## Distance transforms of sampled functions (2004)

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Venue: | Cornell Computing and Information Science |

Citations: | 120 - 11 self |

### BibTeX

@TECHREPORT{Felzenszwalb04distancetransforms,

author = {Pedro F. Felzenszwalb and Daniel P. Huttenlocher},

title = {Distance transforms of sampled functions},

institution = {Cornell Computing and Information Science},

year = {2004}

}

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### Abstract

This paper provides linear-time algorithms for solving a class of minimization problems in-volving a cost function with both local and spatial terms. These problems can be viewed as a generalization of classical distance transforms of binary images, where the binary image is replaced by an arbitrary sampled function. Alternatively they can be viewed in terms of the minimum convolution of two functions, which is an important operation in grayscale mor-phology. A useful consequence of our techniques is a simple, fast method for computing the Euclidean distance transform of a binary image. The methods are also applicable to Viterbi decoding, belief propagation and optimal control. 1

### Citations

7067 |
Probabilistic Reasoning in Intelligence Systems: Networks of Plausible Inference
- Pearl
- 1988
(Show Context)
Citation Context ... transforms of functions arise in the solution of a number of optimization problems. For instance in the widely used Viterbi algorithm for hidden Markov models [17], in max-product belief propagation =-=[16]-=-, in optimal control methods [2] and in resource allocation [1]. In these problems there is a discrete state space S, a cost b(p) for each state p ∈ S, a transition cost a(p,q) for changing from state... |

4286 | A tutorial on hidden Markov models and selected applications in speech recognition
- Rabiner
- 1989
(Show Context)
Citation Context ...alued feature quality measures, distance transforms of functions arise in the solution of a number of optimization problems. For instance in the widely used Viterbi algorithm for hidden Markov models =-=[17]-=-, in max-product belief propagation [16], in optimal control methods [2] and in resource allocation [1]. In these problems there is a discrete state space S, a cost b(p) for each state p ∈ S, a transi... |

526 | Pictorial structures for object recognition
- Felzenszwalb, Huttenlocher
(Show Context)
Citation Context ... for computing distance transforms of functions apply to certain problems of the form in (2). We have recently used these methods to develop improved algorithms for recognition of articulated objects =-=[7]-=-, for inference using large state-space hidden Markov models [9], and for the solution of low-level vision problems such as stereo, image restoration and optical flow using loopy belief propagation [8... |

485 | Comparing Images Using the Hausdorff Distance
- Huttenlocher, Klanderman, et al.
- 1993
(Show Context)
Citation Context ...al role in the comparison of binary images, particularly for images resulting from local feature detection techniques such as edge or corner detection. For example, both the Chamfer [5] and Hausdorff =-=[12]-=- 1smatching approaches make use of distance transforms in comparing binary images. Distance transforms are also used to compute the medial axis of digital shapes [3]. In this paper we consider a gener... |

462 |
Dynamic programming and optimal control. Athena Scientific
- Bertsekas
- 1995
(Show Context)
Citation Context ... the solution of a number of optimization problems. For instance in the widely used Viterbi algorithm for hidden Markov models [17], in max-product belief propagation [16], in optimal control methods =-=[2]-=- and in resource allocation [1]. In these problems there is a discrete state space S, a cost b(p) for each state p ∈ S, a transition cost a(p,q) for changing from state p to state q, and a dynamic pro... |

345 | P.: Efficient belief propagation for early vision
- FELZENSZWALB, HUTTENLOCHER
(Show Context)
Citation Context ...7], for inference using large state-space hidden Markov models [9], and for the solution of low-level vision problems such as stereo, image restoration and optical flow using loopy belief propagation =-=[8]-=-. For instance, in the case of a hidden Markov model with n states the standard computation of the Viterbi recurrence takes O(n 2 ) time which is not practical for large values of n, while the computa... |

323 |
Distance transformations in digital images
- Borgefors
- 1986
(Show Context)
Citation Context ...rary function f(q). Efficient algorithms for computing the distance transform of a binary image using the l1 and l∞ distances were developed by Rosenfeld and Pfaltz [18]. Similar methods described in =-=[4]-=- have been widely used to efficiently compute approximations to the Euclidean distance transform. These algorithms can be easily adapted to compute the distance transform of a function, as we shown in... |

247 |
Hierarchical chamfer matching: a parametric edge matching algorithm
- Borgefors
- 1988
(Show Context)
Citation Context ...forms play a central role in the comparison of binary images, particularly for images resulting from local feature detection techniques such as edge or corner detection. For example, both the Chamfer =-=[5]-=- and Hausdorff [12] 1smatching approaches make use of distance transforms in comparing binary images. Distance transforms are also used to compute the medial axis of digital shapes [3]. In this paper ... |

175 |
Sequential operations in digital picture processing
- Rosenfeld, Pfalts
(Show Context)
Citation Context ... function 1(q) rather than an arbitrary function f(q). Efficient algorithms for computing the distance transform of a binary image using the l1 and l∞ distances were developed by Rosenfeld and Pfaltz =-=[18]-=-. Similar methods described in [4] have been widely used to efficiently compute approximations to the Euclidean distance transform. These algorithms can be easily adapted to compute the distance trans... |

