I hope this is enough of a question (and less of a start of a discussion) to be allowed here. I am trying to get my head around the ideas of inverse and adjoint operators. To keep it simple, let's stay with real, linear ones.

Let's say we have $$ y = A \cdot x, A \in \mathbb{R}^{m \times n} $$ and x and y such that the above makes sense :) Now we are given a measurement y and try to recover x, which in many cases does not work (especially if m < n). So, applying the adjoint to the measurement, $$ A^T \cdot y $$ is what one does in many problems -- but can one do "better"?

Let's suppose that A is the product of some other matrices: $$ A = B \cdot C, B \in \mathbb{R}^{m \times m}, C \in \mathbb{R}^{m \times n} $$

Of course, the transpose is $$ A^T = C^T \cdot B^T $$ and we get $$ A^T \cdot y = C^T \cdot B^T \cdot y $$

However, if B is invertible (especially, if it's diagonal or, even more, a non-trivial multiple of the identity), we could also do something like this: $$ A^T \cdot y = C^T \cdot B^{-1} \cdot y $$ and if B is not self-adjoint, that's a different result.

In a sense, this comes down to either "blindly applying the adjoint" (using the adjoint of full A) or "pre-correcting" the measurements by first computing $$ y' = B^{-1} \cdot y $$ and then applying the adjoint of the non-invertible part of A (which is C -- hence C^T) to y'.

(If anyone is interested in an application: In Positron Emission Tomography, one measures the Radon transform of a 2D distribution. If one considers tissue density, each individual measurement is attenuated by a known transmission factor < 1, hence a diagonal matrix such as B. Usually, precorrection is used to correct for attenuation, and the usual dual Radon transform (the adjoint of the Radon transform without attenuation) is applied to the precorrected measurement. One could also apply the dual of the Radon transform with attenuation (attention: this is not the attenuated Radon transform!), but this is never done.)

My question now is, why and when is "partial inversion" applied, and what benefits (or downsides) does it have? My guess is that a "simpler" operator--adjoint-operator pair produces results which are better suited to approximate the real inverse x. For example, I suspect that the inversion formula for the Radon transform (http://en.wikipedia.org/wiki/Radon_transform#Inversion_formulas) $$ f \propto (-\Delta)^{(n-1)/2}R^*Rf $$ would not be as simple if R was not the "pure" Radon transform.

If anyone has insight into this general problem and/or any helpful literature on this topic of inverse problems and adjoint operators, I would be very happy.



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