# Feature-Preserving Smoothing

László Neumann**Feature-Preserving Smoothing**

TR-186-2-00-24, September 2000 [paper]

## Information

- Publication Type: Technical Report
- Workgroup(s)/Project(s): not specified
- Date: September 2000
- Number: TR-186-2-00-24
- Keywords: Antialiasing., Volume Rendering, Image Processing

## Abstract

The first version of the method presented in this paper is derived from a terrain-modeling problem. The solution of this problem leaded to a general technique, which can be used for image processing purposes like image restoration, noise filtering, and smoothing preserving the important contours. The basic idea is the minimization of a global quadratic penalty function. This error function contains the quadratic global curvature of the unknown image and additionally the difference of the data values and the gradient vectors between the original and the unknown image. The components of the global error are weighted using an appropriate weighting function.The gradient vectors are estimated from the original noisy image using linear regression. Under a certain threshold the gradient or its weight in the quadratic error function is considered to be zero. The method is non-linear because the operations on the gradients and/or the assignment of the weighting factors are non-linear.

The minimization of the penalty function leads to a linear equation system with a large sparse coefficient matrix. It will be shown that it can be solved by a special deconvolution if the weighting of each pixel is the same. The deconvolution can be performed very efficiently in frequency domain using the well-known FFT algorithm. If the weighting function depends on pixel positions then the general conjugated gradient method can be used in order to find the minimum location of the penalty function.

Having position-dependent weights the flexibility is higher since at the positions, where the gradient magnitudes are lower the smoothing effect can be stronger while at the edges smoothing is not performed. The main characteristic feature of the method can be realized when it is used for image restoration, where the missing regions have to be reconstructed. Instead of interpolation of the pixel values the gradient components are interpolated, therefore continuous and non-blurred contours can be obtained. The method calculates densities, which fit best on the gradient field. The basic idea can be applied for gradient interpolation as well, where the component gx and gy of the gradient play the role of the unknown variables. Another special application field is the zooming of low-resolution images preserving the sharp contours, which is rather important in rendering images on the web or in digital photography. The method can be used for color images separately for three components, for instance using CIE Luv coordinates. The 1D and 3D adaptation are obvious. The 1D version can be used for interpolation problems and sound filtering, while the 3D version is capable for efficient processing of medical data sets (CT or MRI files), geographical volumes, arbitrary 3D distributions, or volumes defined by a sequence of frames in a movie file.

Among the special 2D applications one of the most important ones is the continuous reconstruction from binary dithered images, which can be used recursively for image compression. Furthermore, the 2D version is usable for adaptive contrast enhancement. In this case, instead of the deviation from the original density values, the deviation from a constant value or the deviation from the strongly blurred image is built into the penalty function. Last but not least it can be extended to a multi-resolution or hierarchical method, where the different resolution versions of the same image are improved in a parallel way and derivative filters defined by wide finite kernels are used with a ring off weighting.

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## Weblinks

No further information available.## BibTeX

@techreport{Neumann-2000-FPS, title = "Feature-Preserving Smoothing", author = "L\'{a}szl\'{o} Neumann", year = "2000", abstract = "The first version of the method presented in this paper is derived from a terrain-modeling problem. The solution of this problem leaded to a general technique, which can be used for image processing purposes like image restoration, noise filtering, and smoothing preserving the important contours. The basic idea is the minimization of a global quadratic penalty function. This error function contains the quadratic global curvature of the unknown image and additionally the difference of the data values and the gradient vectors between the original and the unknown image. The components of the global error are weighted using an appropriate weighting function. The gradient vectors are estimated from the original noisy image using linear regression. Under a certain threshold the gradient or its weight in the quadratic error function is considered to be zero. The method is non-linear because the operations on the gradients and/or the assignment of the weighting factors are non-linear. The minimization of the penalty function leads to a linear equation system with a large sparse coefficient matrix. It will be shown that it can be solved by a special deconvolution if the weighting of each pixel is the same. The deconvolution can be performed very efficiently in frequency domain using the well-known FFT algorithm. If the weighting function depends on pixel positions then the general conjugated gradient method can be used in order to find the minimum location of the penalty function. Having position-dependent weights the flexibility is higher since at the positions, where the gradient magnitudes are lower the smoothing effect can be stronger while at the edges smoothing is not performed. The main characteristic feature of the method can be realized when it is used for image restoration, where the missing regions have to be reconstructed. Instead of interpolation of the pixel values the gradient components are interpolated, therefore continuous and non-blurred contours can be obtained. The method calculates densities, which fit best on the gradient field. The basic idea can be applied for gradient interpolation as well, where the component gx and gy of the gradient play the role of the unknown variables. Another special application field is the zooming of low-resolution images preserving the sharp contours, which is rather important in rendering images on the web or in digital photography. The method can be used for color images separately for three components, for instance using CIE Luv coordinates. The 1D and 3D adaptation are obvious. The 1D version can be used for interpolation problems and sound filtering, while the 3D version is capable for efficient processing of medical data sets (CT or MRI files), geographical volumes, arbitrary 3D distributions, or volumes defined by a sequence of frames in a movie file. Among the special 2D applications one of the most important ones is the continuous reconstruction from binary dithered images, which can be used recursively for image compression. Furthermore, the 2D version is usable for adaptive contrast enhancement. In this case, instead of the deviation from the original density values, the deviation from a constant value or the deviation from the strongly blurred image is built into the penalty function. Last but not least it can be extended to a multi-resolution or hierarchical method, where the different resolution versions of the same image are improved in a parallel way and derivative filters defined by wide finite kernels are used with a ring off weighting.", month = sep, number = "TR-186-2-00-24", address = "Favoritenstrasse 9-11/E193-02, A-1040 Vienna, Austria", institution = "Institute of Computer Graphics and Algorithms, Vienna University of Technology ", note = "human contact: technical-report@cg.tuwien.ac.at", keywords = "Antialiasing., Volume Rendering, Image Processing", URL = "https://www.cg.tuwien.ac.at/research/publications/2000/Neumann-2000-FPS/", }