Graph Spectral Image Processing - Gene Cheung

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its approximation accuracy gets significantly worse when λ is away from 0. Since the maximum eigenvalue λmax highly depends on the graph used, it is better to use different approximation methods like the Chebyshev approximation, which is introduced in section 1.6.

      1.4.2. Edge-preserving smoothing

      Edge-preserving image smoothing is widely used for various tasks, as well as for image restoration (Nagao and Matsuyama 1979; Pomalaza-Raez and McGillem 1984; Weickert 1998; Tomasi and Manduchi 1998; Barash 2002; Durand and Dorsey 2002; Farbman et al. 2008; Xu et al. 2011; He et al. 2013). Image restoration aims to approximate an unknown ground-truth image from its degraded version(s). In contrast, edge-preserving smoothing is typically used to yield a user-desired image from the original one. The resulting image is not necessarily close to the original one.

      In the graph setting, we need to define pixel-wise or patch-wise relationships as a distance between pixels or patches, and it is used to construct a graph. The following distances are often considered (Milanfar 2013b), where i and j are pixel or patch indices and

is some nonnegative function:

      1) Geometric distance:

, where pi is the ith pixel coordinate.

, where
is the pixel value (often three dimensional) of the ith pixel/patch.

      3) Saliency distance:

where si is the ith saliency value.

      4) Combinations of the above.

      Saliency of the image/region/pixel is designed to simulate perceptual behavior (Itti et al. 1998; Harel et al. 2006). A popular choice of φ(·) is the Gaussian weight

      [1.23]

      where σ controls the spread of the filter kernel.

      Suppose that the filter coefficients are determined based on the above features, and that they are symmetric, i.e. the output pixel value yi is represented as

      [1.24]

      where

      [1.25]

      The Fourier domain representation of such pixel-dependent filters cannot be calculated in a classical sense because it is no longer shift-invariant: the filter matrix W cannot be diagonalized by the DFT matrix. In contrast, GSP provides a frequency-like notion in the graph frequency domain. In general, the weight matrix W in equation [1.24] can be regarded as an adjacency matrix because all dk(·, ·) are assumed to be distances between pixels. Suppose that there is no self-loop in W, for simplicity. In general, the smoothed image in equation [1.24] is represented in the following matrix form:

      [1.26]

      where D = diag(D0,...,DN−1). This can be rewritten by using the relationship L = D − W as (Gadde et al. 2013):

      [1.27]

      [1.28]

      where

is a degree-normalized signal. Let us denote the eigendecomposition of Ln as
. The above filtering in equation [1.29] is further rewritten as:

      [1.30]

      [1.31]

      [1.32]

      where y := D1/2y and the graph spectral filter is defined as

Moreover,
for the symmetric normalized graph Laplacian; therefore, it acts as a linear decay low-pass filter in the graph frequency domain.

      This graph spectral representation of a pixel-dependent filter suggests that the pixel-dependent filter W implicitly and simultaneously designs the underlying graph (and therefore, the GFT basis) and the spectral response of the graph filter. In other words, the GSP expression of the pixel-dependent filter is free to design the spectral response

, apart from the linear decay one, once we determine W. For example, let us consider the following spectral response:

      [1.33]

is an arbitrary graph high-pass filter and η > 0 is a parameter. In this case,
works as a graph low-pass filter and its spectral shape is controlled by
In fact, Gadde et al. (2013) show that equation [1.33] is the optimal solution for the following signal restoration problem:

      [1.34]

      where

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