According to literature, the CS framework operates on the knowledge that most natural signals are sparse in some domain given by a sparsifying transform operation $\Phi$ (Fourier, Haar, WHT, etc.).
Following a single-pixel camera example, if we are trying to image a phenomenon described by a signal $x$, we use a sampling matrix $A$ to obtain a compressed measurement $y=A\Phi x$ of a representation of $x$ in the $\Phi$ domain, which is sparse.
My question is, assuming we have some kind of physical single-pixel device (using a DMD to project linear combinations of $x$ (given by the sampling matrix $A$) onto a detector to obtain a measurement vector $y$), where does the sparsifying transform operation $\Phi$ manifest in this device?
Do the DMD patterns correspond not to the sampling matrix $A$, but to the $A\Phi$ product?
- Must we provide the DMD with the signal in the sparse domain, $\Phi x$? In that case, don't we require previous knowledge of $x$ in its entirety in order to obtain its transform in the $\Phi$ domain? It is my understanding this is not the case, but I fail to grasp the mapping between the mathematical framework and hardware implementation in this matter.