Results Coronal Rat Brain Data Set

The peak picking approach based on Algorithm 10 is applied to the entire rat brain data set with the following input parameters: the slice length and overlap remain as previously defined (D60andoD0:5. The Gabor frame is based on a Hann window of width 15 samples. The regularization parameter is fixed toD1:5e 3.

The resulting sparse outputz then contains only nonzero values where a peak is detected.

The mean spectrumzover all spots is depicted in Fig. 6.3. Hereby, Figure 6.3a shows the mean spectrum of the original rat brain data set. In comparison, the mean spectrum after peak picking is visualized below in Fig. 6.3b. It shows the same prominent features as the original data set.

However, only 48% of allm=z images contain non-trivial coefficients. This means, that for the remaining 52% ofm=zvalues no peak is detected in any of the 20,185 spots. Clearly, this procedure is sensitive to the choice of the regularization parameter. Smaller values increase the sensitivity, resulting in more detected peaks per single spectrum. Larger values, on the other hand, increase the denoising effect by choosing less peaks. Recall from (6.2.13) thatz, and hence the mean spectrum ofz, does not reflect original intensities anymore, but rather significant changes between Gabor coefficients of two consecutive slices. This means the larger a peak in z, the larger the difference between two slices. Despite this drawback, it might still be useful to analyzem=zimages after peak picking.

In the following, let the original approach as summarized in Algorithm 10 be denoted as the basicapproach. The modified approach proposed in (6.2.16) is based on an average filter for a33neighborhoodN. Hence, the filter coefficients arewj D ¹9 for allj 2N. Thebasic as well as theaverageapproach are applied to the rat brain data set with the same parameter settings as previously defined. Four selected m=z images are shown in Figure 6.4, where correspondingm=z-values are indicated with a red triangle in the mean spectrum of Fig. 6.3.

The denoising effect of the proposed algorithm is clearly visible when comparing raw data with

(a)Original mean spectrum

(b)Mean spectrum after peak picking

Figure 6.3:Mean spectra of the original rat brain data set and after peak picking. Red triangles indicatem=zlocations for Fig. 6.4

the basic or average approach. Additionally, the neighborhood based approach smooths peak areas inm=zimages, while preserving edges. This can be nicely seen, for example, in Figures 6.4c and 6.4d. The sensitivity of the proposed peak picking approach is large enough to also detect low intensity peaks. Alexandrov and Bartels (2013) showed that the low intensity peak atm=z D 4385:9, depicted in Fig. 6.4a, is not detected by other spectrum-wise peak picking approaches. Nonetheless, detecting peaks which are present in a small number of spectra is substantial in order to distinguish between matrix and actual protein peaks (Alexandrov and Bartels, 2013, Fig. 5).

Detection of Overlapping Peaks

The detection of overlapping peaks is crucial whenever the sampling rate is low and isotopes are not clearly separated. Figure 6.5 shows part of a rat brain spectrum with overlapping peaks and

(a)m=zD4385:8525

(b)m=zD6223:2627

(c)m=zD6717:3877

(d)m=zD7060:0215

Figure 6.4:Fourm=z images of the rat brain data set showing the raw data, data after basic peak picking and data after spatially aware peak picking using an average filter.

Figure 6.5:Detection of overlapping peaks in the rat brain data set. Local maxima inz are indicated by dashed lines.

the output of the proposed peak picking algorithmz. Local maxima inzindicate the correct positions of overlapping peaks in the original spectrum.

Whenever the proposed algorithm is based on Gabor frames, the capability of separating overlapping peaks strongly depends on the chosen window width of the frame itself. If the window function is too wide, two overlapping peaks might be detected as a single peak. On the other hand, narrower windows might fail to correctly detect wider peaks in the spectrum. One solution to overcome this shortcoming is to use so called multi-window Gabor frames (Zeevi et al., 1998). Such frames consist of multiple single frames, where the window width varies from frame to frame. This allows the combination of estimated masks from different frame multipliers in order to improve the detection of overlapping peaks.

Using wavelet frames, on the other hand, the scaling of the window function simplifies the detection of overlapping peaks. Additional steps, however, have to be taken in order to correctly identify overlapping spikes from noise. If a peak is detected, wavelet coefficients corresponding to scaling factors resulting in narrow wavelets reveal information whether this peak is a single peak or consists of overlapping peaks. This is similar as proposed in the previous Chapter (Song and Li, 2015).

Comparison with Edge-Preserving Denoising

Kobarg (2014) showed that smoothingm=zimages enhances segmentation maps. The hereby used bilateral filter preserves edges and local structures inm=zimages. This bilateral smoothing

(a)Original (b)Bilateral filtered

(c)Proposed algorithm (d)Proposed algorithm bilateral filtered

Figure 6.6:Comparison of bilateral filtering and the proposed algorithm based on the spatially aware approach with an average filter (m=zD6223:2627).

filter has been introduced by Tomasi and Manduchi (1998) and can be summarized as follows.

A Gaussian low-pass filter with standard deviationbsmooths pixels in awwneighborhood.

Additionally, intensity differences of pixels inside this neighborhood are also weighted with a separate Gaussian filter with standard deviation_b. Hence, sharp transitions between intensities inside a neighborhood remain, whereas similar intensities are smoothed.

Figure 6.6 demonstrates the effects of a bilateral smoothing filter on the same data as shown by Kobarg (2014, Fig. 5.2) with similar parameters (w D3,_b D1:5and_b D5=16). Compared to the original image, bilateral filtering spatially smooths the image while preserving local structures as illustrated in Figures 6.6a and 6.6b. Kobarg (2014) improved segmentation maps by clustering spectra with K-means based on the bilateral filtered rat brain data set. It can be assumed that usingm=z images based on the proposed spatially aware peak picking method might further improve segmentation results. Comparing Figures 6.6b and 6.6c demonstrates the advantages of the proposed algorithm. Noise is removed, while preserving edges and local structures. The bilateral filtered image after spatially aware peak picking is shown in Figure 6.6d for illustrational purpose only.

(a)Original mean spectrum

(b)Mean spectrum after peak picking.

Figure 6.7:Part of the mean spectrum of the lung data set before and after peak picking. Red triangles mark positions ofm=zvalues in Figure 6.8.