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Spectral unmixing methods attempt to determine the concentrations of different fluorophores present at each pixel location in an image by analyzing a set of measured emission spectra. Unmixing algorithms have shown great promise for applications where samples contain many fluorescent labels; however, existing methods perform poorly when confronted with autofluorescence-contaminated images.

We propose an unmixing algorithm designed to separate fluorophores with overlapping emission spectra from contamination by autofluorescence and background fluorescence. First, we formally define a generalization of the linear mixing model, called the affine mixture model (AMM), that specifically accounts for background fluorescence. Second, we use the AMM to derive an affine nonnegative matrix factorization method for estimating fluorophore endmember spectra from reference images. Lastly, we propose a semi-blind sparse affine spectral unmixing (SSASU) algorithm that uses knowledge of the estimated endmembers to learn the autofluorescence and background fluorescence spectra on a per-image basis. When unmixing real-world spectral images contaminated by autofluorescence, SSASU greatly improved proportion indeterminacy as compared to existing methods for a given relative reconstruction error.

The source code used for this paper was written in Julia and is available with the test data at https://github.com/brossetti/ssasu.

 

In this paper, we propose an efficient numerical scheme for solving some large‐scale ill‐posed linear inverse problems arising from image restoration. In order to accelerate the computation, two different hidden structures are exploited. First, the coefficient matrix is approximated as the sum of a small number of Kronecker products. This procedure not only introduces one more level of parallelism into the computation but also enables the usage of computationally intensive matrix–matrix multiplications in the subsequent optimization procedure. We then derive the corresponding Tikhonov regularized minimization model and extend the fast iterative shrinkage‐thresholding algorithm (FISTA) to solve the resulting optimization problem. Because the matrices appearing in the Kronecker product approximation are all structured matrices (Toeplitz, Hankel, etc.), we can further exploit their fast matrix–vector multiplication algorithms at each iteration. The proposed algorithm is thus calledstructuredFISTA (sFISTA). In particular, we show that the approximation error introduced by sFISTA is well under control and sFISTA can reach the same image restoration accuracy level as FISTA. Finally, both the theoretical complexity analysis and some numerical results are provided to demonstrate the efficiency of sFISTA.

 

Abstract In this paper, we study the L 1 / L 2 minimization on the gradient for imaging applications. Several recent works have demonstrated that L 1 / L 2 is better than the L 1 norm when approximating the L 0 norm to promote sparsity. Consequently, we postulate that applying L 1 / L 2 on the gradient is better than the classic total variation (the L 1 norm on the gradient) to enforce the sparsity of the image gradient. Numerically, we design a specific splitting scheme, under which we can prove subsequential and global convergence for the alternating direction method of multipliers (ADMM) under certain conditions. Experimentally, we demonstrate visible improvements of L 1 / L 2 over L 1 and other nonconvex regularizations for image recovery from low-frequency measurements and two medical applications of magnetic resonance imaging and computed tomography reconstruction. Finally, we reveal some empirical evidence on the superiority of L 1 / L 2 over L 1 when recovering piecewise constant signals from low-frequency measurements to shed light on future works.