## Injectivity conditions of 2d and 3d uniform cubic b-spline functions (2000)

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Venue: | Graphical Models |

Citations: | 21 - 0 self |

### BibTeX

@ARTICLE{Choi00injectivityconditions,

author = {Yongchoel Choi and Seungyong Lee},

title = {Injectivity conditions of 2d and 3d uniform cubic b-spline functions},

journal = {Graphical Models},

year = {2000},

volume = {62},

pages = {2000}

}

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### Abstract

Uniform cubic B-spline functions have been used for mapping functions in various areas such as image warping and morphing, 3D deformation, and volume morphing. The injectivity (one-to-one property) of a mapping function is crucial to obtaining desirable results in these areas. This paper considers the injectivity conditions of 2D and 3D uniform cubic B-spline functions. We propose a geometric interpretation of the injectivity of a uniform cubic B-spline function, with which 2D and 3D cases can be handled in a similar way. Based on our geometric interpretation, we present sufficient conditions for injectivity which are represented in terms of control point displacements. These sufficient conditions can be easily tested and will be useful in guaranteeing the injectivity of mapping functions in application areas. c ○ 2000 Academic Press 1.

### Citations

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(Show Context)
Citation Context ...nction f3 is defined by UNIFORM CUBIC B-SPLINE FUNCTIONS 415 ⎡ ⎢ J( f3) = ⎢ ⎣ ∂x ∂u ∂y ∂u ∂z ∂u It is known that a square matrix is nonsingular if and only if its row vectors are linearly independent =-=[13]-=-. Hence, function f2 is locally injective if and only if two 2D vectors, ( ∂x ∂x ∂y ∂y , ) and ( , ), are linearly independent all over the domain. Similarly, function f3 ∂u ∂v ∂u ∂v is locally inject... |

618 | Freeform deformation of solid geometric models
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(Show Context)
Citation Context .... In image warping and morphing, an image is distorted by a 2D mapping function that provides a new position for each point in the image [14]. In deformation techniques such as free-form deformations =-=[12]-=-, 3D mapping functions are used to determine the deformed positions of object points. In volume morphing, user-specified features are aligned by distorting given volumes with 3D mapping functions [2, ... |

412 |
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(Show Context)
Citation Context ... that transform certain domains into themselves are widely used. In image warping and morphing, an image is distorted by a 2D mapping function that provides a new position for each point in the image =-=[14]-=-. In deformation techniques such as free-form deformations [12], 3D mapping functions are used to determine the deformed positions of object points. In volume morphing, user-specified features are ali... |

191 | Direct manipulation of free-form deformations
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(Show Context)
Citation Context ...nctions to efficiently generate mapping functions in image morphing [8, 9]. Three-dimensional B-spline functions have been adopted to develop direct manipulation techniques for free-form deformations =-=[5, 6]-=-. Figure 1 gives an example of the application of 2D B-spline functions to image warping. Figure 1a is the original image with a control lattice overlaid on it. In Fig. 1b, the control lattice is chan... |

100 | Three-dimensional distance field metamorphosis
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Citation Context ...[12], 3D mapping functions are used to determine the deformed positions of object points. In volume morphing, user-specified features are aligned by distorting given volumes with 3D mapping functions =-=[2, 10]-=-. In these areas, the injectivity (one-to-one property) of a mapping function is essential to obtaining good results. In image warping and morphing, if a mapping function is not injective, the resulti... |

95 | Feature-Based Volume Metamorphosis - Lerios, Garfinkle, et al. - 1995 |

80 | G.: Image metamorphosis using snakes and free-form deformations
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(Show Context)
Citation Context ..., which is a function from 2D to 3D and is obtained from a 2D control lattice with 3D control points. Lee et al. used 2D B-spline functions to efficiently generate mapping functions in image morphing =-=[8, 9]-=-. Three-dimensional B-spline functions have been adopted to develop direct manipulation techniques for free-form deformations [5, 6]. Figure 1 gives an example of the application of 2D B-spline functi... |

50 | Image morphing using deformation techniques - Lee, Chwa, et al. - 1996 |

38 |
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Citation Context ...) (2) = 3 X i=0 3 X j=0 3 X k=0 B i (u)B j (v)B k (w)OE ijk ; where 0su; v; ws1. Functions f 2 and f 3 are locally injective if and only if their Jacobian matrices are nonsingular all over the domain =-=[1]-=-. The Jacobian matrix of f 2 is defined by J(f 2 ) = 2 6 4 @x @u @x @v @y @u @y @v 3 7 5 : The Jacobian matrix of function f 3 is defined by J(f 3 ) = 2 6 6 6 6 4 @x @u @x @v @x @w @y @u @y @v @y @w @... |

34 | Image metamorphosis with scattered feature constraints
- Lee, Wolberg, et al.
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(Show Context)
Citation Context ..., which is a function from 2D to 3D and is obtained from a 2D control lattice with 3D control points. Lee et al. used 2D B-spline functions to efficiently generate mapping functions in image morphing =-=[8, 9]-=-. Three-dimensional B-spline functions have been adopted to develop direct manipulation techniques for free-form deformations [5, 6]. Figure 1 gives an example of the application of 2D B-spline functi... |

10 |
Foldover-free image warping
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- 1998
(Show Context)
Citation Context ...in undesirable wrinkles because parts of the original image are folded upon nearby parts. Several techniques have been developed to generate injective mapping functions for image warping and morphing =-=[3, 7, 8, 9]-=-. In deformation techniques, the injectivity of a mapping leesy@postech.ac.kr, http://www.postech.ac.kr/leesy, Tel: +82-54-279-2245, Fax: +82-54-279-5699 function guarantees that no self-intersection ... |

8 |
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(Show Context)
Citation Context ...nts are above the bound and other are far below the bound while the resulting function is still injective. Goodman and Unsworth proposed a sufficient condition for a 2D Bézier surface to be injective =-=[4]-=-, which can be applied to a 2D B-spline function. For an m × n lattice of control points, the condition contains 2m(m + 1) + 2n(n + 1) linear inequalities. Unfortunately, when the number of control po... |

6 | Foldover-free image warping,” Graphical Models and - Fujimura, Makarov - 1998 |

2 |
CFFD: A tool for designing flexible shapes
- Hagenlocker, Fujimura
- 1998
(Show Context)
Citation Context ... 2D B-spline functions to efficiently generate mapping functions in image morphing [8, 9]. 3D B-spline functions have been adopted to develop direct manipulation techniques for free-form deformations =-=[5, 6]-=-. Fig. 1 gives an example of the application of 2D B-spline functions to image warping. Fig. 1(a) is the original image with a control lattice overlaid on it. In Fig. 1(b), the control lattice is chan... |

1 |
CFFD: A tool for designing flexible shapes, Visual Comput
- Hagenlocker, Fujimura
- 1998
(Show Context)
Citation Context ...nctions to efficiently generate mapping functions in image morphing [8, 9]. Three-dimensional B-spline functions have been adopted to develop direct manipulation techniques for free-form deformations =-=[5, 6]-=-. Figure 1 gives an example of the application of 2D B-spline functions to image warping. Figure 1a is the original image with a control lattice overlaid on it. In Fig. 1b, the control lattice is chan... |