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## Multi-Label Image Segmentation for Medical Applications Based on Graph-Theoretic Electrical Potentials (2004)

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Venue: | ECCV |

Citations: | 46 - 10 self |

### Citations

3721 | Normalized cuts and image segmentation
- Shi, Malik
(Show Context)
Citation Context ...nction that maps a change in image intensities to weights. Since this is a common feature of graph based algorithms for image analysis, several weighting functions are commonly used in the literature =-=[19, 17, 2]-=-. Additionally, it was proposed in [15] to use a function that maximizes the entropy of the resulting weights. In this work we have preferred (for empirical reasons) the typical Gaussian weighting fun... |

2090 | Fast approximate energy minimization via graph cuts
- Boykov, Veksler, et al.
- 2001
(Show Context)
Citation Context ... the potential obtained in the neutral segmentation.sProof. We may write the potential for node vi as x s i = � � eij∈T T (kij + r) � eij∈T T (kij + r) = |T Ti|rN−2 + O(rN−3 ) |T TG|rN−2 + O(rN−3 , ) =-=(14)-=- T T ∈T Ti � T T ∈T TG where O(·) indicates a term of order no greater than the argument. By l’Hôpital’s Rule, 2.4 Numerical practicalities lim r→∞ xsi = |T Ti| |T TG| = ns i . (15) Many good sources ... |

1768 |
Graph Theory
- Harary
- 1969
(Show Context)
Citation Context ...g the system of equations to solve, proving theoretical results about the algorithm performance and the practical details of implementation. We begin by defining a precise notion for a graph. A graph =-=[18]-=- consists of a pair G = (V, E) with vertices (nodes) v ∈ V and edges e ∈ E ⊆ V × V . An edge, e, spanning two vertices, vi and vj, is denoted by eij. A weighted graph assigns a value to each edge call... |

996 | Interactive graph cuts for optimal boundary and region segmentation of objects in N-D images
- Boykov, Jolly
- 2001
(Show Context)
Citation Context ...bution of these new variables. After combining terms, the numerator of (12) is given by E[ST Ti (|T Ti| + |T TC|) − |T Ti|(ST Ti + ST TC )] = µ|T Ti|(|T Ti| + |T TC|) − |T Ti|(µ|T Ti| + µ|T TC|) = 0, =-=(13)-=- and the denominator of (12) must be strictly positive, since all the wij are guaranteed to be positive by construction. Therefore, the conditions of Lemma 1 are satisfied for the left hand side of (1... |

763 | Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image segmentation
- Zhu, Yuille
- 1996
(Show Context)
Citation Context ...he following relationship for the potential at node vi in the presence of unit voltage sources (tied to ground) is given in [6, 26] � � x s i = eij∈T T w(eij) T T ∈T Ti � � T T ∈T TG eij∈T T w(eij) , =-=(10)-=-swhere T Ti is the set of 2-trees present in the graph, such that node vi is connected to a seed with label s, and T TG is the set of all possible 2-trees in the graph. A 2-tree is defined as a tree w... |

414 |
Numerical Recipes in C: The Art of Scientific Computing 2 nd Edition
- Press, Teukolsky, et al.
- 1994
(Show Context)
Citation Context ...nt. By l’Hôpital’s Rule, 2.4 Numerical practicalities lim r→∞ xsi = |T Ti| |T TG| = ns i . (15) Many good sources exist on the solution to large, sparse, symmetric, linear systems of equations (e.g., =-=[27, 28]-=-). A direct method, such as LU decomposition with partial pivoting has the advantage that the computation necessary to solve (6) is only negligibly increased over the amount of work required to solve ... |

261 | Multiclass spectral clustering
- Yu, Shi
- 2003
(Show Context)
Citation Context ...t to prove some interesting results about the behavior of xs i under different choices of weights. If the weights are uniform, the neutral case, by (10), yields potentials satisfying x s i = |T Ti| . =-=(11)-=- |T TG| Now we are in position to prove a series of propositions about xs under different conditions. We also note that if the weights are all multiplied by a constant, k, there will be no effect on x... |

203 |
Iterative Solution of Large Sparse Systems of Equations
- Hackbusch
- 1995
(Show Context)
Citation Context ... not have enough memory to allow an LU decomposition with that number of equations. The standard alternative to the class of direct solvers for large, sparse systems is the class of iterative solvers =-=[29]-=-. These solvers have the advantages of a small memory requirement and the ability to represent the matrix-vector multiplication as a function. In particular since, for a lattice, the matrix LU has a c... |

