Denoising images: non-linear leap-frog for shape and light-source recovery

  • Authors:

  • Lyle Noakes;Ryszard Kozera

  • Affiliations:

  • The University of Western Australia, School of Mathematics and Statistics, Crawley, WA, Australia;The University of Western Australia, School of Computer Science and Software Engineering, Crawley, WA, Australia

  • Venue:
  • Proceedings of the 11th international conference on Theoretical foundations of computer vision
  • Year:
  • 2002

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Abstract

In 3-source photometric stereo, a Lambertian surface is illuminated from 3 known independent light-source directions, and photographed to give 3 images. The task of recovering the surface reduces to solving systems of linear equations for the gradients of a bivariate function u whose graph is the visible part of the surface [9], [16], [17], [24]. In the present paper we consider the same task, but with slightly more realistic assumptions: the photographic images are contaminated by Gaussian noise, and light-source directions may not be known. This leads to a non-quadratic optimization problem with many independent variables, compared to the quadratic problems resulting from addition of noise to the gradient of u and solved by linear methods in [6], [10], [20], [21], [22], [25]. The distinction is illustrated in Example 1 below. Perhaps the most natural way to solve our problem is by global Gradient Descent, and we compare this with the 2-dimensional Leap-Frog Algorithm [23]. For this we review some mathematical results of [23] and describe an implementation in sufficient detail to permit code to be written. Then we give examples comparing the behaviour of Leap-Frog with Gradient Descent, and explore an extension of Leap-Frog (not covered in [23]) to estimate light source directions when these are not given, as well as the reflecting surface.