## Stochastic Differential Equations and Geometric Flows

Venue: | IEEE TRANSACTIONS ON IMAGE PROCESSING |

Citations: | 10 - 1 self |

### BibTeX

@ARTICLE{Unal_stochasticdifferential,

author = {Gozde Unal and Hamid Krim and Anthony Yezzi},

title = {Stochastic Differential Equations and Geometric Flows},

journal = {IEEE TRANSACTIONS ON IMAGE PROCESSING},

year = {},

volume = {11},

pages = {1405--1417}

}

### OpenURL

### Abstract

In recent years curve evolution, applied to a single contour or to the level sets of an image via partial differential equations, has emerged as an important tool in image processing and computer vision. Curve evolution techniques have been utilized in problems such as image smoothing, segmentation, and shape analysis. We give a local stochastic interpretation of the basic curve smoothing equation, the so called geometric heat equation, and show that this evolution amounts to a tangential diffusion movement of the particles along the contour. Moreover, assuming that a priori information about the shapes of objects in an image is known, we present modifications of the geometric heat equation designed to preserve certain features in these shapes while removing noise. We also show how these new flows may be applied to smooth noisy curves without destroying their larger scale features, in contrast to the original geometric heat flow which tends to circularize any closed curve.

### Citations

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Citation Context ...isely these insights that led us to the generalizations presented in Section IV. Remark: Another popular approach to anisotropic diffusion is based upon models first introduced by Perona and Malik in =-=[23]-=-. Since then, these models have received a tremendous amount of attention, as have the models (4)sbased upon curve evolution theory. Perona and Malik extended the linear heat equation by considering d... |

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Citation Context ...n, and shape analysis, but have also been considered for their simultaneous action on the level sets of an image in a number of geometrically based anisotropic smoothing algorithms. Osher and Sethian =-=[1, 2]-=- extended this latter perspective to the treatment of individual curves through a set of algorithms, known as level set methods, which enable the implementation of curve and surface evolution on a fix... |

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Citation Context ...he geometric heat equation which corresponds to the simplest case of this class with � � Ø� Ü � � �Ø� �Ü , has been shown to be well-posed, and its existence and uniqueness propert=-=ies may be found in [9, 28, 29]. The operator of the geometric heat equation is given by where Ä�Ù℄ � ��-=-� ��� ��� Ù Ü� Ü� Ä�Ù℄ �Ä�Ù℄sÙ Ø 13 � (17) �×�Ò �ÙÜÜs×�Ò � Ó× �ÙÜÝ Ó× �ÙÝÝ� (18) is the principal part of the operator Ä. The mat... |

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Citation Context ...icroscopic process of an evolution of a pixel or a point. The dynamics of this evolution at a macroscopic level are captured by a PDE, also referred to as a generator (infinitesimal) of the diffusion =-=[13, 14, 24]. Suppose w-=-e want to describe the motion of a small particle suspended in a moving liquid, subject to random molecular bombardments. If � Ø� Ü Ê Ò is the velocity of the fluid at a point Ü Ê Ò and tim... |

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Citation Context ...tated theorem, and to intuitively illustrate our flows, we next present simulation results. In our experiments with contours, we use the narrowband implementation of the level set method developed in =-=[32]. The time step is, ÆØ � -=-� . Starting with a circular shape, the flow �Ø � � � �Æ evolves it towards a specific polygon, i.e. it produces an Òs�ÓÒ� shape depending on the specific function � � . Sever... |

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Citation Context ...ation, which diffuses isotropically, the following anisotropic model, which diffuses along the boundaries of image features but not across them, ÙØ � Ù�� � ÙÝÙÜÜsÙÜÙÝÙÜÝ ÙÜÙ�=-=�Ý � (2) ÙÜ ÙÝ is obtained [9]. -=-We may obtain this same equation in a completely different and much more geometric manner by specifying the evolution of each level curve in the image. Let � denote a particular iso-intensity contou... |

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Citation Context ...in a curve from finer to coarser scales as the evolution proceeds. An affine invariant scale space can be obtained from a related curvature flow which depends upon the cube root of the curvature (see =-=[8, 11, 12]-=-). When applied to the level sets of an image, these flows have a powerful denoising effect when run for a short amount of time. If run for too long, however, even large scale features will be destroy... |

229 |
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Citation Context ...for treating certain types of singularities such as shocks and topological transitions [1, 3]. Much of the research in curve evolution theory has centered around the so called geometric heat equation =-=[4] i-=-n which a curve is evolved along the normal direction in proportion to its signed curvature. This flow is well known for its smoothing properties [5–7] and the fact that it corresponds to the gradie... |

