Optimal approximations by piecewise smooth functions and associated variational problems
David Mumford,Jayant Shah +1 more
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TLDR
In this article, the authors introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision, and study their application in computer vision.Abstract:
: This reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision. In computer vision, a fundamental problem is to appropriately decompose the domain R of a function g (x,y) of two variables. This problem starts by describing the physical situation which produces images: assume that a three-dimensional world is observed by an eye or camera from some point P and that g1(rho) represents the intensity of the light in this world approaching the point sub 1 from a direction rho. If one has a lens at P focusing this light on a retina or a film-in both cases a plane domain R in which we may introduce coordinates x, y then let g(x,y) be the strength of the light signal striking R at a point with coordinates (x,y); g(x,y) is essentially the same as sub 1 (rho) -possibly after a simple transformation given by the geometry of the imaging syste. The function g(x,y) defined on the plane domain R will be called an image. What sort of function is g? The light reflected off the surfaces Si of various solid objects O sub i visible from P will strike the domain R in various open subsets R sub i. When one object O1 is partially in front of another object O2 as seen from P, but some of object O2 appears as the background to the sides of O1, then the open sets R1 and R2 will have a common boundary (the 'edge' of object O1 in the image defined on R) and one usually expects the image g(x,y) to be discontinuous along this boundary. (JHD)read more
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References
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Handbook of Mathematical Functions
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Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images
Stuart Geman,Donald Geman +1 more
TL;DR: The analogy between images and statistical mechanics systems is made and the analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations, creating a highly parallel ``relaxation'' algorithm for MAP estimation.
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A Course of Modern Analysis
TL;DR: The volume now gives a somewhat exhaustive account of the various ramifications of the subject, which are set out in an attractive manner and should become indispensable, not only as a textbook for advanced students, but as a work of reference to those whose aim is to extend the knowledge of analysis.
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Elliptic Problems in Nonsmooth Domains
TL;DR: Second-order boundary value problems in polygons have been studied in this article for convex domains, where the second order boundary value problem can be solved in the Sobolev spaces of Holder functions.