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
Secure smartcardbased fingerprint authentication
WBMA '03 Proceedings of the 2003 ACM SIGMM workshop on Biometrics methods and applications
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(MATH) Motivated by applications in vision and pattern detection, we introduce the following problem. We are given pairs of datapoints $(x_1, y_1)$, $(x_2, y_2)$, $\ldots,(x_m, y_m)$, a noise parameter $\delta 0$, a degree bound $d$, and a threshold $\rho0$. We desire "every" degree $d$ polynomial $h$ satisfying h(x_i) \in [y_i -\delta, y_i +\delta] & \qquad \nonumber for at least &rgr; fraction of i's.(MATH) We assume by rescaling the data that each $x_i, y_i \in [-1, 1]$.(MATH) If $\delta =0$, this is just the list decoding problem that has been popular in complexity theory and for which Sudan gave a $\poly(d,1/\rho)$ time algorithm.We show a few basic results about the problem. We show that there is no polynomial time algorithm for this problem as defined; the number of solutions can be as large as exp(d0.5 -&egr;) even if the data is generated using a 50-50 mixture of two polynomials. We give a rigorous analysis of a brute force algorithm for the version of this problem where the data is generated from a mixture of polynomials. Finally, in surprising contrast to our "lower bound", we describe a polynomial-time algorithm for reconstructing mixtures of O(1) polynomials when the mixing weights are "nondegenerate.The tools used include classical theory of approximations.