Can ICA be applied, when the number of mixture signal is less than number of source signal?

Question

I am referring to the the following paper : Non-contact, automated cardiac pulse measurements using video imaging and blind source separation

In the above article, the authors are able to extract cardiac pulse signal out from RGB components. I try to visualize the process as follow.

R' = R + cardiac pulse
G' = G + cardiac pulse
B' = B + cardiac pulse

R', G' and B' are the colour components observed by camera. R, G, B are the colour components for a person, by assuming that he doesn't have any cardiac pulse.

It seems that we will be having 4 sources (R, G, B, Cardiac pulse). We are now trying to obtain 1 of the 4 sources (Cardiac pulse) from 3 mixture signals (R', G', B'), by using ICA.

Does it make sense? Am I missing some techniques? Or, am I making a wrong assumption on the process?

Answer 1

You might also want to consider Principal Component Analysis (PCA) or an extension of it known as Independent Subspace Analysis which is PCA followed by ICA. These techniques work very well for extracting pitch stationary signals from a single observation signal. I'm an audio specialist but have discussed biomedical signals with colleagues in the past and from recollection cardiac pulses from a single observation are pretty well characterised and thus would be suitable sources for extraction using ISA. I have used it to great avail to separate drums from full musical polyphonies.

Answer 2

You are making a wrong assumption on the process. In ICA, the number of mixtures must be at least as many as the number of components. The paper you cite does in fact, acknowledge this:

These observed signals from the red, green and blue color sensors are denoted by $x_1(t)$, $x_2(t)$ and $x_3(t)$ respectively, which are amplitudes of the recorded signals (averages of all pixels in the facial region) at time point $t$. In conventional ICA the number of recoverable sources cannot exceed the number of observations, thus we assumed three underlying source signals, represented by $s_1(t)$, $s_2(t)$ and $s_3(t)$.

The conversion $x_i^'=(x_i-\mu_i)/\sigma_i$ is merely centering and sphering of the data, which I explain in another answer on this site.

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The cases considered in the paper are the noiseless ICA model and noisy ICA. In other words, the heart rate measurements considered at rest (not a pulseless model as you suggested) is the ICA model:

$$\mathbf{x}(t)=\mathbf{As}(t)$$

where $\mathbf{x}$ is the observed vector, $\mathbf{s}$ is the underlying component vector and $\mathbf{A}$ is the mixing matrix.

On the other hand, heart rate measurements when in motion can be considered as

$$\mathbf{x}(t)=\mathbf{As}(t)+\mathbf{n}(t)$$

where $\mathbf{n}(t)$ is a noise vector (in this case the motions).

Answer 3

When there are more sources than sensors the problem is referred to as over-complete ICA or under-determined ICA. You can google that. Your case is more tractable than for instance the case of one sensor and two sources and if your model is really correct you already know the mixing matrix. It might be worth to look further into. Cheers