171 |
Spectral k-way ratio-cut partitioning and clustering
- Chan, Schlag, et al.
- 1993
(Show Context)
Citation Context ... it can be shown that any connected subset, P ⊆ VU, assigned to segment s must be connected to at least one node that is also labeled s. A block matrix form of (7) may be written LP x s P = −RP x s , =-=(8)-=- P where xs = [xs P , xs P ]T ,L has been decomposed into the block form � LP RP L = RT P L � , (9) P and P denotes the set complement of P in V . For example, in the case of P = {vi} in (7), LP = di ... |

144 |
Matrix Computations, 3rd edn
- Golub, Van
- 1996
(Show Context)
Citation Context ...nt. By l’Hôpital’s Rule, 2.4 Numerical practicalities lim r→∞ xsi = |T Ti| |T TG| = ns i . (15) Many good sources exist on the solution to large, sparse, symmetric, linear systems of equations (e.g., =-=[27, 28]-=-). A direct method, such as LU decomposition with partial pivoting has the advantage that the computation necessary to solve (6) is only negligibly increased over the amount of work required to solve ... |

140 | Learning segmentation by random walks
- Meila, Shi
(Show Context)
Citation Context ...ral methods are not guaranteed to produce a unique solution and, indeed, may lead to a fully degenerate problem [2]. One should not be confused by the random walker interpretation of spectral methods =-=[16]-=- and the present method. In the former, it is illustrated that spectral methods may be interpreted as finding a subset of the nodes with cardinality equal to half the nodes in the graph, such that a r... |

138 |
Laplacian Matrices of Graphs: A Survey, Linear Alg
- Merris
- 1994
(Show Context)
Citation Context ...e literature [20, 4] and a convenient form for the solution in the context that we are concerned with is given in [21]. We will now review the method of solution. Define the discrete Laplacian matrix =-=[22]-=- as Lvivj = ⎧ ⎪⎨ dvi if i = j, −wij if vi and vj are adjacent nodes, (2) ⎪⎩ 0 otherwise, where Lvivj is used to indicate that the matrix L is indexed by vertices vi and vj. Partition the vertices into... |

120 | Combining active learning and semisupervised learning using gaussian fields and harmonic functions
- Zhu, Ghahramani, et al.
- 2003
(Show Context)
Citation Context ...binary (i.e., 2-way) segmentation is obtained, without resorting to recursion or clustering on the spectral coefficients. Semi-automatic methods with a similarity to our proposed algorithm also exist =-=[17, 15]-=-. The K-way maxflow/min-cut algorithm of [17] attempts to find the cuts with smallest value (as defined by image gradients) that separate each labeled region from all others using K-way graph cuts. Al... |

102 | How good is recursive bisection
- Simon, Teng
- 1993
(Show Context)
Citation Context ...find the last entry in the full K-tuple. There exists a small literature that addresses the problem of multi-label image segmentation without resorting to recursively applied binary segmentation (see =-=[7]-=- for a discussion on some concerns about recursive bisection) using both automated [8–12] and semi-automated [13–15] methods. Existing automated methods operate by defining K-way clustering heuristics... |

92 |
Black Box Multigrid
- Dendy
- 1982
(Show Context)
Citation Context ...to a finite differences approach to solving the Dirichlet problem on a hypercube domain, the techniques of numerical solution to PDEs may also be applied. Most notably, the algebraic multigrid method =-=[32, 33]-=- achieves near-optimal performance for the solution to equations like (5). For simplicity, we have implemented the standard conjugate gradients algorithm with Jacobi preconditioning, representing the ... |

88 |
Algebraic potential theory on graphs
- Biggs
- 1997
(Show Context)
Citation Context ... The reader can verify that the probabilities at each node sum to unity (up to rounding).s2.2 Discrete Dirichlet problem The discrete Dirichlet problem has been discussed thoroughly in the literature =-=[20, 4]-=- and a convenient form for the solution in the context that we are concerned with is given in [21]. We will now review the method of solution. Define the discrete Laplacian matrix [22] as Lvivj = ⎧ ⎪⎨... |

