3. COLOR PHOTO DENOISING VIA HSI DIFFUSION

The HSI color system is adopted in our algorithm because it represents colors similarly as how the human eye senses colors. Given a color photo, the first step is to convert the image from the RGB space to the HSI space. After the iterative diffusion process on the hue, saturation and intensity channels, the final result is obtained by converting the diffusion result back to the RGB space.

3.1. Intensity Diffusion

The intensity component (I) of a color image is the average of the red, green and blue values, which corresponds to the gray version of the color image. Thus it transfers the most important information for human visual “feeling” on the image. Following the original PM equation (Eq. (3)) and many subsequent approaches, the proposed intensity diffusion also adopts the divergence format. The common diffusion directions such as η-ξ direction have difficulty in robust noise removal and boundary gap closing, which is critical for edge identification and preservation in image denoising. To address this problem, we restrict the diffusion flow along the GVF direction: v(x) = (u(x), v(x))T, x 2, which was initially proposed for image segmentation and more robust on noise and small boundary gaps. Note that our proposed method shares the same objective for more stable and accurate denoising performance as the GVF-based denoising method in [33], but uses a different framework more targeted for the color photo denoising.

In order to restrict the diffusion along the GVF direction, we construct a diffusion tensor (D) similar to the coherence-enhancing diffusion in [10], which sets the diffusion directions along the eigenvectors of D by constructing it as the outer product of image gradients. In our algorithm, the tensor D is also built as a 2 × 2 matrix, with its two eigenvectors (p, q) parallel and perpendicular to the GVF direction (v), i.e., p || v and q v. The two eigenvalues (f1 and f2) correspond to the g( ) functions to adaptively control the diffusions in the p and q directions. With the embedded GVF field, the diffusion has the advantage of more accurate noise removal and edge preservation than those along the η-ξ directions. However, it still suffers from the staircase effect due to the piecewise constant image modeling. To overcome this staircase effect, a fourth-order term in [32], is specifically added to the diffusion framework to overcome the staircase effect. This fourth-order term is derived from the minimization of a second-order functional, which models an image as a piecewise linear signal. More details and illustrations about this high order filter can be seen in [32]. Given these considerations, we define our proposed intensity diffusion formulation as follows:

where IGVF is the GVF-based diffusion component, and IH represents the fourth-order term, α>0 and η>0.

where 255 ≥ LI ≥ 0 and ZI > 0 are constants to control the smoothing degree according to image brightness. Examples of this term with different ZI and LI are shown in Figure 3. It can be seen that this term inhibits diffusion at bright locations and enhances diffusion at dark locations, which is inspired by observations that color photos usually have smaller signal to noise ratio in dark regions. The gradient-based terms are given as

Note this is only one possible empiric choice (from the g( ) function overview in [6,27]) for the adaptive smoothing purpose.

Figure 3. Intensity-based coefficient of Eq. (18).

3.2. Hue Diffusion

The hue component (H) is an attribute to describe a pure color and is measured as an angle. As indicated by Perona [41], the direct application from intensity diffusion to anglevalued quantities would cause ambiguity in the updating process due to the angle periodicity. Therefore, we implement the hue component denoising as an orientation diffusion. In [41], the orientation diffusion is defined as

where λ > 0 is to control the diffusion strength. The first component is the diffusion term and Nb(x, y) is the 3 × 3 neighborhood of a point (x, y). The constant β > 0 is to balance the diffusion term and the data term (θ - θ0).

With a constant λ to adjust the diffusion strength, the original orientation diffusion equation (Eq. (20)) is an isotropic filter that cannot adaptively control the smoothing, i.e. edges cannot be preserved in diffusion. To solve this problem, we propose an anisotropic orientation diffusion for the hue denoising, with a spatially varying weighting function incorporated in the equation.

positive constants (K1 ≥ 1 ≥ K2 > 0) to control the diffusion strength. Note that other forms

Figure 4 compares the original orientation diffusion and the proposed weighted orientation diffusion performance on color edges. Figure 4(a) is an artificial hue image with constant values along the y direction and gradually changed values in the x direction (a slope to include all values), except an edge in the middle. Thus any horizontal cross section along the x direction yields a step type function in the middle. Figure 4(b) is the corresponding color image with the hue component from Figure 4(a) and a maximum value of 255 assigned to the saturation and intensity components. With the original orientation diffusion (Eq. (20), λ = 1, β = 0.03), the resultant image is shown in Figure 4(c) after 10 iterations. It can be seen that the center edge becomes blurry. Figure 4(d) is the result generated by the weighted orientation diffusion (Eq. (21), λ = 1, β = 0.03, K1 = 1, K2 = 0.01) with 20 iterations, and the color edge is preserved well.

Figure 4. Weighted orientation diffusion.

3.3. Saturation Diffusion

The saturation component (S) gives a measure of the degree to which a pure color is diluted by white light and is represented as a distance map. The saturation diffusion here is based on a modified curvature flow [15]: