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Information about sicily03

Published on June 19, 2007

Author: Mentor


Application of Fisher Information to Line Detection:  Application of Fisher Information to Line Detection S.J. Maybank Department of Computer Science University of Reading 10 September 2003 Questions:  Questions How many lines are there in an image? Is it possible to search exhaustively for lines by checking a finite list of candidates? What about other image structures? Answers:  Answers Let 2t = σ2 = noise variance. The number of lines in the unit disc is n=π/(4√3 t) n candidates (=models) suffice for exhaustive search. Similar results apply to projective transformations of the line. Geometric Model:  Geometric Model α ρ ρ=x1 cos(α)+x2 sin(α) The points (ρ,α), 0≤ρandlt;1,0≤αandlt;2π parameterise the lines in the unit disk. x1 x2 Probabilistic Model:  Probabilistic Model The probability of a measurement x=(x1, x2) given the true value x’ is p(x|x’)=Gaussian(x’,2t) p(x’|line) is uniform on the line. Parameterised Family of Densities:  Parameterised Family of Densities Let =(ρ,α) Measurement space, D=unit disk Parameter space, T= [0,1)×[0,2π) Probability of x ε D given  ε T is p(x|) Fisher Information J():  Fisher Information J() Formal definition Jij()=-∫∂2ijlog(p(x|)) p(x|) dx Meaning: J() defines a Riemannian metric on T. Points , ψ are close if a measurement x is unlikely to distinguish between them. Properties of Fisher Information:  Properties of Fisher Information det(J) d= Volume form which defines a Bayesian prior density on T. (NJ())-1 = Covariance of Maximum Likelihood estimate of , based on N measurements. Asymptotic Approximation:  Asymptotic Approximation Let w(x,)= min distance from x to line(). The leading order term K() in the asymptotic expansion of J() is Kij()=(4t)-1∫line()∂2ij w(x,)2 d(line) Expression for K():  Expression for K() Under K, T embeds isometrically as a surface in R3 Number of models:  Number of models B() is the set of models indistinguishable from . The area of B() under K is π. The number n of models is n = area(T)/area(B()) = π(4√3 t)-1 Line Detection:  Line Detection Let N measurements be given in the unit disk D. Let r be a threshold. A line is detected if r or more measurements are close enough to a model  False Detection:  False Detection Suppose N measurements are uniformly distributed in D. How high must r be to give only a small probability of false detection? Example: If N=20, r=5, t=(1/2)×10-4, then there is approximately 1 false detection. Example:  Example Measurements for N=20, r=5, t=(1/2)×10-4,with a single false detection

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