Distance Transforms of Sampled Functions

We describe linear-time algorithms for solving a class of problems that involve transforming a cost function on a grid using spatial information. These problems can be viewed as a generalization of classical distance transforms of binary images, where the binary image is replaced by an arbitrary function on a grid. Alternatively they can be viewed in terms of the minimum convolution of two functions, which is an important operation in grayscale morphology. A consequence of our techniques is a simple and fast method for computing the Euclidean distance transform of a binary image. Our algorithms are also applicable to Viterbi decoding, belief propagation, and optimal control.

[1]  R. Bellman,et al.  Functional equations in the theory of dynamic programming XII: An application of the maximum transform , 1963 .

[2]  Azriel Rosenfeld,et al.  Sequential Operations in Digital Picture Processing , 1966, JACM.

[3]  H. Blum.  Biological shape and visual science (part I) , 1973 .

[4]  H. Blum.  Biological shape and visual science. I. , 1973, Journal of theoretical biology.

[5]  Alok Aggarwal,et al.  Geometric applications of a matrix-searching algorithm , 1987, SCG '86.

[6]  Gunilla Borgefors,et al.  Distance transformations in digital images , 1986, Comput. Vis. Graph. Image Process..

[7]  Gunilla Borgefors,et al.  Hierarchical Chamfer Matching: A Parametric Edge Matching Algorithm , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

[8]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems - networks of plausible inference , 1991, Morgan Kaufmann series in representation and reasoning.

[9]  Lawrence R. Rabiner,et al.  A tutorial on hidden Markov models and selected applications in speech recognition , 1989, Proc. IEEE.

[10]  D. Eppstein.  Efficient algorithms for sequence analysis with concave and convex gap costs , 1989 .

[11]  Daniel J. Kleitman,et al.  An Almost Linear Time Algorithm for Generalized Matrix Searching , 1990, SIAM J. Discret. Math..

[12]  Michael Werman,et al.  Computing 2-D Min, Median, and Max Filters , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[13]  Daniel P. Huttenlocher,et al.  Comparing Images Using the Hausdorff Distance , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[14]  Olivier Devillers,et al.  Incremental Algorithms for Finding the Convex Hulls of Circles and the Lower Envelopes of Parabolas , 1995, Inf. Process. Lett..

[15]  David G. Kirkpatrick,et al.  Linear Time Euclidean Distance Algorithms , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[16]  Dimitri P. Bertsekas,et al.  Dynamic Programming and Optimal Control, Two Volume Set , 1995 .

[17]  Petros Maragos,et al.  Differential morphology , 2000 .

[18]  Jon M. Kleinberg,et al.  Fast Algorithms for Large-State-Space HMMs with Applications to Web Usage Analysis , 2003, NIPS.

[19]  Calvin R. Maurer,et al.  A Linear Time Algorithm for Computing Exact Euclidean Distance Transforms of Binary Images in Arbitrary Dimensions , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[20]  Daniel P. Huttenlocher,et al.  Efficient Belief Propagation for Early Vision , 2004, Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004..

[21]  Daniel P. Huttenlocher,et al.  Pictorial Structures for Object Recognition , 2004, International Journal of Computer Vision.

[22]  Nando de Freitas,et al.  Fast maximum a-posteriori inference on Monte Carlo state spaces , 2005, AISTATS.

[23]  Andrew Blake,et al.  Contour-based learning for object detection , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[24]  Christophe Restif,et al.  Towards Safer, Faster Prenatal Genetic Tests: Novel Unsupervised, Automatic and Robust Methods of Segmentation of Nuclei and Probes , 2006, ECCV.

[25]  Petros Maragos,et al.  Multigrid Geometric Active Contour Models , 2007, IEEE Transactions on Image Processing.

[26]  Harry J. P. Timmermans,et al.  A Metric of Compactness of Urban Change Illustrated to 22 European Countries , 2008, AGILE Conf..

[27]  Siddhartha S. Srinivasa,et al.  CHOMP: Gradient optimization techniques for efficient motion planning , 2009, 2009 IEEE International Conference on Robotics and Automation.

[28]  Hideki Koike,et al.  EdgeSonic: image feature sonification for the visually impaired , 2011, AH '11.