Feature-Preserving Smoothing

László Neumann
Feature-Preserving Smoothing
TR-186-2-00-24, September 2000 [paper]

Information

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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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/",
}