Computer Science Honours and Graduate Diploma Projects for 1999

Dr Ryszard Kozera (ryszard@cs.uwa.edu.au)

The following preliminary topics specified below are offered and can be supervised by Dr R. Kozera. These honours/postgraduate projects can also be alternatively co-supervised together with:

• A./Prof. Lyle Noakes, Department of Mathematics, The University of Western Australia
• Dr Neville Fowkes, Department of Mathematics, The University of Western Australia

1. Shape recovery from three light-source photometric stereo
   This project embarks on implementing an algorithm to reconstruct an unknown surface from its multiple images obtained by consecutive illumination of the unknown surface from three different directions. This computer vision technique, called three light-source Photometric Stereo technique, can be applied in medical image interpretation, robotics, or in applied optics. The method is based on theoretical studies done by Kozera in the recent few years. The whole reconstruction process for Photometric Stereo is split into two independent steps: gradient computation and gradient integration. The first one, in the case of three light-sources, renders a unique vector field explicitly expressed in terms of three images and three light-source directions. The latter step requires numerical integration of the computed gradient vector field. Many gradient integration methods can be here tested (e.g. Newton-Cotes schemes) in order to evaluate an efficient numerical algorithm for shape reconstruction in three light-source Photometric Stereo. A development of a user friendly package as well as testing the algorithms on real image data might also be a part of this project. This project can be an introductory work for a possible PhD studies on Photometric Stereo.

2. Shape recovery from two-source photometric stereo
   As with the above project (see three light-source Photometric Stereo), the task of this project and its application are similar. Here the shape is reconstructed from a pair of images instead of a triplet of images. There exist however some other essential differences as compared with the three light-source Photometric Stereo. Namely, the gradient vector field may not be unquely determined in terms of a pair of images and a pair of light-source directions. It can be shown that, in case of ambiguity there exist generically two solutions. The integration of both vector fields can be carried out as in the case of three light-sources or by using an alternative decomposition algorithm introduced Kozera. A possible supplementary subtask would be to localise the bifurcation curve along which different solutions can be glued together thus increasing the magnitude of the appearing ambiguity (should such curve occurs). A development of a user friendly package as well as testing the algorithms on real image data could also be a part of this project. This project can be an introductory work for a possible PhD studies on Photometric Stereo.

3. Shape reconstruction from a single image based sequential algorithms
   In case of a single image the shape reconstruction based on pre-computing gradient vector field (as opposed to above mentioned Photometric Stereo technique) is not feasible There are inifinitely many possible fake gradient candidates to be analyzed and integrated. The unknown surface can be, in turn, recovered by using other alternative available techniques like e.g. the method of chracteristic strips. Finding an analytic solution to the corresponding system of five ordinary differential equation constitutes, in general an impossible task. The alternative, is to resort to numerical analysis and use the finite-difference technique in solving the corresponding discretized problem. This technique can be applied not only for a single-image shape-from-shading problem but also in many other areas modelled by differential equations (a common case in physics, geophysics, meteorology, mechanics, engineering and computer science). The purpose of the project is to implement and test few algorithms based on the method of characteristic strips. Analysis of stability and convergence or/and a development of a user friendly package could be an additional part of the project. This project can be an introductory work for a possible PhD studies on shape-from-shading, numerical analysis or other research domains using differential equations.

4. Graph based optimization algorithms applied to a single image shape reconstruction
   The sequential methods applied e.g. to shape-from-shading problem (for more detail see the previous project) rely in a crucial way on provision of prior information such as certain boundary conditions. These boundary conditions can, in certain cases be easily obtainable, if e.g. Photometric Stereo technique, introduced above, is additionally applied. The alternative is to apply the direct optimization method rendering the special type of surfaces (e.g. convex/concave in the case of one singular point in the image). The algorithm based on Dijkstra's shortest path method and Chojnacki et. al. work is to be evaluated and implemented. More that one singular point within the image can also be considered. This project can be an introductory work for a possible PhD studies on shape-from-shading area, numerical analysis or optimization theory. The latter can also be applied in many engineering areas (e.g. in control theory) modelling a real life problems.

5. A leap-frog optimization algorithm applied to a single image shape reconstruction
   As in the previous project the application of the leap-frog algorithm developed by Noakes can be performed in many areas (e.g. in optimization and control theory or shape-from-shading problem). We will apply the algorithm in question in order to find a minimum of a pertinent functional rendering the unknown Lambertian shape. The evaluation and the performance of the proposed algorithm shall be discussed. The issue of the stability of the corresponding scheme might also be rediscussed. In addition, a development of a user friendly package as well as testing the algorithms on real image data containing more than one sigular point could also be a part of this project. This project can be an introductory work for a possible PhD studies on shape-from-shading area, numerical analysis or optimization theory.

6. Evaluation of finite-difference based algorithms
   This project embarks on developing and testing different finite-difference schemes which could be applied, among all, in shape reconstruction process in the case of the so-called linear reflectance map (e.g. appearing in the light reflection from the Moon). This approach can, however can be also used to solve many other problems modelled by the ordinary or partial differential equations (e.g. appearing in phycisc, geophysics, mechanics, meteorolgy, engineering or computer science). The evaluation of single-layer and double-layer finite-difference based schemes is intended to be performed. Possibly, a pertinent analysis of stability and convergence of each scheme in question, already introduced by Kozera and Klette can also be re-discussed. This project can be an introductory work for a possible PhD studies on shape-from-shading area, numerical analysis or variety of other engineering problems.

7. Can the sun direction be determined from the single or multiple images?
   In this project the implementation of the algorithm of Chojnacki et al. recovering the light-source direction is intended to be accomplished. This problem is vital for the so-called dynamic vision when prior to the shape reconstruction process the direction of the light has to be first established. In the first step an appropriate known sample surface is illuminated in order to render the illumination direction. In this project a sample surface to be originally considered is a Lambertian hemisphere. The evaluation of the algorithm is one of the ultimate project task. In addition a generalization of this algorithm for other sample surface can be considered. Alternatively the direction of the illumination can be attempted by using Photometric Stereo method (for more detailed information see above).

8. Banach fixed-point theorem and its numerics
   In this project, a student is requested to implement different possible applications of the Banach Fixed Point Theorem. It can be used in solving some non-linear and linear equations, differential and integral equations, in finding an implicit function etc. A possible comparison with the classical Newton's method (for solving non-linear equation(s)) is also intended to be accomplished. A user friendly interface visualizing the important aspects of the Banach Fixed Point Theorem can also be implemented.

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