86 |
Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird, Annalen der Physik und Chemie 72 no
- Kirchhoff
(Show Context)
Citation Context ...er inequality [24], E[A] = E[XB] ≤E[X]E[B]. By the same inequality, E[X] = E[ A B ] ≤ E[A]E[ 1 E[A] B ]. Therefore, E[B] ≤E[X] ≤ E[A]E[ 1 B ], and the result is proved. ⊓⊔ Since the time of Krichhoff =-=[25]-=-, it has been known that there is a relationship between the potentials solved for in (5) and the weighted tree structure of the graph. The following relationship for the potential at node vi in the p... |

74 | Spectral partitioning: The more eigenvectors, the better
- Alpert, Kahng, et al.
- 1999
(Show Context)
Citation Context ...east one node that is also labeled s. A block matrix form of (7) may be written LP x s P = −RP x s , (8) P where xs = [xs P , xs P ]T ,L has been decomposed into the block form � LP RP L = RT P L � , =-=(9)-=- P and P denotes the set complement of P in V . For example, in the case of P = {vi} in (7), LP = di and −RP x s P = � eij∈E w(eij)x s j . If xs P > xf P ∀f �= s, then xsP − xf P > 0 and −L−1 − xf P P... |

64 | Combinatorial Preconditioners for Sparse, Symmetric, Diagonally Dominant Linear Systems
- GREMBAN
- 1996
(Show Context)
Citation Context ...f memory is at a premium) and the operation LU x s U may be performed very cheaply. Furthermore, sparse matrix operations (like those required for conjugate gradients) may be efficiently parallelized =-=[30, 31]-=-, e.g., for use on a GPU. Because of the relationship of (5) to a finite differences approach to solving the Dirichlet problem on a hypercube domain, the techniques of numerical solution to PDEs may a... |

48 | Space variant image processing
- Wallace, Ong, et al.
- 1994
(Show Context)
Citation Context ... (e.g., a weight of zero means that the walker may not move along that edge). Formulation of the algorithm on a graph allows the application of the algorithm to surface meshes or space-variant images =-=[1, 2]-=-. Regardless of the dimensions of the data, we will use the term pixel throughout this paper to refer to the basic picture element in the context of its intensity values. In contrast, the term node wi... |

43 |
Special matrices and their applications in numerical mathematics, chapter Tensor Product of Matrices
- Fiedler
- 1986
(Show Context)
Citation Context ...e entries of RP are nonpositive by definition of L. Since L is an M-matrix, any block diagonal submatrix of an M-matrix is also an Mmatrix, and the inverse of an M-matrix has nonnegative entries (see =-=[23]-=- for the previous three facts), then −L −1 P R has nonnegative entries and therefore, some xsi ∈ P must be greater than xfi ∈ P . Furthermore, since the entries of RP are zero for nodes not connected ... |

34 |
An introduction to probability theory and its applications, Second Edition
- Feller
- 1957
(Show Context)
Citation Context .... ⊓⊔ P RP (x s Proof of the remaining propositions rest on following lemma Lemma 1. For random variables, X, A and B, such that X = A B , E[X]= 0 if E[A]= 0 and B > 0. Proof. By the Hölder inequality =-=[24]-=-, E[A] = E[XB] ≤E[X]E[B]. By the same inequality, E[X] = E[ A B ] ≤ E[A]E[ 1 E[A] B ]. Therefore, E[B] ≤E[X] ≤ E[A]E[ 1 B ], and the result is proved. ⊓⊔ Since the time of Krichhoff [25], it has been ... |

25 | Space-Variant Computer Vision: A Graph-Theoretic Approach
- Grady
- 2004
(Show Context)
Citation Context ... (e.g., a weight of zero means that the walker may not move along that edge). Formulation of the algorithm on a graph allows the application of the algorithm to surface meshes or space-variant images =-=[1, 2]-=-. Regardless of the dimensions of the data, we will use the term pixel throughout this paper to refer to the basic picture element in the context of its intensity values. In contrast, the term node wi... |

24 |
Applied Graph Theory: Graphs and Electrical Networks
- Chen
- 1976
(Show Context)
Citation Context ...potentials solved for in (5) and the weighted tree structure of the graph. The following relationship for the potential at node vi in the presence of unit voltage sources (tied to ground) is given in =-=[6, 26]-=- � � x s i = eij∈T T w(eij) T T ∈T Ti � � T T ∈T TG eij∈T T w(eij) , (10)swhere T Ti is the set of 2-trees present in the graph, such that node vi is connected to a seed with label s, and T TG is the ... |

