Directional enhancement

The spatial coordinates of similar blocks may indicate whether the blocks contain an edge or not. In fact, if the blocks are aligned, it is reasonable

to assume that the they contain an edge, and that the direction of the edge is aligned with the blocks. An example can be seen in the first row of Figure 4.1, where we show the basis of a group of blocks (purple area) whose reference one (white block) lies on an edge. Here, it is clear that the blocks are aligned with the direction of the edge. On the bottom row, instead, we report similar blocks from a uniform region; note how the blocks are scattered, and a principal orientation cannot be discerned.

We propose a directional algorithm that exploits the above spatial infor-mation to enhance the content of similar blocks by increasing or decreasing, individually, each spectrum coefficients; this results in, respectively, sharp-ening or softsharp-ening of the corresponding basis functions. Let us consider the set of blocks spatial coordinates S; we compute its PCA decomposition as S = U ΛV^T. By definition, the columns of U indicate the two principal directions along which the cloud of points is oriented; at the same time, the diagonal elements of Λ are the length of these components: the larger the value, the more the cloud is aligned with the corresponding principal component. Based on the ratio between the first and second component, we can decide whether to sharpen or soften a block. In fact, if the ratio between the first and second component is large, then the set of coordinates is mostly oriented on a thin line. On the contrary, if the ratio is close to one, the blocks coordinates are spread uniformly on a broad area, suggest-ing that the blocks are extracted from a regular region. In this way we can sharpen the blocks from an edge, and we can soften the blocks from a uni-form area. This concept is similar to the tensor methods developed byFeng and Milanfar [2002] and Weickert [1999], that estimate weather an image region is an edge or a smooth area based on the principal components of its gradient.

Since we use the alpha-rooting sharpening method [Dabov et al.,2007c], we also adapt the sharpening and softening to specific 2-D transform coef-ficients. In particular, for each 2-D spectrum coefficient, we compute the energy (in terms of `₂ norm) of the derivative of its corresponding basis function in the directions of the two principal components of the coordi-nates of the group. Based on the ratio between them, we decide whether to sharpen the coefficient, if the ratio is larger than one, or to soften it, if the

4.1. Collaborative filtering based on coordinates of similar features

ratio is smaller than one. In this way we perform at the same time softening along an edge, while sharpening the edge itself. In our implementation, we adjust the sharpening/softening strength with the ratio between first and second principal component of the set of coordinatesS: if the ratio is large we sharpen more compared to a case in which the ratio is close to 1.

4.1.3 Results

In Figure 4.2, reproduced from Publication III, we report a detail of Lena denoised with the proposed method compared to the standard BM3D (Haar 1-D transform); the image has been corrupted by additive Gaussian noise with standard deviation σ = 35. We specifically report the shoulder detail because it is where the proposed algorithm visibly outperforms its canonical counterpart.

In Figure 4.3, also reproduced from Publication III, we denoise and sharpen Peppers (we show only a detail) corrupted by AWGN with stan-dard deviation σ = 20. On the left-bottom we report the results from the proposed enhancement scheme, while on the right-bottom we show the con-ventional alpha-rooting result, with constant alpha coefficient. As a mean of comparison, on the top-right position of the same figure we also report the result of the proposed denoising algorithm with no sharpening applied.

Note how the proposed enhancing algorithm sharpens the edges of the image while softening the piecewise smooth areas. In contrast, the standard alpha-rooting algorithm sharpens all the image details indiscriminately, enhancing in some cases artifacts introduced by the denoising algorithm itself.

Finally, to show the adaptive sharpening strength of the proposed alpha-rooting filter, we report in Figure 4.4 the ratio between the principal compo-nents of each coordinates set of similar blocks. Red areas indicate high ratio (edges), while blue areas approach the value 1 (smooth regions). We remind the reader that we do not perform directly sharpening on the red areas, but for the red areas we perform sharpening only on the transform coefficients whose corresponding basis functions have most variations orthogonal to the edge, while we soften the coefficients whose corresponding basis functions mostly vary along the edge.

Detail of Lena Noisy detail

PSNR_Haar = 29.72 PSNR_OP = 29.90

Figure 4.2: Denoising of Lena corrupted by i.i.d. Gaussian noise withσ = 35. From left to right, top to bottom: original image, noisy observation, image denoised by the standard BM3D algorithm, images denoised by the proposed algorithm based on adaptive orthonormal polynomials. Notice the improvement, particularly in smooth regions, such as the shoulder area.