Shape offsets via level sets

Kimmel, Ron; Bruckstein, Alfred M.

doi:10.1016/0010-4485(93)90040-u

Cited by 67 publications

(41 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The constant velocity νN in (6), which is related with classical mathematical morphology [30,54] and shape offsetting in CAD [31], acts as the balloon force in [12]. Actually this velocity pushes the curve inwards and it is crucial in the model in order to allow convex initial curves to become non-convex, and thereby detect non-convex objects.…”

Section: Deformable Models Based On Mean Curvature Motionmentioning

confidence: 99%

“…On each region R i we consider the problem: (31) where T i will be specified later. We want to construct a family of subsolutions of (31) which becomes positive as t → ∞.…”

Section: Lemmamentioning

confidence: 99%

“…We want to construct a family of subsolutions of (31) which becomes positive as t → ∞. Obviously u(t, x ), the viscosity solution of (26) given by Theorem 1, is a supersolution of the problem (31), and by (30) u 0 is a subsolution of (26). We shall use the following comparison principle [15].…”

Section: Lemmamentioning

confidence: 99%

See 2 more Smart Citations

Minimal surfaces: a geometric three dimensional segmentation approach

Caselles

Kimmel

Sapiro

et al. 1997

Numerische Mathematik

View full text Add to dashboard Cite

Summary.A novel geometric approach for three dimensional object segmentation is presented. The scheme is based on geometric deformable surfaces moving towards the objects to be detected. We show that this model is related to the computation of surfaces of minimal area (local minimal surfaces). The space where these surfaces are computed is induced from the three dimensional image in which the objects are to be detected. The general approach also shows the relation between classical deformable surfaces obtained via energy minimization and geometric ones derived from curvature flows in the surface evolution framework. The scheme is stable, robust, and automatically handles changes in the surface topology during the deformation. Results related to existence, uniqueness, stability, and correctness of the solution to this geometric deformable model are presented as well. Based on an efficient numerical algorithm for surface evolution, we present a number of examples of object detection in real and synthetic images.

show abstract

Section: Deformable Models Based On Mean Curvature Motionmentioning

confidence: 99%

“…On each region R i we consider the problem: (31) where T i will be specified later. We want to construct a family of subsolutions of (31) which becomes positive as t → ∞.…”

Section: Lemmamentioning

confidence: 99%

See 1 more Smart Citation

Minimal surfaces: a geometric three dimensional segmentation approach

Caselles

Kimmel

Sapiro

et al. 1997

Numerische Mathematik

View full text Add to dashboard Cite

show abstract

“…We consider a volume bounded by a NURBS surface of degree (2,3) given by a control net of 4 × 6 control points, a plane and two canal surfaces (see Fig. 10 (left), the single patches of the NURBS surface are distinguished by color; the visible parts of the canal surfaces are displayed without parametric lines to improve contrast).…”

Section: Examplementioning

confidence: 99%

“…Hence, shape (curve, surface or volume) offsets and corresponding algorithms have been widely studied in Computer-Aided Design and Manufacturing, see [1,2,3,4,5,6,7,8] and references therein for more details. However, free-form NURBS shapes do not possess rational offsets in general and thus suitable approximation techniques, often based on rational curves or surfaces of relatively low degree, are used -see e.g.…”

Section: Introductionmentioning

confidence: 99%

Volumes with piecewise quadratic medial surface transforms: Computation of boundaries and trimmed offsets

Bastl

Jüttler

Kosinka

et al. 2010

Computer-Aided Design

View full text Add to dashboard Cite

MOS surfaces are rational surfaces in R 3,1 which provide rational envelopes of the associated two-parameter family of spheres. Moreover, all the offsets admit rational parameterizations as well. Recently, it has been proved that quadratic triangular Bézier patches in R 3,1 are MOS surfaces. Following this result, we describe an algorithm for computing an exact rational envelope of a two-parameter family of spheres given by a quadratic patch in R 3,1 . The main focus of this paper is given to geometric aspects of the algorithm. Since these patches are capable of producing C 1 smooth approximations of medial surface transforms of spatial domains, we use this algorithm to generate rational approximations of envelopes of general medial surface transforms. One of the main advantages of this approach to offsetting is the fact that the trimming procedure becomes considerably simpler.

show abstract

Eikonal equation‐based front propagation for arbitrary complex configurations

Wang

Guibault

Camarero

2007

Numerical Meth Engineering

View full text Add to dashboard Cite

SUMMARYThis paper presents a front propagation method using the Eikonal equation, ∇ · ∇ = 1, in which, represents the smallest Euclidean distance field to the front to be propagated. The offset capturing approach consists in first calculating the field over a uniform Cartesian grid fully covering the front to be propagated, and then constructing the iso-curves or surfaces as the propagated result. The calculation of uses a 3D numerical scheme, the Fast Sweeping Scheme. Validation for accuracy of the method is presented using academic test cases. A real 3D industry application, draft tube with two piers, is successfully propagated and demonstrated using special boundary conditions to cope with inlet and outlet planes during front propagtion. This application involves the propagation of a front that exhibits both concave and convex shape regions, sharp corners, and local tangent plane surface discontinuities as well as a multi-connected domain.

show abstract

Shape offsets via level sets

Cited by 67 publications

References 18 publications

Minimal surfaces: a geometric three dimensional segmentation approach

Minimal surfaces: a geometric three dimensional segmentation approach

Volumes with piecewise quadratic medial surface transforms: Computation of boundaries and trimmed offsets

Eikonal equation‐based front propagation for arbitrary complex configurations

Contact Info

Product

Resources

About