19 |
Markoff process and the Dirichlet problem
- Kakutani
- 1945
(Show Context)
Citation Context ...s paper to refer to the basic picture element in the context of its intensity values. In contrast, the term node will be used in the context of a graph-theoretical discussion. It has been established =-=[3, 4]-=- that the probability a random walker first reaches a seed point exactly equals the solution to the Dirichlet problem [5] with boundary conditions at the locations of the seed points and the seed poin... |

17 |
Random Walks and Electric Networks. Number 22
- Doyle, Snell
- 1984
(Show Context)
Citation Context ...s paper to refer to the basic picture element in the context of its intensity values. In contrast, the term node will be used in the context of a graph-theoretical discussion. It has been established =-=[3, 4]-=- that the probability a random walker first reaches a seed point exactly equals the solution to the Dirichlet problem [5] with boundary conditions at the locations of the seed points and the seed poin... |

16 | A new algorithm for energy minimization with discontinuities
- Boykov, Veksler, et al.
- 1999
(Show Context)
Citation Context ...binary (i.e., 2-way) segmentation is obtained, without resorting to recursion or clustering on the spectral coefficients. Semi-automatic methods with a similarity to our proposed algorithm also exist =-=[17, 15]-=-. The K-way maxflow/min-cut algorithm of [17] attempts to find the cuts with smallest value (as defined by image gradients) that separate each labeled region from all others using K-way graph cuts. Al... |

13 | Anisotropic interpolation on graphs: The combinatorial dirichlet problem
- Grady, Schwartz
- 2003
(Show Context)
Citation Context ...e Dirichlet problem The discrete Dirichlet problem has been discussed thoroughly in the literature [20, 4] and a convenient form for the solution in the context that we are concerned with is given in =-=[21]-=-. We will now review the method of solution. Define the discrete Laplacian matrix [22] as Lvivj = ⎧ ⎪⎨ dvi if i = j, −wij if vi and vj are adjacent nodes, (2) ⎪⎩ 0 otherwise, where Lvivj is used to in... |

10 |
Matrix-Based Multigrid: Theory and Applications
- Shapira
- 2003
(Show Context)
Citation Context ...to a finite differences approach to solving the Dirichlet problem on a hypercube domain, the techniques of numerical solution to PDEs may also be applied. Most notably, the algebraic multigrid method =-=[32, 33]-=- achieves near-optimal performance for the solution to equations like (5). For simplicity, we have implemented the standard conjugate gradients algorithm with Jacobi preconditioning, representing the ... |

8 | Learning with equivalence constraints, and the relation to multiclass learning
- Bar-Hillel, Weinshall
- 2003
(Show Context)
Citation Context ... complement of T Ti in T TG as T TC, such that T Ti ∪ T TC = T TG and T Ti ∩ T TC = ∅. For brevity of notation, denote ST Ti = � � T T ∈T Ti eij∈T T w(eij). E[x s i − n s � � ST Ti |T Ti| i ] = E − . =-=(12)-=- |T Ti| + |T TC| ST Ti + ST TC Since each of the 2-trees will contain an equal number of edges, (n − 2), and all of the weights are identically distributed, ST Ti will contain the sum of |T Ti| identi... |

4 |
Methods of mathematical physics, volume 2
- ourant, Hilbert
- 1991
(Show Context)
Citation Context ...d in the context of a graph-theoretical discussion. It has been established [3, 4] that the probability a random walker first reaches a seed point exactly equals the solution to the Dirichlet problem =-=[5]-=- with boundary conditions at the locations of the seed points and the seed point in question fixed to unity, while the others are set to zero. The solution to the discrete Dirichlet problem on an arbi... |

3 |
Algebraic Graph Theory. Number 67
- Biggs
- 1974
(Show Context)
Citation Context ...ulate the probability that a random walker starting at pixel vi first reaches a seed with label s, by solving the circuit theory problem that corresponds to a discrete analog of the Dirichlet problem =-=[6]-=-. Ground (i.e., fix the potential to zero) all seed points belonging to labels other than s and establish a unit voltage source with ground that fixes the s-labeled seeds to have a unit potential. The... |