144 |
Biological shape and visual science
- Blum
(Show Context)
Citation Context ...oth the Chamfer [5] and Hausdorff [12] 1smatching approaches make use of distance transforms in comparing binary images. Distance transforms are also used to compute the medial axis of digital shapes =-=[3]-=-. In this paper we consider a generalization of distance transforms to arbitrary functions on a grid rather than binary-valued ones (i.e., real valued images rather than binary images). There is a sim... |

62 | M.: Linear time Euclidean distance transform algorithms
- Breu, Gil, et al.
- 1995
(Show Context)
Citation Context ...chnique for computing the exact Euclidean distance transform of a binary image. There are a number of algorithms for computing the Euclidean distance transform of a binary image in linear time (e.g., =-=[13, 6, 15]-=-), however these methods are quite involved and are not widely used in practice. In contrast, our algorithm is relatively simple, easy to implement and very fast in practice. We use the terminology di... |

52 |
A linear time algorithm for computing exact euclidean distance transforms of binary images in arbitrary dimensions
- Jr, Qi, et al.
- 2003
(Show Context)
Citation Context ...chnique for computing the exact Euclidean distance transform of a binary image. There are a number of algorithms for computing the Euclidean distance transform of a binary image in linear time (e.g., =-=[13, 6, 15]-=-), however these methods are quite involved and are not widely used in practice. In contrast, our algorithm is relatively simple, easy to implement and very fast in practice. We use the terminology di... |

26 | Fast algorithms for large-state-space hmms with applications to web usage analysis, in
- Felzenszwalb, Huttenlocher, et al.
- 2003
(Show Context)
Citation Context ... problems of the form in (2). We have recently used these methods to develop improved algorithms for recognition of articulated objects [7], for inference using large state-space hidden Markov models =-=[9]-=-, and for the solution of low-level vision problems such as stereo, image restoration and optical flow using loopy belief propagation [8]. For instance, in the case of a hidden Markov model with n sta... |

19 |
Data structures for operations on digital images
- Rutovitz
- 1968
(Show Context)
Citation Context ...econd, there already are methods for computing distance transforms of gray level images based on minimum distances along paths, where the cost of a path is the sum of gray level values along the path =-=[19]-=-. We want to avoid confusion with these methods which compute something quite different from what we consider here. 1.1 Minimum Convolution The distance transform of a sampled function is closely rela... |

18 |
A euclidean distance transform using grayscale morphology decomposition
- Huang, Mitchell
- 1994
(Show Context)
Citation Context ...onvolution algorithms. In the case of the squared Euclidean distance we are computing the minimum convolution of a sampled function and a parabola. This is a problem that has been studied before (see =-=[11]-=-) but our algorithm is more efficient than previous techniques. In the case of the l1 distance we compute the minimum convolution of a function and a diamond shaped cone using the classical algorithm ... |

4 | Differential morphology
- Maragos
- 2001
(Show Context)
Citation Context ...tion The distance transform of a sampled function is closely related to the minimum convolution operation. This operation and its continuous counterpart play an important role in grayscale morphology =-=[14]-=-. The minimum convolution of two discrete signals f and g is defined as, (f ⊗ h)(p) = min q (f(q) + h(p − q)). 3sJust like standard convolution this operation is commutative and associative, f ⊗ h = h... |

3 |
Computing 2D Min, Max and Median Filters
- Gil, Werman
- 1993
(Show Context)
Citation Context ...e is the distance transform under the box distance defined by d(p,q) = 0 when |p − q| < a and ∞ otherwise. This transform can be computed using a linear time algorithm for the min-filter described in =-=[10]-=-. Another way to obtain fast algorithms is to use the relationships described below. For example, the distance d(p,q) = min(c(p − q) 2 ,a|p − q| + b) is commonly used in robust estimation and is very ... |

3 |
algorithm for determining the distances from the points of the given subset of an integer lattice to the points of its complement
- Quick
- 1992
(Show Context)
Citation Context ...chnique for computing the exact Euclidean distance transform of a binary image. There are a number of algorithms for computing the Euclidean distance transform of a binary image in linear time (e.g., =-=[13, 6, 15]-=-), however these methods are quite involved and are not widely used in practice. In contrast, our algorithm is relatively simple, easy to implement and very fast in practice. We use the terminology di... |

1 |
Functional equations in the theorey of dynamic programming XII: An application of the maximum transform
- Bellman, Karush
- 1963
(Show Context)
Citation Context ...timization problems. For instance in the widely used Viterbi algorithm for hidden Markov models [17], in max-product belief propagation [16], in optimal control methods [2] and in resource allocation =-=[1]-=-. In these problems there is a discrete state space S, a cost b(p) for each state p ∈ S, a transition cost a(p,q) for changing from state p to state q, and a dynamic programming equation, δ ′ (q) = b(... |