Using prior information for speeding up the measurement of fiber transmission matrices
[S. Li et al., arxiv, 2007.15891, (2020)]
Due to disorder and dispersion, knowing the transmission matrix of a multimode fiber is usually required to reconstruct an input image for endoscopic applications. In the general case, its characterization for a fiber allowing \(N\) guided modes requires at least \(N\) complex measurements. However, we usually have additional information, the most common one being that the matrix is never totally random, and usually sparse, when expressed in the mode basis. In this study, the authors use such prior information to reduce drastically the number of measurements for the transmission matrix estimation using the framework of compressed sensing. They demonstrate the validity of such an approach for endoscopic imaging through multimode fibers.
In typical situations, we are almost never in a situation in which the transmission matrix is totally random in the mode basis. Even in the worst situation in which you mess up the fiber on purpose by, for instance, pressing on the fiber on multiple locations, mode mixing also comes with mode-dependent losses, which introduces some sparsity in the transmission matrix. The opposite extreme case is the one of a perfect straight fiber. By definition of the propagation modes, in the basis of defined by those modes, the transmission matrix is purely diagonal. Because losses are negligible if the fiber is short enough, the absolute values of the diagonal elements are very close to one. You then have only \(N\) unknown variables (the phase on the diagonal) instead of \(2 N\) (amplitude and phase of each matrix component) in the general case. This means that if you illuminate the fiber with one illumination that excites all the modes, by measuring the output field, you can estimate accurately the transmission matrix with this only measurement. That is the idea behind the previous paper of the same team [S. Li et al., arxiv, 2005.06445 (2020)]. The present paper explores the cases in between those extreme ones using compressed sensing.
By definition of the transmission matrix \(\mathbf{H}\), the output field \(y\) is related to the input one \(x\) by
$$
y = \mathbf{H} x
$$
For a set of \(m\) input vectors \(x\) in the mode basis forming a \(M\) by \(m\) matrix \(\mathbf{X}\), we measure an output set of fields represented by the \(M\) by \(m\) matrix \(\mathbf{Y}\) with
$$
\mathbf{Y} = \mathbf{H} \mathbf{X}
$$
If we assume low noise, the system has a unique solution if the rank of \(\mathbf{X}\) is \(N\). This in particular the case if we sent \(m\) orthogonal inputs or if \(m \gg M\). In other words, this matrix equation represents a set of \(m \times N\) linear equations with \(N^2\) unknown variables. If you have as many equations as unknown variables (\(m=N\)) and these equations are linearly independent or if you have more equations than unknown variables (\(m \gg N\)), you will be able to solve the system.
If \(m < M\), the system is ill-posed, it has an infinite number of solutions that minimizes the quadratic error :
$$
\eta = ||\mathbf{Y} - \mathbf{H} \mathbf{X}||_ 2^2
$$
The idea behind compressed sensing is to add other terms to the cost function we try to minimize in order to regularize the system, i.e. to come back to a system that converges to one solution.
$$
\eta = ||\mathbf{Y} - \mathbf{H} \mathbf{X}||_ 2^2 + \lambda f\left(H\right)
$$
The term \(f\left(H\right)\) depends upon the prior knowledge you have about the system. In general, the prior knowledge you have tells you that your unknown signal, here the transmission matrix, can be represented by only a few parameters in a given representation basis. So you want to choose \(f\left(H\right)\) so that minimizing it enforces the sparsity of your system in a given representation.
In the paper, the authors isolate three cases:
- A direct sparsity constraint. It physically means the knowledge of a basis in which the light is little scattered. In this basis, only a few parameters will be non-zero. Off-diagonal terms representing conversion from one mode to another one, they will have mostly small values.
- Approximate model, when you have an accurate model parameterized by only a few parameters.
- A previous estimation of the matrix. When the matrix changes over time and if those changes are small, you only need to estimate the parameters characterizing this change. It is then the deformation of the matrix that is sparse.
If one knows for instance that the matrix is sparse in a given basis, in the case of a multimode fiber it will be the mode basis, we can add a term that, when minimized, enforces the sparsity of the matrix in this basis. Then, while there is an infinite number of solutions that minimizes the error, there is only one that minimizes the error and maximizes the sparsity (kind of).
Another important problem in compressed sensing is the choice of the measurement basis. Let's take an example decorrelated from this one, and say we want to recover an image that we know is sparse in the pixel basis. If you want to use compressed sensing to perform fewer measurements than the total number of pixels, you cannot choose the pixel basis as the measurement basis. If you miss one pixel, you have no way of knowing if it was on or off. You would want to choose a basis that is totally different, like performing projections on cosine functions. You know that projecting a signal onto a cosine function will give at least a bit of information about almost all the pixels. In the general case, as stated by the authors, "a good measurement basis is incoherent with respect to the predicted sparse basis".
Figure 1. Experimental focusing using transmission matrices reconstructed using sparse measurements (compressed ratio of 10%). Image from [S. Li et al., arxiv, 2007.15891, 5 (2020)].
In slightly disordered multimode fibers, the transmission matrix is sparse in the mode basis. As this basis is delocalized over the entire core of the fiber, we are in a situation somewhat opposite to the previous example. So choosing a localized basis as a measurement basis seems to be a good choice. Here the authors chose a set of diffraction-limited spots as input excitations.
Figure 2. Imaging using compressively recorded transmission matrices with a compressed ration of 10%, 5%, and 3%. Image from [S. Li et al., arxiv, 2007.15891, 5 (2020)].
The authors also show that one mode, determined by its radial and azimuthal numbers \(m\) and \(l\), would not couple to modes that have too different numbers. So we do not just know that the matrix is sparse, we also know where to expect the non-zero values, i.e. we know its support. In Figure 1, the authors show the results of focusing using transmission matrices reconstructed with 10 times fewer measurements than the number of modes. They compare the case where they add the support constraint or not.
Because of dispersion and disorder, when we illuminate a multimode fiber with coherent light, the input information is scrambled and we observe at the distal end a speckle pattern. However, when we know the transmission matrix of any given linear system, we can inverse the system and recover the input information/image. In Figure 2. the authors used the matrices reconstructed using compressed sensing to recover images. Even with a number of measurements representing only 3% of the size of the basis (so we theoretically lose 97% of the information if we do not use compressed sensing), the image can be recovered with good fidelity.