Poisson Noise Reduction with Non-local PCA

Abstract

Photon-limited imaging arises when the number of photons collected by a sensor array is small relative to the number of detector elements. Photon limitations are an important concern for many applications such as spectral imaging, night vision, nuclear medicine, and astronomy. Typically a Poisson distribution is used to model these observations, and the inherent heteroscedasticity of the data combined with standard noise removal methods yields significant artifacts. This paper introduces a novel denoising algorithm for photon-limited images which combines elements of dictionary learning and sparse patch-based representations of images. The method employs both an adaptation of Principal Component Analysis (PCA) for Poisson noise and recently developed sparsity-regularized convex optimization algorithms for photon-limited images. A comprehensive empirical evaluation of the proposed method helps characterize the performance of this approach relative to other state-of-the-art denoising methods. The results reveal that, despite its conceptual simplicity, Poisson PCA-based denoising appears to be highly competitive in very low light regimes.

Appendices

Appendix A: Biconvexity of Loss Function

Lemma 1

The function L is biconvex with respect to (U,V) but not jointly convex.

Proof

The biconvexity argument is straightforward; the partial functions U↦L(U,V) with a fixed V and V↦L(U,V) with a fixed U are both convex. The fact that the problem is non-jointly convex can be seen when U and V are in $\mathbb{R}$ (i.e., ℓ=m=n=1), since the Hessian in this case is

$$ H_L(U,V) = \begin{pmatrix} V^2 e^{UV} \ \ & UV e^{UV}+e^{UV}-Y\\ UV e^{UV}+e^{UV}-Y \ \ & U^2 e^{UV} \end{pmatrix} . $$

Thus at the origin one has , which has a negative eigenvalue, −1. □

Appendix B: Gradient Calculations

We provide below the gradient computation used in Eqs. (13) and (3):

Using the component-wise representation this is equivalent to

Appendix C: Hessian Calculations

The approach proposed by [19, 35] consists in using an iterative algorithm which sequentially updates the jth column of V and the ith row of U. The only problem with this method is numerical: one needs to invert possibly ill conditioned matrices at each step of the loop.

The Hessian matrices of our problems, with respect to U and V respectively are given by

$$ \frac{\partial^2 L(U,V)}{\partial U_{a,b} \partial U_{c,d}}=\left \{ \begin{array}{l@{\ \ }l} \sum_{j=1}^N \exp(UV)_{a,j} V^2_{b,j}, & \mbox{if } (a,b)=(c,d), \\[3pt] 0 & \mbox{otherwise,} \end{array} \right . $$

and

$$ \frac{\partial^2 L(U,V)}{\partial V_{a,b} \partial V_{c,d}}=\left \{ \begin{array}{l@{\ \ }l} \sum_{i=1}^M U^2_{i,a} \exp(UV)_{i,b}, &\mbox{if } (a,b)=(c,d), \\[3pt] 0&\mbox{otherwise.} \end{array} \right . $$

Notice that both Hessian matrices are diagonal. So applying the inverse of the Hessian simply consists in inverting the diagonal coefficients.

Appendix D: The Newton Step

In the following we need to introduce the function $\operatorname{Vect}_{C}$ that transforms a matrix into one single column (concatenates the columns), and the function $\operatorname{Vect}_{R}$ that transforms a matrix into a single row (concatenates the rows). This means that

and

Now using the previously introduced notations, the updating steps for U and V can be written

(25)

(26)

We give the order used to concatenate the coefficients for the Hessian matrix with respect to U, H _U : (a,b)=(1,1),…,(M,1),(1,2),…(M,2),…(1,ℓ),…,(M,ℓ). We concatenate the column of U in this order.

It is easy to give the updating rules for the kth column of U, one only needs to multiply the first equation of (25) from the left by the M×Mℓ matrix

$$ F_{k,M,\ell,} = \begin{pmatrix} 0_{M,M}, & \ldots,& I_{M,M},&\ldots,& 0_{M,M} \end{pmatrix} $$

(27)

where the identity block matrix is in the kth position. This leads to the following updating rule

$$ {U_{t+1,\cdot,k}=U_{t,:,k} -D_k^{-1} \bigl( \exp(U_tV_t) -Y\bigr)V_{t,k,:}^\top } ~, $$

(28)

where D _k is a diagonal matrix of size M×M:

This leads easily to (13).

By the symmetry of the problem in U and V, one has the following equivalent updating rule for V:

$$ {V_{t+1,k,:}=V_{t,k,:} -U_{t,:,k}^\top\bigl( \exp(U_tV_t) -Y\bigr) E_{k,M}^{-1} } ~, $$

(29)

where E _k is a diagonal matrix of size N×N: