P. Olver, G. Sapiro, and A. Tannenbaum, "Invariant geometric evolutions of surfaces and volumetric smoothing", SIAM J. Applied Math. 57 (1997), pp. 176-194.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....by evolving in the normal direction with speed equal to the mean curvature. However, the 2D results regarding the behavior of non convex level curves do not generalize to 3D. Under mean curvature flow, non convex surfaces can develop singularities during evolution or even change their topology [12]. 3 Blob characterization by extremal paths Local minima or maxima of the image intensity function generated by the nonlinear evolution process correspond to level curves which are contracted to a single point, whereas the attributed intensities define the gray value of the corresponding level ....

Peter.J. Olver, Guillermo Sapiro, and Allen Tannenbaum. Invariant Geometric Evolutions of Surfaces and Volumetric Smoothing. Technical Report LIDS-P-2238, Dept. of Mathematics, Minneapolis / MIT, 1994.

....of curves and surfaces can be used to remove noise and enhance the quality of digital images. Much better results are obtained when this evolution is performed in an invariant, geometric manner. Such evolutions can be described by group invariant evolutionary differential equations [Faugeras94] [Olver Sapiro Tannenbaum97]. ....

Olver, P.J., Sapiro, G., and Tannenbaum, A., Invariant geometric evolutions of surfaces and volumetric smoothing, SIAM J. Appl. Math. 57 (1997) 176-194.

....to solving the heat flow equation with certain boundary conditions. In this case, however, we do not keep the solution fixed at the surface, but allow points to travel along the surface. This is a dual approach to the geometry driven curve and surface diffusion by Olver, Sapiro, and Tannenbaum [8] and others. We keep only the tangential part of the diffusion along the surface whereas they diffuse the geometry of the surface maintaining only the normal flow. The surface normal nS2 may simply be obtained as a length normalization of nS1 JnS1 where J is the Jacobean of D. In this way the ....

P. J. Olver, G. Sapiro, and A. Tannenbaum. Invariant geometric evolutions of surfaces and volumetric smoothing. SIAM Journal on Applied Mathematics, 57(1):176-- 194, 1997.

....image. Catte et al. 6] Nitzberg and Mumford [26] and Alvarez et al. 2] recognized the ill posedness of the Perona Malik diffusion and proposed modifications to overcome the same. Since then, several nonlinear diffusion methods have been developed and a good account of these can be found in [27, 30, 33]. In [17] Kimia et al. proposed an elegant reaction diffusion based theory which describes the shapes of objects in an entropy scale space . This theory was later used by Tek and Kimia [31] for image segmentation applications. Image segmentation can also be achieved by approaches based on curve ....

P. J. Olver, G. Sapiro, and A. Tannenbaum. Invariant geometric evolutions of surfaces and volumetric smoothing. Preprint, 1995.

.... only on linear scale spaces, but later many non linear and geometric scale space methods were also developed (for example, the anisotropic diffusion proposed by Perona et al. 4, 5] the level set method described in Osher Sethian [6] and Olver et al. s work based on differential invariants [7]. See also Brakke [8] Gage Hamilton [9] Alvarez Lions Morel [10] etc. Descriptions of a shape at different scales are obtained by continuously deforming it to smoother ones. Ideally, this deformation process should be causal [1] in the sense that it should maintain a hierachical structure of ....

Olver, P., Sapiro, G., and Tannenbaum, A., "Invariant geometric evolutions of surfaces and volumetric smoothing", SIAM J. Appl. Math., 57, pp.176-194, 1997.

....highly interesting. Among its applications, the study and classification of differential invariants and invariant evolutions of curves and surfaces under certain groups (Euclidean, affine, projective and conformal) has become relevant in the subject of image enhancing and image processing (see [18] and references within) The classification of differential invariants of reparametrizations of R n or parametrized submanifolds is directly related to the classification of invariant cocycles of some infinite dimensional Lie algebras, which is itself related to quantization. There is, of ....

Olver, P.J., Sapiro, G., and and Tannenbaum, A., Invariant geometric evolutions of surfaces and volumetric smoothing, SIAM J. Appl. Math. 57, (1997), pp. 176--194 .

....energy function is also transformed; the normal directions of C can be quite different from those of A(C) But once the surface energy is so transformed, then provided the mobility is also transformed, the motion which is affine invariant uses Phi , not 1=3 Phi . As surveyed and extended in [OST] the [ST] result is a special case of motions of the form where the time derivative in the normal direction is equal to the second spatial derivative in a metric which is invariant under some group action. The interpretation of this paper in those terms is that one has a metric defined only on ....

Peter J. Olver, Guillermo Sapiro, and Allen Tannenbaum, Invariant geometric evolution of surfaces and volumetric smoothing, SIAM J. Appl. Math. 57 (1997), 176-194.

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P. Olver, G. Sapiro, and A. Tannenbaum, "Invariant geometric evolutions of surfaces and volumetric smoothing", SIAM J. Applied Math. 57 (1997), pp. 176-194.

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P. J. Olver, G. Sapiro, and A. Tannenbaum, "Invariant geometric evolutions of surfaces and volumetric smoothing," SIAM J. of Appl. Math., to appear.

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Olver, P.J., Sapiro, G., and Tannenbaum, A., Invariant geometric evolutions of surfaces and volumetric smoothing, SIAM J. Appl. Math. 57 (1997), 176--194.

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Olver, P.J., Sapiro, G., and Tannenbaum, A., Invariant geometric evolutions of surfaces and volumetric smoothing, SIAM J. Appl. Math., to appear.

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Olver, P.J., Sapiro, G., and Tannenbaum, A., Invariant geometric evolutions of surfaces and volumetric smoothing, SIAM J. Appl. Math., to appear.

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P. J. Olver, G. Sapiro, and A. Tannenbaum, "Invariant geometric evolutions of surfaces and volumetric smoothing," SIAM J. of Appl. Math., February 1997.

....sp ecial, de similitudes et projectif, alors le flux suivant est le seul invariant d ordre minimal: o u c est une constante. La premi ere partie du Th eor eme 1 ne n ecessite pas que G soit un sous groupe de SL(R; 3) Cette condition est n ecessaire pour la deuxi eme partie (voir [10] pour extension) La classification peut etre construite pour d autres groupes, mais les relations seront plus compliqu ees. L unicit e de l equation de la chaleur g eom etrique pour le groupe Euclidien et le groupe affine, est d emontr ee dans [1] avec une autre m ethodologie et sous certaines ....

....invariant arc lengths, invariant evolution equations, etc. for any group of transformations in the plane, but the interconnections are more complicated. See Lie [6] and Olver [8] for the details of the complete classification of all groups in the plane and their differential invariants. See [10] for extensions of invariant flows to other groups and dimensions. The uniqueness of the Euclidean and affine heat flows (see [11, 12] was also proven in [1] using a completely different approach, and based on a number of requirements, whereas we only need to impose the conditions of ....

[Article contains additional citation context not shown here]

P. J. Olver, G. Sapiro, and A. Tannenbaum, "Invariant geometric evolutions of surfaces and volumetric smoothing," MIT Report -- LIDS, April 1994.

....extending the earlier classification theorems, 19] and then discuss recent applications to the study of invariant evolution equations, which is of great interest in image processing April 8, 1994 Supported in part by NSF Grants DMS 91 16672 and DMS 92 04192. and computer vision, cf. 21] [22]. Space considerations preclude the inclusion of proofs and significant examples here. I shall assume the reader is familiar with the fundamentals of the Lie theory of symmetry groups of differential equations, as discussed, for instance in my book, 18] I shall employ the same basic notation ....

....in his study of integrability and solitons. The classification has received added impetus from recent work, done in collaboration with G. Sapiro and A. Tannenbaum, on applications to computer vision and image processing, including connections with geometric curve shortening flows; see [23] 21] [22]. Let G be a transformation group acting on M ae X Theta U , and consider a scalar evolution equation u t = K(x;u ) 15) in which t is an additional independent variable (the time) and the right hand side depends only on the spatial (x) derivatives of u. The group action is extended to M ....

[Article contains additional citation context not shown here]

Olver, P.J., Sapiro, G., and Tannenbaum, A., Invariant geometric evolutions of surfaces and volumetric smoothing, preprint, University of Minnesota, 1994.

.... initiated; see also [1] In particular, using the methods of [1, 31] a shape can be smoothed in an affine invariant manner before the computation of invariant descriptors such as those reported in corresponding chapters in [21] This work was partially extended for other groups and dimensions in [10, 24, 25, 32]. The purpose of this paper is to derive simple geometric object detectors which incorporate affine in This work was partially supported by the National Science Foundation ECS 91 22106, DMS 92 04192 and DMS 95 00931, by the Air Force Office of Scientific Research F49620 941 00S8DEF, by ....

....segmentation algorithm. The work of Lindeberg is related to our definition of affine gradient, as will be explained in Section 3. 2 Affine edges from affine scale space We begin by deriving the first affine invariant edge detector. It is based on the theory of invariant scalespaces developed in [1, 24, 25, 30, 31, 33]. We first introduce some preliminary notation. For planar column vectors, X = x 1 ; x 2 ) Y = y 1 ; y 2 ) 2 IR , we let [X; Y ] x 1 y 2 Gamma x 2 y 1 be the area of the parallelogram spanned by X;Y . We also define Y : Gammay 2 ; y 1 ) by [X; Y ] hX; Y i; where hX; ....

[Article contains additional citation context not shown here]

P. J. Olver, G. Sapiro, and A. Tannenbaum, "Invariant geometric evolutions of surfaces and volumetric smoothing," SIAM J. of Appl. Math., to appear.

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Olver, P.J., Sapiro, G., and Tannenbaum, A. ---. Invariant geometric evolutions of surfaces and volumetric smoothing. SIAM J. of Appl. Math., (to appear). Osher, S.J. and Sethian, J.A. 1988. Fronts propagation with curvature dependent speed: Algorithms based on HamiltonJacobi formulations. Journal of Computational Physics, 79:12-- 49.

.... prefer to propagate in the curvature direction (see, e.g. 31] This model was heuristically justified in [4] 22] It can be described as the composition of: 1) A smoothing term: Twice the mean curvature in the case of (1) More efficient smoothing velocities as those proposed in [2] 7] [24] can be used instead of H. 4 Note again that unlike classical energy based models, this component of the deforming surface is intrinsic. 2) A constant balloon type force (n u ) Similar to the energy based models, this term is necessary in this case for the detection of nonconvex objects. 3) ....

....contours, leading to accurate numerical implementations and topology free object segmentation. Furthermore, the following results can be proved for this flow: 4. Although curvature flows smooth 2D curves [14] 15] 28] a 3D geometric flow that smoothes all possible surfaces was not found [24]. Frequently used are mean curvature or the positive part of the Gaussian curvature flows [2] 7] 396 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 19, NO. 4, APRIL 1997 THEOREM 1. 6] Assume that g # 0 is sufficiently smooth. Then, for any Lipschitz initial condition ....

P.J. Olver, G. Sapiro, and A. Tannenbaum, "Invariant Geometric Evolutions of Surfaces and Volumetric Smoothing," SIAM J. Applied Math., forthcoming.

....of the normalizations on the horizontal components of the moving coframe forms during the computation. The moving coframe itself will also include invariant contact forms, which vanish upon restriction, but which, nevertheless, play an important role in other aspects of the geometry. See [40] [42], 21] for applications of invariant contact forms to the study of invariant evolution equations, with applications to image processing. Applications to the computation of the invariant cohomology of the variational bicomplex, cf. 2] are also of particular importance in the analysis of ....

Olver, P.J., Sapiro, G., and Tannenbaum, A., Invariant geometric evolutions of surfaces and volumetric smoothing, SIAM J. Appl. Math. 57 (1997), 176--194.

....et projectif, alors le flux suivant est le seul invariant d ordre minimal: u t = c g 2 2 u x 2 ; o u c est une constante. La premi ere partie du Th eor eme 1 ne n ecessite pas que G soit un sous groupe de SL(R; 3) Cette condition est n ecessaire pour la deuxi eme partie (voir [10] pour extension) La classification peut etre construite pour d autres groupes, mais les relations seront plus compliqu ees. L unicit e de l equation de la chaleur g eom etrique pour le groupe Euclidien et le groupe affine, est d emontr ee dans [1] avec une autre m ethodologie et sous certaines ....

....invariant arc lengths, invariant evolution equations, etc. for any group of transformations in the plane, but the interconnections are more complicated. See Lie [6] and Olver [8] for the details of the complete classification of all groups in the plane and their differential invariants. See [10] for extensions of invariant flows to other groups and dimensions. The uniqueness of the Euclidean and affine heat flows (see [11, 12] was also proven in [1] using a completely different approach, and based on a number of requirements, whereas we only need to impose the conditions of simplicity ....

[Article contains additional citation context not shown here]

P. J. Olver, G. Sapiro, and A. Tannenbaum, "Invariant geometric evolutions of surfaces and volumetric smoothing," MIT Report -- LIDS, April 1994.

.... geometric diffusion based smoothing, 40] which, in the Euclidean case, means the Euclidean curve shortening flow, 43] and, in the affine case, means affine curve shortening, 47] See [17] 39] 40] for generalizations of these flows to other subgroups of the projective group, and [41] for further generalizations to arbitrary transformation groups in arbitrary dimensions. Noisy data would produce high rates of change of curvature, and hence spurious outlying portions of the signature curve. One can use various statistical approaches to replace a noisy discrete signature curve ....

Olver, P.J., Sapiro, G., and Tannenbaum, A., Invariant geometric evolutions of surfaces and volumetric smoothing, SIAM J. Appl. Math., to appear.

....need to select 6= 0 even if the surface is close to the object s boundary. This model was heuristically justified in [4, 22] It can be described as the composition of: 1. A smoothing term: Twice the mean curvature in the case of (1) More efficient smoothing velocities as those proposed in [2, 7, 24] can be used instead of H. 4 Note again that in contrast with classical energy based models, this component of the deforming surface is intrinsic. 2. A constant balloon type force (jruj) This is exactly as in the energy based models, and is fundamental to detect non convex objects here. 3. A ....

.... based and curve evolution based models are mathematically related to the minimization of a weighted length of the form R g(I)ds, where g(I) is a decreasing function 4 Although curvature flows smooth 2D curves [14, 15, 28] a 3D geometric flow that smoothes all possible surfaces was not found [24]. Frequently used are mean curvature or the positive part of the Gaussian curvature flows [2, 7] of the image gradient as before and s is the Euclidean arc length. This idea can be extended to 3D. In this case, length is replaced by surface area A : R R da; and weighted length by weighted ....

P. J. Olver, G. Sapiro, and A. Tannenbaum, "Invariant geometric evolutions of surfaces and volumetric smoothing," SIAM J. of Appl. Math., to appear.

....the earlier classification theorems, 19] and then discuss recent applications to the study of invariant evolution equations, which is of great interest in image processing April 8, 1994 y Supported in part by NSF Grants DMS 91 16672 and DMS 92 04192. and computer vision, cf. 21] [22]. Space considerations preclude the inclusion of proofs and significant examples here. I shall assume the reader is familiar with the fundamentals of the Lie theory of symmetry groups of differential equations, as discussed, for instance in my book, 18] I shall employ the same basic notation ....

....in his study of integrability and solitons. The classification has received added impetus from recent work, done in collaboration with G. Sapiro and A. Tannenbaum, on applications to computer vision and image processing, including connections with geometric curve shortening flows; see [23] 21] [22]. Let G be a transformation group acting on M ae X Theta U , and consider a scalar evolution equation u t = K(x;u (n) 15) in which t is an additional independent variable (the time) and the right hand side depends only on the spatial (x) derivatives of u. The group action is extended to M ....

[Article contains additional citation context not shown here]

Olver, P.J., Sapiro, G., and Tannenbaum, A., Invariant geometric evolutions of surfaces and volumetric smoothing, preprint, University of Minnesota, 1994.

No context found.

P. J. Olver, G. Sapiro, and A. Tannenbaum. Invariant geometric evolutions of surfaces and volumetric smoothing. SIAM Journal on Applied Mathematics, 57(1):176--194, 1997.

No context found.

P. J. Olver, G. Sapiro, and A. Tannenbaum. Invariant geometric evolutions of surfaces and volumetric smoothing. J. Diff. Geom., 22:117--138, 1985.

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