Multiple Paths Extraction in Images Using a Constrained Expanded Trellis Changming Sun and Ben Appleton Abstract—Single shortest path extraction algorithms have been used in a number of areas such as network flow and image analysis. In image analysis, shortest path techniques can be used for object boundary detection, crack detection, or stereo disparity estimation. Sometimes one needs to find multiple paths as opposed to a single path in a network or an image where the paths must satisfy certain constraints. In this paper, we propose a new algorithm to extract multiple paths simultaneously within an image using a constrained expanded trellis (CET) for feature extraction and object segmentation. We also give a number of application examples for our multiple paths extraction algorithm. Index Terms—Multiple paths extraction, constrained expanded trellis, feature extraction, object segmentation. æ 1 INTRODUCTION S INGLE shortest path extraction techniques have been used for a number of applications such as crack detection in borehole images [1], road extraction in satellite images [2], disparity estimation for stereo images [3], [4], [5], optical flow estimation [6], object boundary extraction in images [7], and optimization problems in network and transporta- tion analysis [8]. The result of ordinary shortest path extraction is usually a single optimal path. In some applications, such as high-performance commu- nication, fault-tolerant routing, transportation networks, and radar multitarget tracking, it is often necessary to obtain multiple paths which are in some sense significantly distinct from each other. In earlier work, such paths have been variously called mutually exclusive paths, disjoint paths, completely unmerged paths, or K best paths [9], [10], [11], [12], [13], [14]. In this paper, we are interested in multiple paths in images where the total sum of values along all the K paths is a minimum. These paths could correspond to multiple features in images or boundaries of objects in images. Successive application of a single shortest path extraction algorithm to obtain multiple paths, although a simple solution, clearly will not produce the desired optimal solution in general. Nikolopoulos and Samaras proposed solutions to the multiple track detection problem using graph theoretic concepts for detecting discontinuous lines and for achieving an efficient solution [15]. However, their solutions are suboptimal. For optimal multiple paths finding, there are mainly two classes of algorithms: one based on dynamic programming or the Viterbi algorithm (as described in [14]) and one based on minimum cost network flow (MCNF) algorithms (as described in [12], [13]). The computation cost of the first class of algorithms grows exponentially with the number of paths due to the large network or trellis generated. In the MCNF approach, an input image is converted into an augmented graph and an MCNF algorithm is applied to this graph to obtain the multiple paths which are mutually exclusive (see [13] for details). This approach takes a topological rather than a geometric viewpoint of the paths and, so, manages to avoid explicitly representing the configuration space. As a result, it requires OððmnÞ 3 logðmnÞÞ computation and Oðn 2 mÞ mem- ory. Although this class of algorithms is relatively efficient, it is difficult to impose necessary constraints on the structure or geometric relationship of the multiple paths. The application of the MCNF method to image analysis has several difficulties which will be discussed in Section 4.3. In this paper, we propose a new algorithm using constrained expanded trellises (CETs) for feature extraction and object segmentation in images. Our algorithm is much more computationally efficient than the first class of algorithms, as exemplified in [14], and we can easily apply constraints to control the geometry of the multiple paths where the second class of algorithms is unable. The paper is organized in the following sections: Section 2 gives the problem definition and cost functions. Section 3 describes multiple paths extraction problems and the initial expanded trellis. Section 4 develops a number of geometric and heuristic methods which greatly reduce the computation and memory requirements. Section 5 gives our efficient implementation of dynamic programming in the constrained expanded trellis. We show the results of our algorithm on a range of real images from different applications in Section 6. Section 7 gives concluding remarks. 2 PROBLEM DEFINITION AND COST FUNCTIONS 2.1 Finite Sequences and Paths In this paper, Z Z m denotes the sequence of integers ð1; 2;  ;mÞ. A path is represented by a finite sequence P :Z Z m ! R which is a function labeling each element of Z Z m with an element from the range R. In this paper, the range is a discrete set consisting of a finite number of points. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 27, NO. 12, DECEMBER 2005 1923 . C. Sun is with CSIRO Mathematical and Information Sciences, Locked Bag 17, North Ryde, NSW 1670, Australia. E-mail: changming.sun@csiro.au. . B. Appleton is with Electromagnetics and Imaging, School of ITEE, The University of Queensland, QLD 4067, Australia. E-mail: appleton@itee.uq.edu.au. Manuscript received 20 Dec. 2004; revised 28 Feb. 2005; accepted 30 Mar. 2005; published online 13 Oct. 2005. Recommended for acceptance by A. Srivastava. For information on obtaining reprints of this article, please send e-mail to: tpami@computer.org, and reference IEEECS Log Number TPAMI-0674-1204. 0162-8828/05/$20.00 ß 2005 IEEE Published by the IEEE Computer Society