Setup II Domain-Specific Representation Spaces Based on KPCA

In figure 8.10, the Fisher’s score is plotted as a function of the kernel bandwidth σ. The Fisher’s score was evaluated for principal component values computed for examples in ΓFisher using the

principal axes ξ₁, . . . , ξ₃ derived from KPCA of the data from the training cases excluding M005A. The images below the plot depict the fusion image KPCA1 of the excluded case M005A computed

2 1 3 RGB KPCA KPCA KPCA KPCA 2 1 3 RGB KPCA KPCA KPCA KPCA M005A M009A

Figure 8.11:Fusion images KPCA1, . . . , KPCA3 and KPCARGB computed for cases M005A (left column) and M009A (right column). All three principal axes are computed using the same kernel bandwidth. The bandwidth value is chosen according to the maximum of the Fisher’s score curve computed for the first principal axis.

with different bandwidth values. The curve of the Fisher’s score for ξ₁ reaches a maximum for σ ∈ [0.2; 0.3], while the curve of ξ₂ has its minimum at this point. According to the Fisher’s score, normal and suspicious signals are best separated by ξ₃for a kernel bandwidth σ ∈ [0.5; 0.6]. The four fusion images of the excluded case M005A displayed below the plot illustrate that the Fisher’s score is a reasonable criterium for tuning the presentation of lesion masses. The fusion image KPCA1 computed with the bandwidth value yielding the maximum Fisher’s score (σ = 0.2,

Bandwidth B) depicts the lesion mass and the surrounding normal tissue with high visual contrast, i.e. with significantly different intensity levels. This visual contrast decreases for increasing values of σ (Bandwidths C,D), which is also reflected by the decreasing Fisher’s score.

The kernel bandwidth σ used for computing the principal axes ξ₂ and ξ₃ can either be se- lected dependently or independently from the kernel bandwidth used for computing ξ₁. In figure

2 1 3 RGB KPCA KPCA KPCA KPCA 2 1 3 RGB KPCA KPCA KPCA KPCA M005A M009A

Figure 8.12:Fusion images KPCA1, . . . , KPCA3 and KPCARGB computed for cases M005A (left column) and M009A (right column). All three principal axes are computed using a different kernel bandwidth. The bandwidth value is chosen according to the maximum of the Fisher’s score individually computed for each principal axis on a subset of the training data.

8.11, the principal axes ξ₁, . . . , ξ₃, i.e. the fusion images KPCA1, . . . , KPCA3 and KPCARGB, are

determined using the same kernel bandwidth for all three axes. Thus, all three principal axes are computed within the same kernel-induced feature space, leading to uncorrelated principal component values. The value of σ was chosen according to the maximum of the course of the Fisher’s score computed for ξ1. Hence, the lesions are conspicuously displayed in KPCA1. Since

the maximum of the Fisher’s score course computed for ξ₁ does not necessarily coincide with the maximum determined for ξ₂ and ξ₃, the corresponding fusion images KPCA2 and KPCA3

may provide a less conspicuous visualisation of the lesions. As illustrated in figure 8.10 for case M005A, the optimal value for the bandwidth of ξ₁ may even coincide with the minimum of the Fisher’s score for ξ₂ and with a low Fisher’s score for ξ₃, leading to a less prominent depiction of the lesion mass in the fusion images KPCA2, KPCA3 and KPCARGB as depicted in figure 8.11.

Table 8.1: Az for fusion images PCA1, PCA2 and PCA3 based on case-specific representation spaces spanned by the first three principal axes of PCA.

Case ID PCA M005A 0.547/0.935/0.759 M007A 0.653/0.902/0.684 M009A 0.781/0.904/0.829 B015A 0.702/0.945/0.781 M094A 0.820/0.980/0.625 M104A 0.810/0.860/0.856

In this case, ξ₁ lies in the direction which is most sensitive to the characteristic signal courses of suspicious tissue, whereas the orthogonality constraint causes the following principal axes to be less sensitive to lesion signals. Therewith, the lesions are predominantly encoloured with shadings of red in the composite images KPCARGB.

In order to enhance the display of lesion masses in the fusion images KPCA2 and KPCA3, the

corresponding bandwidth values can be chosen independently for all three principal axes. In the case that all three principal axes are computed for different values of σ, they relate to different feature spaces F . Therewith, they are not orthogonal to each other and the intensity values of the different fusion images may be correlated and reflect redundant information. Figure 8.12 presents the same coronal slices as figure 8.11 but fused by projecting the corresponding temporal kinetic signals onto individually tuned principal axes. Each principal axis was computed by KPCA with the kernel bandwidth value for which the corresponding Fisher’s score curve reaches its maximum. In all three fusion images KPCA1, . . . , KPCA3, lesions are displayed as bright masses with medium

to high visual contrast to normal tissue. The fusion images KPCA2 and KPCA3 of case M005A

exhibit similar image characteristics and represent redundant information. In KPCA2 of M009A,

the lesion mass as well as glandular tissue can be identified, whereas KPCA1 depicts the lesion

and fat tissue. Due to the image characteristics of KPCA1, . . . , KPCA3, the two lesion masses

are encoloured intense white in the corresponding composite image KPCARGB. Furthermore,

structures such as blood vessels or glandular tissue are depicted more clearly than in the colour images computed by KPCA with a single bandwidth value (Fig. 8.11).