where t is a threshold parameter, which is proposed to be calculated by mean and standard deviation of the amplitude through t = µl+ 2σl.
3.2.2. Results of Decompositions and the EOB Algorithm
The standard test image Barbara12 is famous among researchers in the field of image processing. It is used as a prototype of a natural image to show several results in this section.
The decomposition and reconstruction was tested to guarantee the correctness of the algorithm in the beginning. A set of 15 natural uint8 greyscale images of differ- ent sizes between 256x256 and 1024x124 of the SIPI image database13 built the test dataset. First, the images where transcoded from uint8 to floating point numbers of the interval [0 . . . 1] and decomposed with according granularities (see Table 3.1) using the modified stationary monogenic wavelet transform in Algorithm 3.5. Afterwards, the approximation on each scale was calculated through amplitude and phase, com- puted by the monogenic signal as shown in the last Section 3.2.1. The reconstructed images had a l2 norm with upper bound below 10−15, which is explained by double precision data-types used by Matlab[58] and which are neglectable after transcoding into uint8 again.
Figure 3.8 shows an example of a decomposition of the Barbara test image. Here, the lowest granularity is used to give an impression of the idea of multi scaling14. While
on the highest scale finest details are present, on lower scales more coarser structures influence the bulk of the image. In the amplitudes, there are still structural information left, which is a side effect of the low granularity which was used. A pure Reconstruction of the image is, as already mentioned, of less interest. The decomposition into Riesz
12The source of the Barbara test image is, according to [71], Allen Gersho, who found it
during his time at the University of California, Santa Barbara (UCSB). It is intended to be used in wavelet analysis due to its striped structures and is available in uint8 greyscale coding.
13available at http://sipi.usc.edu/database/(February 15th, 2018)
14The decompositions which are made under high levels of granularity usually create images
3.2. Applications in Image Processing 79
Fig. 3.8.: Decomposition of the famous Barbara test image. The single image in the top row is the original. The left Column shows the amplitudes on the highest three scales and the right one the according phase of the image. While the amplitude is scaled from 0 to 0.5 (from possible 0 to 1) to increase the visibility, the phase is shown in its original scale between 0 and π.
80 3. Applications of Monogenic Wavelet Frames
Fig. 3.9.: Application of the equalization of brightness and corresponding histograms. The values of the original image on the left side were scaled to the interval [−15 . . . 15] to show the difference of the output of the image which was purely reconstructed by phase values and amplitude set to 1. No regularizations were applied.
3.2. Applications in Image Processing 81
Fig. 3.10.: This plot shows the variances of results of the equalization of brightness algorithm. The dataset consisted of a collection of 15 natural images from the SIPI image database. The new regularization consistently gives better results.
82 3. Applications of Monogenic Wavelet Frames
Fig. 3.11.: Comparison of different Regularizations on the Barbara test image is applied. On top, no regularization was applied. The second histogram shows the regularization which was introduced by Held (see (3.20)). The last histogram shows the results from the new Tikhonov influenced regularization.
3.2. Applications in Image Processing 83
Fig. 3.12.: EOB with both regularization strategies in comparison on a part of the Barbara test image. In the left image, the Held regularization is shown. There is still illumination information on the fold of the trouser leg. Also, the edge between trousers and ground has artefacts considering signal jumps. These do not occur on the new algorithm on the right side.
The plot shows the image values along the imprinted line from left to right. The one also sees lower values in the second half of the line. This is an area with no directional information. Because the regularization always applies in case of the new algorithm there is improvement as well while the threshold value is not surpassed yet in case of Held regularization.
84 3. Applications of Monogenic Wavelet Frames
components and their corresponding values of the original signal can be used explicitly to construct special constellations like equalization of brightness (EOB).
From a statistical point of view, EOB equalizes the wide spread intensity of the image to a small range of values which can be reflected as a decrease of the variance. Thus, the results can be illustrated and benchmarked by histograms and distribution of the intensity values. Figure 3.9 shows a first impression of the procedure. The intensity values are merged closer together and so the variance shrinks from σoriginal= 30.8348 to
σEOB = 17.0654. While in the original paper [41] a regularization as shown in equation
(3.20) is proposed, a better possibility is to control the amplitude values continuously with a method which is influenced by Tikhonov regularization:
Dl=
A◦22
A◦22 + t2 cos (ϕl) (3.21)
A comparison of both approaches in contrast to no regularization is shown in Figure 3.10. The according threshold value was found through quantiles of the image. In strictly non binary images, using the first 10-quantile (decile) has turned out to be a good choice for an initial value of t.
The new algorithm is less influenced by the original data of the given image which is affirmed by less oscillations in the variance data in the plot. Also the new algorithm is of a more continuous nature which slowly starts to regulate the phase influence down in areas of low amplitude. Further differences are shown in Figure 3.12, where a part of the test image is zoomed in and investigated. Here, improvements can be seen along sharp edges and along modest illumination areas.