Functional images

Suppose that only projection of lung and heart can be seen in the images of a sequence, but all the details of them function in the same way. Then the images from the heart and the lung can be obtained from the images of the sequence and the function of them.

Figure 38.

L and H images illustrate the ratio of the lung and the heart in each pixel. I(t) and h(t) shows how lung and heart “work”. L and H are physiological (factor) images. I(t) and h(t) are physiological factors (factors weight). If the F_k image has been done at t_k (in the time interval with t_k at its middle), then .

With the help of factor analysis or principal component analysis (PCA) such a Ω1, Ω2, … Ωn factor images and such a ω1, ω2, … ωn can be found for an arbitrary F1, F2, … Fn image sequence that $F_{k}\approx\sum{\Omega_{i}\omega_{i}(k)}$ , and $(F-\Omega\omega)^{2}$ is minimal among the possible sequences with m element.

The pictures made by factor analysis generally cannot be regarded as physiological images, since they contain negative elements. The explanation for this is that application of factor analysis provides orthogonal factors (here factor images and factor weights), whose scalar product is 0. This would mean that only one organ can be projected into one pixel.

The factors can be transformed: If T is invertible, then with $\Phi=\Omega T$ and $\phi=T^{-1}\omega$ definitions, $(F-\Omega\omega)^{2}=(F-\Phi\phi)$ . If Φ and φ have not got negative elements, they can be considered to be philological.

Criteria used for searching T transformation are the followings:

• Positivity (every pixel of Φ_i and every point of φ_i > 0),
• At certain area certain organ does not appear (at ROI, Φ_i = 0)

Due to noise, usually the criteria can be accomplished only partially.
Questions, unsolved problems:
• Number of factors. Solution is influenced by the number of the required factors.
• Uniqueness of physiological factors. It is conceivable that repeating a measurement, different “physiological factors” are received using the same data.
• Stability of physiological factors. It is possible that repeating a recording (only noise is changing), significantly different factors are received.

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