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Citation Context ...he geometric heat equation which corresponds to the simplest case of this class with � � Ø� Ü � � �Ø� �Ü , has been shown to be well-posed, and its existence and uniqueness propert=-=ies may be found in [9, 28, 29]. The operator of the geometric heat equation is given by where Ä�Ù℄ � ��-=-� ��� ��� Ù Ü� Ü� Ä�Ù℄ �Ä�Ù℄sÙ Ø 13 � (17) �×�Ò �ÙÜÜs×�Ò � Ó× �ÙÜÝ Ó× �ÙÝÝ� (18) is the principal part of the operator Ä. The mat... |

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Citation Context ...n, and shape analysis, but have also been considered for their simultaneous action on the level sets of an image in a number of geometrically based anisotropic smoothing algorithms. Osher and Sethian =-=[1, 2]-=- extended this latter perspective to the treatment of individual curves through a set of algorithms, known as level set methods, which enable the implementation of curve and surface evolution on a fix... |

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Citation Context ...rves that link occluded edges. For a more general situation, e.g. curves in Ê , Mumford used other sorts of stochastic processes such as an Uhlenbeck process to find the elastica. Williams and Jacobs=-= [26], later -=-in their “Stochastic Completion Fields” work, define the same SDE as Mumford’s, for a particle’s position and the orientation, and through this model of diffusion incorporate the prior assumpt... |

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Citation Context ... VOL. 11, NO. 12, DECEMBER 2002 a dual perspective to our contour-based approach to shape representation, skeletonization approaches may also allow shape analysis without displacement of corners [7], =-=[17]-=-–[21]. In this paper, we develop a new class of curve evolutions, which are obtained by a modification of the geometric heat equation. Given an initial shape in the form of a continuous curve, the cla... |

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Citation Context ... VI. II. BACKGROUND AND FORMULATION It is known that the low pass Gaussian filter from signal processing can be implemented by evolving the intensities of an image Ù Ü� Ý via the linear heat equa=-=tion [10], Ù �Ü�Ý � Ù Ü� Ý � ÙØ Ø� Ü-=-� Ý � Ö¡sÖÙ Ø� Ü� Ý ¡ � Ø� (1) 3swhere the gradient operator, Ö, and the divergence operator, Ö¡, involve only the spatial variables Ü and Ý. The solution to this equation y... |

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Citation Context ...umber of researchers in pushing the application of curve evolution to new limits by providing a simple framework for treating certain types of singularities such as shocks and topological transitions =-=[1, 3]-=-. Much of the research in curve evolution theory has centered around the so called geometric heat equation [4] in which a curve is evolved along the normal direction in proportion to its signed curvat... |

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67 |
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Citation Context ...ns we will obtain, deform it into a pre-specified final polygonal shape. The problem of deforming an input shape into a different form has been of interest in various fields such as computer graphics =-=[22]-=-. The contents of this paper are outlined as follows. In Section II, we review some theoretical concepts associated with the curve shortening flow, including its connection to a nonlinear, directional... |

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Citation Context ...xists an operator � such that Ø � �� Ø� � Ü Ê � � Ü � � Ü � Ü Ê � SDEs and stochastic processes, most commonly the Brownian motion, have previously been used in curve=-= and image analysis. Mumford [25]-=- used it to model completion curves of occluded edges, the so-called elastica. By taking the curvature function (of arc length) as a Gaussian process, and the tangent direction on the curve then as a ... |

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Citation Context ... Proof: Assume that a family of curves � Ø� Ô ,whereÔ is any parameter along the curve, evolves according to the evolution equation The evolution equation for the angle of the unit normal is gi=-=ven in [5] as � Ø � « Ø� Ô Ì ¬ Ø� Ô Æ (21) � Ø �s� �¬�-=-�s«��℄ Õ where � � ���Ô�� � �Ô �Ô is the length along the curve (metric). If we consider the case « � and ¬ �s� � � (following the convention used by the aut... |

46 |
Partial differential equations
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Citation Context ...e with the eigen values 1 and 0. If we multiply this matrix by a positive function, it remains positive semi-definite. Such elliptic-parabolic operators satisfy a maximum principle (see, for example, =-=[30]). In our case we mult-=-iply by a non-negative function � � which can be made strictly positive by adding a very small number, ¯� , �� � ¯℄ÄÙ � � This results in a family of nonlinear parabolic equation... |

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Citation Context ...ulation. Similarly, a Kalman filter which produces estimates of a system as it evolves in time and affected by noise, (which is indeed an SDE written for the system and its observations), was used in =-=[27]-=- for grouping of contour segments. Our use of SDEs is along a different line of thought in that our inspiration starts with a desired effect of a nonlinear filter. Specifically, the theory of SDEs pro... |

21 |
Affine invariant scale space
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Citation Context ...in a curve from finer to coarser scales as the evolution proceeds. An affine invariant scale space can be obtained from a related curvature flow which depends upon the cube root of the curvature (see =-=[8, 11, 12]-=-). When applied to the level sets of an image, these flows have a powerful denoising effect when run for a short amount of time. If run for too long, however, even large scale features will be destroy... |

