Geometric Deformations of Discriminants and Apparent Contours

This book presents new and original results on the deformations of apparent contours of surfaces in Euclidean 3-space and the discriminants of plane-to-plane map-germs. Given a viewing direction, the apparent contour (also called the profile or outline) is the projection of the set of points on the surface where the viewing direction is tangent to the surface. Apparent contours are extensively used in computer vision and image analysis and pose significant mathematical challenges. 

 As the viewing direction varies, the apparent contour deforms, with emerging and vanishing inflections and vertices. The book provides a complete catalog of these bifurcations for generic surfaces as the viewing direction changes. Additionally, it explores geometric invariants that determine the maximum number of inflections and vertices that may appear in such deformations of an apparent contour. Aimed at researchers working in differential geometry, singularity theory, computer vision, and related areas, the text can also serve as material for an undergraduate reading course.

Chapter 1. Map-germs from the plane to the plane.- Chapter 2. Geometric deformations of discriminants.- Chapter 3. Geometric deformations of the fold and cusp.- Chapter 4. Ae-codimension 1 singularities.- Chapter 5. Ae-codimension 2 singularities.- Chapter 6. Apparent contours.- Chapter 7. Geometric invariants.

Farid Tari is a Full Professor at the University of São Paulo. He has published over 60 research papers, including some in leading mathematical journals, authored one book, and edited another. He has supervised several postdoctoral fellows, PhD, and MSc students. Currently, he serves as an Associate Editor of the Journal of Singularities.

Mostafa Salarinoghabi is currently a postdoctoral researcher at the Federal University of Viçosa, Brazil. He earned his PhD from the University of São Paulo, Brazil. His expertise lies in differential geometry, singularity theory, and random complex dynamics.

Masaru Hasegawa is currently an Assistant Professor at Iwate Medical University, Japan. He earned his PhD from Saitama University, Japan, where he also held a postdoctoral position. Additionally, he completed another postdoctoral position at the University of São Paulo, Brazil. His research focuses on applying singularity theory to differential geometry.