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The transform coefficients can be positioned at the same place in the lattice as the original pixels resulting in the

same division of the space

as in the original image.

Such a set of transforms produces a new set of coefficients

that also belong to the original space

. If a new set

of unitary orthogonal transforms is applied along a different direction, the same equations hold again. Therefore, orthog- onality holds for any particular direction and for any series

of directions chosen.

Since orthogonality holds in any particular direction, a frame built from several directions is tight. The frame can be separated into two parts containing scaling and wavelet bases. Now, we can write:

where

. For multiple directions, this sepa-

ration principle allows us to write:

as it was proved above. Therefore, such a frame is tight regardless of the number of directions.

Since iteration is possible to be done on any sub-band channel, orthogonality and tightness hold for any step of iteration. Thus, they hold for the whole transform including all steps of iteration.

5. APPLICATION IN NON-LINEAR APPROXIMATION AND DENOISING

Application of the described method is possible in many ar- eas of image processing where the standard wavelet trans- form is applied. Below, applications in non-linear approxi- mation and denoising are presented.

5.1. Non-linear approximation

The number of significant coefficients produced by a hori- zontal and vertical transform depends on dominant orienta- tion of objects in an image. If this direction is not matched with either the horizontal or vertical direction the number will increase. On the other hand, if some other directional transforms are applied on the channels, the number of sig- nificant coefficients related to those objects may decrease.

This property is exploited in non-linear approximation, where a fixed number of the largest coefficients is kept (the vectors that approximates the signal are adaptive and there- fore the approximation is non-linear). A new decomposition tree is being compared to a standard one. The first is shown in Figure 3a) and the latter in Figure 3b). The quality of the reconstructed images in both cases are compared and the result of the comparison is expressed in terms of the PSNR

factor. Figure 4b) shows the result of approximation of the image shown in Figure 4a), which is a synthetic randomly generated image that contains lines with different orienta- tions. These results indicate that the new method can be attractive in non-linear approximation.

5.2. Denoising

The standard denoising process is usually done by thresh- olding of coefficients obtained by a transform along hori- zontal and vertical direction. Some difficulties may appear since visibility of objects in an image depends on their ori- entations.

Also, one-dimensional denoising of an image is not able to catch most of the two-dimensional interdependencies pre- sent in images. This is the reason why independent de- noising in only two directions (line-by-line, or column-by- column) of an image does not necessarily give the best re- sult.

However, such an approach has the main advantage of simplicity. In trying to retain this simplicity, but still mak- ing a method that better exploits the two-dimensional char- acteristics of an image, we introduce multiple directional denoising. An image is processed by taking sets of pixels in different orientations and denoising them by hard thresh- olding [6] of wavelet coefficients. The result is compared to the result of a standard method where horizontal and ver- tical wavelet decomposition was done in several levels and the coefficients were hard thresholded as well.

Figure 5 shows an example of denoising of the image ’Cameraman’. The original image is affected by additive Gaussian noise and both methods were applied. The depen- dence of the performance of the denoising on the number of directions in the analysis is shown in the Figure 6a). The comparison between the new and the standard method is shown in the Figure 6b). The new method strongly outper- forms the standard one.

6. CONCLUSION

The standard method of the wavelet decomposition of im- ages using separable bases involves only two directions of analysis. Our new method goes beyond this limitation and calculates separable wavelet transform along sets of differ- ent directions. Such an approach takes into account two- dimensional interdependencies among image pixels better than the standard method. Simplicity and low computa- tional complexity are still maintained. Application of the new method is possible in various areas of image process- ing. Outperforming results are shown in non-linear approx- imation and denoising.

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