This document proposes a 3D deconvolution method for wide field fluorescence microscopy images using sparse image priors. It uses a maximum a posteriori approach with two priors: 1) a global hyper-laplacian prior to preserve edges, and 2) a local smooth mask to suppress ringing artifacts. The method is tested on synthetic and real zebrafish embryo images, outperforming total variation and Tikhonov regularization in reducing blur and artifacts.

1. Three-Dimensional Deconvolution of Wide Field Microscopy with Sparse Priors:
Application to Zebrafish Imagery
Bo Dong, Ling Shao, Alejandro F Frangi, Oliver Bandmann, Marc Da Costa
The University of Sheffield, UK
bdong2@sheffield.ac.uk & ling.shao@ieee.org
3. Methodology
Convolution process: 𝑔 = 𝑛(ℎ ∗ 𝑓)
Poisson distribution: 𝑝 𝑘 =
𝜆 𝑘 𝑒−𝜆
𝑘!
, 𝑘 = 0,1,2, …
Maximum a-Posteriori (MAP):
𝑝 𝑓|𝑔, ℎ =
𝑝 𝑔 𝑓, ℎ 𝑝(𝑓)
𝑝(𝑔)
∝ 𝑝 𝑔 𝑓, ℎ 𝑝(𝑓)
Where 𝑝 𝑔(𝑥)|𝑓, ℎ =
(𝑓∗ℎ)(𝑥) 𝑔(𝑥) 𝑒−(𝑓∗ℎ)(𝑥)
𝑔 𝑥 !
Sparse image priors: 𝑝 𝑓 = 𝑝 𝑔(𝑓) 𝑝𝑙(𝑓)
1) Global prior 𝑝 𝑔 𝑓 ∝ 𝑒−𝜏 𝛻𝑓 𝑖
𝛼
, (0 < 𝛼 < 1)
(a) One frame of Z-stack image (b) log-gradient distribution
2) Local prior 𝑝𝑙(𝑓) = 𝑁(∇𝑓𝑖 − ∇𝑔𝑖|0, 𝜎)𝑖∈ϕ (similar to [1])
(a) Original frame (b) Texture area (c) Ringing map
Optimization: 𝑓𝑘+1 𝑥 =
𝑓 𝑘(𝑥)
1−λ 𝑔ѱ 𝑔−λ 𝑙ѱ 𝑙
∙ [ℎ −𝑥 ∗
𝑔(𝑥)
(𝑓 𝑘∗ℎ)(𝑥)
]
ѱ 𝑔 = 𝑑𝑖𝑣(
𝛻𝑓 𝑘
|𝛻𝑓 𝑘|2−𝛼) and ѱ𝑙 = 𝑑𝑖𝑣( 𝛻𝑓𝑘 − 𝛻𝑔 𝑀(𝑥))
PSF: Model in [2] is used, and the microscopy parameters are required.
2. Contribution
• The MAP method with sparse priors is applied to 3D deconvolution for Poisson
noise WF microscopy.
• The Hyper-Laplacian distribution as a global prior is used in the deconvolution
framework. The Hyper-Laplacian distribution is a better model to describe the
log-gradient distribution of the image compared with previously used models,
especially for WF microscopy.
• A local mask as the second prior is applied to the 3D deconvolution problem for
reducing ringing artifacts.
4. Results: Synthetic Data – Hollow Bar
5. Synthetic Data –Hela Cell Nucleus
6. Real Data – Zebrafish Embryo
Fig.3. Deconvolution results of 4 different methods. The top row indicates the central xy-section of the 3D image data, and the bottom
row shows the central xz-section. (a): The ground truth data. (b): The blurred images. (c): Result with TM regularization after 120
iterations. (d): Result with TV regularization. (e): Result with the proposed HL. (f): Result with HL+Mask. (g): The trend of the RMSE
during iterations ( Blue line: TM, Green line: TV, Black line: HL, