Browse Books

Reviewer: Louis Hodes

This is a treatise on the problem of converting a two-dimensional line drawing of a polyhedral scene into a three-dimensional spatial representation. The solution is unfolded systematically over the length of the book in a series of four modules. The first module uses the Huffman-Clowes line-labeling scheme [1,2]. A line-labeled drawing shows which lines are a result of convex or concave adjoining faces and which sides of edges contain matter. This allows constraints at the junctions to limit the number of possible three-dimensional interpretations. However, a second module is required to narrow down the correct interpretations. Here a necessary and sufficient condition is derived in terms of linear inequalities. A coordinate system is assumed with the z axis perpendicular to the x, y plane upon which the line drawing appears as a projection. Thus each vertex gives one unknown, its z value; and each face is determined by three coefficients, ax + by + z = c. With this normalization of the z coefficient to one, it becomes apparent that all constraints of vertices on faces are linear equalities. Further, labeling of the edges from the first module yields inequalities from points lying in front of or behind various faces. In the case where hidden lines are included in the drawing, there is an added complication in order to avoid ambiguities. Each polygonal region is supplied with a sequence of faces, a face-layer structure, that effectively orders the objects in the scene. There can be different interpretations with different sequences. In this manner a set of inequalities is algorithmically obtained such that a solution to this set is equivalent to a correct interpretation. Also, a matrix is obtained with rank indicating the degrees of freedom of the variables in the interpretation, for example, the number of vertices that must be fixed to completely specify the three-dimensional construct. Matroids are introduced to deal with the complex relations among the vertex and face variables. Interesting properties of drawings with four degrees of freedom are uncovered. The entire work so far has assumed that the line drawings are absolutely precise. For example, three faces in general position must, if extended, meet in a point. However a hand-drawn image or even digitization errors usually preclude the structures from being realized in three dimensions. The third module shows how images may be corrected by adjusting the vertices to provide a feasible interpretation. A relationship among the number of vertices, faces, and incidences of vertices on faces is proved such that a construction is possible from a generic position, that is, a generic position assumes the errors are correctable. Thus, a substructure that is feasible can be found. If the entire structure is correctable, then the remaining incidences can be adjusted. Finding a maximal, generically-reconstructable substructure can be combinatorially expansive, so an efficient algorithm by Imai [3] based on network flow theory is presented. The fourth and final module deals with the determination of unique shape either from specifications in the drawing or from surface information such as texture. The final chapter discusses reciprocal diagrams and the rigidity of polyhedrons. This work includes references to J. C. Maxwell. This book brings together a great deal of information on a neat, well-specified problem in artificial intelligence. The majority of the effort yields an integrated approach to the solution. Thus, the book fulfills its purpose well. Although the problem is solved in over 200 pages, the presentation seemed just right. The theory is supported by well-chosen examples, and the results are applied to more interesting examples. The best features are the appropriate use of intellectually-stimulating basic mathematical tools, and the thorough manner in which the problem is analyzed. The worst feature is the thin font that forces the reader to use strong lighting to read the subscripts. My overall evaluation is that it is an excellent and unusual book, probably the definitive work on this subject. It is intended for workers in machine vision or students of the subject. People interested in computational geometry may also find interesting ideas here. The illustrations, some dealing with familiar optical illusions, are well done. There are over 100 pertinent and varied references. The index seems adequate. This book is highly recommended to anyone interested in the subject.