21 | M-reps: A new object representation for graphics - Pizer, Thall, et al. |

19 | The geometry of Wulff crystal shapes and its relations with Riemann problems
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Citation Context ... points � and � . This, in conjunction with the above lemma completes the proof of the theorem. 1 Note added in Proof: We recently found out per Dr. Osher at UCLA that Peng, Osher, Merriman and Zh=-=ao, [31]-=-, have also independently proposed flows similar to those described in this paper, albeit from a totally different perspective, and with a convective trend. 15sA. Examples in Polygonization V. EXPERIM... |

17 | Shapes, shocks, and deformations - Kimia, Tannenbaum, et al. - 1995 |

16 |
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Citation Context ...of curvature-based flows and anisotropic diffusion filters which preserve desired features in the shape of an object. Under these new flows, evolving curves take 2sthe limiting form of a polygon (see =-=[15]-=- for evolutions of polygons related to the geometric and affine geometric heat flows, and [16] for evolutions of polygons globally through an electric field concept). The resulting diffusion models ma... |

10 |
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Citation Context ...ermore, it is not well understood how these curvature-based filters are affected by different noise distributions and when this sort of problem may occur. To the best of our knowledge, and aside from =-=[13, 14]-=-, nonlinear diffusion in the previous literature was discussed from a purely deterministic perspective. In this paper we provide a stochastic formulation of the geometric heat equation and use the res... |

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5 |
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Citation Context ...the shape of an object. Under these new flows, evolving curves take 2sthe limiting form of a polygon (see [15] for evolutions of polygons related to the geometric and affine geometric heat flows, and =-=[16]-=- for evolutions of polygons globally through an electric field concept). The resulting diffusion models may therefore be applied for much longer periods of time without distorting the shapes of polygo... |

3 |
A stochastic approach to signal denoising
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Citation Context ...ermore, it is not well understood how these curvature-based filters are affected by different noise distributions and when this sort of problem may occur. To the best of our knowledge, and aside from =-=[13, 14]-=-, nonlinear diffusion in the previous literature was discussed from a purely deterministic perspective. In this paper we provide a stochastic formulation of the geometric heat equation and use the res... |

3 |
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Citation Context ...s of curvature-based flows and anisotropic diffusion filters which preserve desired features in the shape of an object. Under these new flows, evolving curves take the limiting form of a polygon (see =-=[15]-=- for evolutions of polygons related to the geometric and affine geometric heat flows, and [16] for evolutions of polygons globally through an electric field concept). The resulting diffusion models ma... |

2 |
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Citation Context ...in a curve from finer to coarser scales as the evolution proceeds. An affine invariant scale space can be obtained from a related curvature flow which depends upon the cube root of the curvature (see =-=[8, 11, 12]-=-). When applied to the level sets of an image, these flows have a powerful denoising effect when run for a short amount of time. If run for too long, however, even large scale features will be destroy... |

1 |
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Citation Context ...from finer to coarser scales as the evolution proceeds. An affine invariant scale space can be obtained from a related curvature flow which depends upon the cube root of the curvature (see [8], [11], =-=[12]-=-). When applied to the level sets of an image, these flows have a powerful denoising effect when run for a short amount of time. If run for too long, however, even large scale features will be destroy... |

1 |
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(Show Context)
Citation Context ...ermore, it is not well understood how these curvature-based filters are affected by different noise distributions and when this sort of problem may occur. To the best of our knowledge, and aside from =-=[13]-=-, [14], nonlinear diffusion in the previous literature was discussed from a purely deterministic perspective. In this paper we provide a stochastic formulation of the geometric heat equation and use t... |

1 |
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(Show Context)
Citation Context ..., it is not well understood how these curvature-based filters are affected by different noise distributions and when this sort of problem may occur. To the best of our knowledge, and aside from [13], =-=[14]-=-, nonlinear diffusion in the previous literature was discussed from a purely deterministic perspective. In this paper we provide a stochastic formulation of the geometric heat equation and use the res... |

1 |
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- 1999
(Show Context)
Citation Context ... 11, NO. 12, DECEMBER 2002 a dual perspective to our contour-based approach to shape representation, skeletonization approaches may also allow shape analysis without displacement of corners [7], [17]–=-=[21]-=-. In this paper, we develop a new class of curve evolutions, which are obtained by a modification of the geometric heat equation. Given an initial shape in the form of a continuous curve, the class of... |

1 |
Warping and morphing of graphical objects, course notes
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(Show Context)
Citation Context ...ns we will obtain, deform it into a pre-specified final polygonal shape. The problem of deforming an input shape into a different form has been of interest in various fields such as computer graphics =-=[22]-=-. The contents of this paper are outlined as follows. In Section II, we review some theoretical concepts associated with the curve shortening flow, including its connection to a nonlinear, directional... |