Unmixing biological fluorescence image data with sparse and low-rank Poisson regression

Abstract Motivation Multispectral biological fluorescence microscopy has enabled the identification of multiple targets in complex samples. The accuracy in the unmixing result degrades (i) as the number of fluorophores used in any experiment increases and (ii) as the signal-to-noise ratio in the recorded images decreases. Further, the availability of prior knowledge regarding the expected spatial distributions of fluorophores in images of labeled cells provides an opportunity to improve the accuracy of fluorophore identification and abundance. Results We propose a regularized sparse and low-rank Poisson regression unmixing approach (SL-PRU) to deconvolve spectral images labeled with highly overlapping fluorophores which are recorded in low signal-to-noise regimes. First, SL-PRU implements multipenalty terms when pursuing sparseness and spatial correlation of the resulting abundances in small neighborhoods simultaneously. Second, SL-PRU makes use of Poisson regression for unmixing instead of least squares regression to better estimate photon abundance. Third, we propose a method to tune the SL-PRU parameters involved in the unmixing procedure in the absence of knowledge of the ground truth abundance information in a recorded image. By validating on simulated and real-world images, we show that our proposed method leads to improved accuracy in unmixing fluorophores with highly overlapping spectra. Availability and implementation The source code used for this article was written in MATLAB and is available with the test data at https://github.com/WANGRUOGU/SL-PRU.


The Proposed Algorithms for Solving PNMF and SL-PRU
We first present the developed algorithms for solving the endmember extraction problem through PNMF and for solving the abundance estimation problem through SL-PRU, respectively. To this end, we first recall that the endmember m ∈ R C + extracted from a reference image Y m ∈ R C×N + and the corresponding abundances a ∈ R N + are obtained through Poisson Nonnegative Matrix Factorization (PNMF): where 1 C and 1 N are vectors of length C and N whose entries are all 1 and • denotes element-wise multiplication. PNMF (1) can be solved by multiplicative update algorithm (Lee and Seung, 2000) which is a diagonally rescaled version of gradient descent. Denoting any variable X at the t-th, iteration as X (t) and the maximum norm as ∥ · ∥ ∞ , the pseudocode of the multiplicative update algorithm is provided in Algorithm 1. The abundance vector a and the endmember vector m are initialized with random values that follow the standard uniform distribution U (0, 1). At each iteration, the endmember vector is standardized by dividing by its maximum for uniqueness. The algorithm stops when the relative change of the standardized endmember m between the (t − 1)-th and the t-th iterations given by is less than a small threshold value.
until the stopping criterion is satisfied ; Algorithm 1: Multiplicative update algorithm for PNMF (1) We are now in a position to detail the proposed algorithm for solving the proposed regularized sparse and low-rank Poisson regression unmixing approach (SL-PRU) that reads Inspired by the work in Giampouras et al. (2016), an alternating direction method of multipliers (ADMM) technique (Boyd et al., 2011) is adopted in our study by first letting all elements of w p be equal to ensure the convexity of the low-rankness regularization term in SL-PRU (2). Similar to the work in Giampouras et al. (2016), we introduce auxiliary variables V 1 ∈ R C×N , V 2 , V 3 , V 4 ∈ R R×N and reformulate SL-PRU (2) as follows min U,V1,V2,V3,V4 where I R+ (·) is the indicator function which is zero if all the entries are nonnegative and infinity otherwise. The augmented Lagrangian function for the constrained optimization problem (3) is given as follows: where D 1 ∈ R C×N , D 2 , D 3 , D 4 ∈ R R×N denote the Lagrange multipliers, tr(·) denotes matrix trace, µ > 0 is a Lagrange multiplier regularization parameter, and ∥ · ∥ F denotes the Frobenius norm.
iii Denoting the identity matrix of size k × k as I k and the scaled Lagrange multipliers as D ′ i = D i /µ, i = 1, 2, 3, 4, the augmented Lagrangian function L 1 can be rewritten as The proposed ADMM-type algorithm for solving SL-PRU sequentially optimizes (4) or (5) with respect to each variable while the other variables remain as the latest values. Note that the augmented Lagrangian (5) is convex w.r.t. U, V 1 , V 2 , V 3 , and V 4 , respectively, due to the assumption that all elements of w p are equal and the fact that all entries of W q are nonnegative. Therefore, at the t-th iteration, the updates of the abundance matrix U and the auxiliary variables V 1 , V 2 , V 3 , and V 4 can be deduced, respectively, as follows: Updating U: The minimization of L 2 w.r.t. U at the t-th iteration is equivalent to As a result, we have Updating V 1 : Denoting the (c, n)-th entry of V 1 as v cn with c = 1, · · · , C and n = 1, · · · , N , we have Since where √ · denotes the element-wise square root.
which can be solved by a soft-thresholding operation (Cai et al., 2010) on the singular values of V 2 .
Updating D ′ 1 , D ′ 2 , D ′ 3 , and D ′ 4 : These scaled Lagrange multipliers are updated as follows D The stopping criteria adopted in the algorithm are based on the primal and dual residuals (Boyd et al., 2011) r p and r d given by that go to 0, respectively, as t → ∞. The algorithm terminates whenever any of the ℓ 2 norms of r p or r d is less than a small threshold value or some number of iterations is reached. To enhance the performance of the algorithm, as done in Giampouras et al. (2016), we also update the weights w p and W q based on U (t) at the t-th iteration as follows r denotes the r-th row of U (t) and ε > 0 is assigned a small value to avoid singularities. The pseudo-code of the proposed algorithm for solving SL-PRU is presented in Algorithm 2.   (Amann et al., 1990) Supplementary   Magenta arrows in (C) identify peaks with a full-width-at-half-maximum of approximately 4-5 pixels (0.7-0.9 µm), the known diameter of oral Streptococcus cells (Baron, Samuel and Patterson, Maria Jevitz, 1996).
x 6 Supplementary Movie: SL-RPU Unmixed Dental Plaque Smear 3-D volume-rendered movie of a dental plaque smear labeled with 8 taxon-specific FISH probes (See Figure 6 in Main Text for color legend). Spectral image was unmixed with SL-RPU.