## Gradient Estimation in Volume Data using 4D Linear Regression (2000)

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Citations: | 33 - 4 self |

### BibTeX

@MISC{Neumann00gradientestimation,

author = {László Neumann and Balázs Csébfalvi and Andreas König and Eduard Gröller},

title = {Gradient Estimation in Volume Data using 4D Linear Regression},

year = {2000}

}

### Years of Citing Articles

### OpenURL

### Abstract

. In this paper a new gradient estimation method is presented which is based on linear regression. Previous contextual shading techniques try to fit an approximate function to a set of surface points in the neighborhood of a given voxel. Therefore, a system of linear equations has to be solved using the computationally expensive Gaussian elimination. In contrast, our method approximates the density function itself in a local neighborhood with a 3D regression hyperplane. This approach also leads to a system of linear equations but we will show that it can be solved with an efficient convolution. Our method provides at each voxel location the normal vector and the translation of the regression hyperplane which are considered as a gradient and a filtered density value respectively. Therefore, this technique can be used for surface smoothing and gradient estimation at the same time. 1 Introduction In direct volume rendering the quality of the generated image is strongly influ...

### Citations

65 |
Three-Dimensional Display of Human Organs from Computed Tomograms
- Herman
- 1979
(Show Context)
Citation Context ...count. The basic idea of this approach can be applied to object space techniques too[3]. Object space methods estimate the normal according to the 3D neighborhood of the given voxel. Constant shading =-=[6]-=- which is based on the cuberille method is an early example of this category. Thevoxels are considered as unit cuboids, and the normals at each point of a boundary surface are the true normals of the ... |

43 |
Shading 3D-images from CT using gray-level gradients
- Höhne, Bernstein
- 1986
(Show Context)
Citation Context ...ion of the adjacent visible faces into account increasing the number of possible normal vectors and giving a better impression about the inclination of a boundary surface. Gray-level gradient shading =-=[9]-=-[10] is used for volumes, where each voxel represents a gray-level value. The gradient vectors are estimated according to the neighborhood of the voxels using traditional derivative lters [1]. Context... |

38 | A comparison of normal estimation schemes
- Möller, Machiraju, et al.
- 1997
(Show Context)
Citation Context ... shading [9][10] is used for volumes, where each voxel represents a gray-level value. The gradient vectors are estimated according to the neighborhood of the voxels using traditional derivative lters =-=[1]-=-. Contextual shading ts a local approximate plane [11] or a biquadratic [12][13] function to the set of points that belong to the same isosurface. These methods are timeconsuming and limited to a cert... |

20 |
Surface Shading in the Cuberille Environment
- Chen, Herman, et al.
- 1985
(Show Context)
Citation Context ...this category. Thevoxels are considered as unit cuboids, and the normals at each point of a boundary surface are the true normals of the corresponding cuboid faces. Normal-based contextual shading [7]=-=[8]-=- is based on the cuberille method as well. This technique takes also the orientation of the adjacent visible faces into account increasing the number of possible normal vectors and giving a better imp... |

12 | Normal estimation in 3D discrete space
- Yagel, Cohen, et al.
- 1992
(Show Context)
Citation Context ...tours of di erent organs are displayed smoothly with reduced staircase artifacts. In order to ful ll this requirement our method integrates the ltering and the normal computation into one process. In =-=[3]-=- Yagel overviews several methods for discrete normal estimation and analyses their performance. Depending on the neighborhood considered, these techniques can be classi ed into two fundamentally di er... |

6 | tracing voxel data via biquadratic local surface interpolation - Ray - 1990 |

5 |
Display of Three-dimensional Discrete Surfaces
- Herman, Udupa
- 1981
(Show Context)
Citation Context ...of this category. Thevoxels are considered as unit cuboids, and the normals at each point of a boundary surface are the true normals of the corresponding cuboid faces. Normal-based contextual shading =-=[7]-=-[8] is based on the cuberille method as well. This technique takes also the orientation of the adjacent visible faces into account increasing the number of possible normal vectors and giving a better ... |

5 |
Rendering Tomographic Volume Data: Adequacy of Methods for Different Modalities and Organs. 3D Imaging in Medicine
- Höhne, Bomans, et al.
- 1990
(Show Context)
Citation Context ... of the adjacent visible faces into account increasing the number of possible normal vectors and giving a better impression about the inclination of a boundary surface. Gray-level gradient shading [9]=-=[10]-=- is used for volumes, where each voxel represents a gray-level value. The gradient vectors are estimated according to the neighborhood of the voxels using traditional derivative lters [1]. Contextual ... |

3 |
Normal Computation for Discrete Surfaces
- Thürmer, Wüthrich
- 1997
(Show Context)
Citation Context ... based on central di erences is the special case of our method using the following weighting function: 8>< >: wk = 1ifk =4ork =22 or k =10ork =16 or k =12ork =14 0 otherwise. (13) Thurmer`s technique =-=[2]-=- which has been proposed for normal estimation in binary volumes is also a special case of our method. According to this approach the Nx, Ny, andNz components of the estimated normal vector are calcul... |

3 |
Display of 3D Medical Images
- Tam, Davis
- 1988
(Show Context)
Citation Context ...gories as image space and object space methods.sImage space techniques, take only the 2D neighborhood in the projected image into account, therefore they are view-dependent. Depth-gradient shading [4]=-=[5]-=- as a representative of image space methods, approximates the gradient from the z-bu er calculating the di erences between the depths of the given and the neighboring pixels. This approach produces sh... |

3 |
tracing voxel data via biquadratic local surface interpolation
- Webber
- 1990
(Show Context)
Citation Context ...vel value. The gradient vectors are estimated according to the neighborhood of the voxels using traditional derivative lters [1]. Contextual shading ts a local approximate plane [11] or a biquadratic =-=[12]-=-[13] function to the set of points that belong to the same isosurface. These methods are timeconsuming and limited to a certain neighborhood. Bryant and Krumvieda [11] solve a set of linear equations ... |

2 |
Shading Voxel Data via Local Curved-surface Interpolation. Volume Visualization
- Webber
- 1991
(Show Context)
Citation Context ...value. The gradient vectors are estimated according to the neighborhood of the voxels using traditional derivative lters [1]. Contextual shading ts a local approximate plane [11] or a biquadratic [12]=-=[13]-=- function to the set of points that belong to the same isosurface. These methods are timeconsuming and limited to a certain neighborhood. Bryant and Krumvieda [11] solve a set of linear equations by G... |

1 |
Display of3DBinary Objects: I-shading
- Bryant, Krumvieda
- 1989
(Show Context)
Citation Context ...l represents a gray-level value. The gradient vectors are estimated according to the neighborhood of the voxels using traditional derivative lters [1]. Contextual shading ts a local approximate plane =-=[11]-=- or a biquadratic [12][13] function to the set of points that belong to the same isosurface. These methods are timeconsuming and limited to a certain neighborhood. Bryant and Krumvieda [11] solve a se... |

1 | Display of 3D Binary Objects: Ishading - Bryant, Krumvieda |