A Computational Approach to Edge Detection
IEEE Transactions on Pattern Analysis and Machine Intelligence
A survey of the Hough transform
Computer Vision, Graphics, and Image Processing
CVGIP: Image Understanding
Decoding of Reed Solomon codes beyond the error-correction bound
Journal of Complexity
Reconstructing Algebraic Functions from Mixed Data
SIAM Journal on Computing
An Operator Which Locates Edges in Digitized Pictures
Journal of the ACM (JACM)
Use of the Hough transformation to detect lines and curves in pictures
Communications of the ACM
An Efficient PAC Algorithm for Reconstructing a Mixture of Lines
ALT '02 Proceedings of the 13th International Conference on Algorithmic Learning Theory
Determining state transition probabilities using multi-objective optimisation
MS '08 Proceedings of the 19th IASTED International Conference on Modelling and Simulation
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We introduce the following problem which is motivated by applications in vision and pattern detection: We are given pairs of datapoints (x1,y1), (x2,y2),...,(xm, ym) ∈ [- 1, 1] × [- 1, 1], a noise parameter δ 0, a degree bound d, and a threshold ρ 0. We desire an algorithm that enlists every degree d polynomial h such that |h(xi) - yi| ≤ δ for at least δ fraction of the indices i. If δ = 0, this is just the list decoding problem that has been popular in complexity theory and for which Sudan gave a poly(m,d) time algorithm. However, for δ 0, the problem as stated becomes ill-posed and one needs a careful reformulation (see the Introduction). We prove a few basic results about this (reformulated) problem. We show that the problem has no polynomial-time algorithm (our counterexample works for ρ = 0.5). This is shown by exhibiting an instance of the problem where the number of solutions is as large as exp(d0.5-ε) and every pair of solutions is far from each other in ℓ∞ norm. On the algorithmic side, we give a rigorous analysis of a brute force algorithm that runs in exponential time. Also, in surprising contrast to our lowerbound, we give a polynomial-time algorithm for learning the polynomials assuming the data is generated using a mixture model in which the mixing weights are "nondegenerate."