I am an experimental physicist who is collecting a series of coherent imaging of trapped gas. If you are familiar with phase contrast imaging, you may understand what I mean by coherent imaging. The information about the sample is retrieved as interference between a phase-shifted version illumination and light scattered by the sample. The light I am using is monochromatic. Since the imaging technique is coherent, the point-spread function (PSF) I need to use is $J_1(x)/x$ instead of the usual $(J_1(x)/x)^2$ for a circular aperture. This means that PSF has both positive and negative values. (It may be even complex in real-life situations.) I will describe the details of what I actually observe in an image below.

Also, since I am working with a low light-level condition, I thought it is best to use the Richardson-Lucy (RL) algorithm, which seems tailored for deconvolving images with Poisson noise. My understanding of the RL algorithm is based on Gordon Wetzstein - Stanford EE367/CS448I: Computational Imaging and Display - Lecture 10 (slide numbers 26-30). The algorithm assumes that you have an input $\vec{x}$, which goes through a linear operator $\hat{A}$ (convolution with PSF), and the output is a Poisson random variable $\vec{b}$ with average $\langle\vec{b}\rangle = \hat{A}\vec{x}$.

The assumptions used for the RL algorithm are consistent with typical incoherent imaging, where the PSF and the image are everywhere non-negative, but they are not in general for coherent imaging which may have negative values (because negative values wouldn't make much sense for a Poisson distribution).

Question: would there be a way to adopt the RL algorithm for coherent imaging, or more in general, PSF with negative values?

More detailed description of a typical image. $I$ denotes intensity. $E$ denotes complex electric field, and $I = |E|^2/2$.

$$ \begin{align*} I_{\text{image}} &= \frac{|E_{\text{scattered}} + i E_{\text{illumination}}|^2}{2} - I_{\text{illumination}}\\ &= I_{\text{scattered}} + \mathrm{Im}(E_{\text{scattered}} E_{\text{illumination}}^*)\\ E_{\text{illumination}} &= \text{plane wave}\\ E_{\text{scattered}} &\sim\text{spherical wave focused to a spot}\\ |E_{\text{scattered}}| &<< |E_{\text{illumination}}| \end{align*} $$ the last condition allows us to ignore $I_{\text{scattered}}$ and focus on the $\mathrm{Im}(E_{\text{scattered}} E_{\text{illumination}}^*)$. Since $E_{\text{illumination}}$ is more-or-less flat, you could say we are imaging a magnified version of $E_{\text{scattered}}$ times its relative phase shift $\phi$ (small) with respect to the illumination.

$$ \begin{align*} E_{\text{scattered}} &= E_{\text{illumination}} (e^{i\phi_{gas}}-1)\\ &\approx E_{\text{illumination}}i\phi_{gas}\\ I_{\text{image}}[x,y] &\approx |E_{\text{illumination}}|^2\cdot\phi_{gas}[x,y] \end{align*} $$

Here is a sample image that shows fringes. sample image

  • $\begingroup$ What about adding large value to the output so all values are non negative, apply the algorithm both to the image and to the added value and then remove it? $\endgroup$
    – Royi
    Oct 31 '20 at 13:30
  • $\begingroup$ @Royi, I think the problem is that it is unclear what to do with the normalization factor in the RL algorithm. See the term $\mathrm{diag}(A^T\,1)$ in bigwww.epfl.ch/deconvolution/challenge2013/…. EDIT: never mind, I think as long as the integral of kernel over all space is positive, it should be okay. $\endgroup$
    – wcc
    Nov 2 '20 at 18:01

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