# Recovering Three Dimensional Shape From a Single Image of Curved Objects

### Jitendra Malik

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-87-340

January 1987

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1987/CSD-87-340.pdf

We present an algorithm to recover three-dimensional shape i.e. surface orientation and relative depth from a single segmented image. It is assumed that the scene is composed of opaque regular solid objects bounded by piecewise smooth surfaces with no markings or texture. Also it is assumed that the reflectance map E=R(n) is known. For the canonical case of lambertian surfaces illuminated by a point light source, this implies knowing the light source direction.

Solutions for simplified versions of this problem have been proposed by Sugihara for the case of polyhedra, and by Horn et al for the case of a single smooth surface patch given the surface orientation on the patch boundary. This work presents the first solution for the general case.

A variational formulation of line drawing and shading constraints in a common framework is developed. Finding the 3-D shape which best satisfies these constraints is a difficult global optimization problem. The problem is made tractable by precomputing line labels at junctions using the algorithm developed in [Malik '85]. The global constraints are partitioned into constraint sets corresponding to the faces, edges and vertices in the scene. For a face the constraints are given by Horn's image irradiance equation. We develop a variational formulation of the constraints at an edge -- both from the known direction of the image curve corresponding to the edge and shading. The associated Euler-Lagrange differential equation completely captures the local information. At a vertex, the constraints are modelled by a system of non-linear equations.

An algorithm has been developed to solve this system of constraints. We are now studying various refinements intended to make the algorithm robust in the presence of image noise.

BibTeX citation:

@techreport{Malik:CSD-87-340, Author = {Malik, Jitendra}, Title = {Recovering Three Dimensional Shape From a Single Image of Curved Objects}, Institution = {EECS Department, University of California, Berkeley}, Year = {1987}, Month = {Jan}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1987/5408.html}, Number = {UCB/CSD-87-340}, Abstract = {We present an algorithm to recover three-dimensional shape i.e. surface orientation and relative depth from a single segmented image. It is assumed that the scene is composed of opaque regular solid objects bounded by piecewise smooth surfaces with no markings or texture. Also it is assumed that the reflectance map E=R(n) is known. For the canonical case of lambertian surfaces illuminated by a point light source, this implies knowing the light source direction. <p> Solutions for simplified versions of this problem have been proposed by Sugihara for the case of polyhedra, and by Horn et al for the case of a single smooth surface patch given the surface orientation on the patch boundary. This work presents the first solution for the general case. <p> A variational formulation of line drawing and shading constraints in a common framework is developed. Finding the 3-D shape which best satisfies these constraints is a difficult global optimization problem. The problem is made tractable by precomputing line labels at junctions using the algorithm developed in [Malik '85]. The global constraints are partitioned into constraint sets corresponding to the faces, edges and vertices in the scene. For a face the constraints are given by Horn's image irradiance equation. We develop a variational formulation of the constraints at an edge -- both from the known direction of the image curve corresponding to the edge and shading. The associated Euler-Lagrange differential equation completely captures the local information. At a vertex, the constraints are modelled by a system of non-linear equations. <p> An algorithm has been developed to solve this system of constraints. We are now studying various refinements intended to make the algorithm robust in the presence of image noise.} }

EndNote citation:

%0 Report %A Malik, Jitendra %T Recovering Three Dimensional Shape From a Single Image of Curved Objects %I EECS Department, University of California, Berkeley %D 1987 %@ UCB/CSD-87-340 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1987/5408.html %F Malik:CSD-87-340