder et al [KD04] and Bonfort [BS03] use stereo vision to uniquely recover the surface. Some approaches based on a local surface representation have also been discussed: Savarese, Chen, and Perona [SCP05] use a special pattern to find a set of surface patches by locally neglecting higher-order surface properties. In [BWB06], the Lambertian behavior of a certain class of surfaces is employed.…”

Technique for measuring the three-dimensional shapes of telescope mirrors

“…We determine the homography between camera c1 and the horizontal reference plane z ¼ 0 using the MATLAB calibration toolbox. The determined homography could be formulated as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 6 ; 3 2 6 ; 3 4 …”

“…To improve the manufacturing accuracy, techniques of measuring the three-dimensional (3-D) shapes of mirrors become important because the manufacturing errors of mirrors could be computed directly after their shapes are known. Although many noncontact techniques [1][2][3][4][5][6][7][8][9][10] have been developed to measure the mirrors in the past decades, none of them are capable of measuring the shapes of mirrors with adequate accuracy.…”

Specular Flow and Shape in One Shot:

“…In practice, this so called parabolic singularity [3] drives virtually all optical flow algorithms into global qualitative errors across the image plane. The same problem is revealed in related methods also, and is coped with by assuming no parabolic regions (e.g., [16,18]) or by explicit surface modeling (e.g., [15,17]). …”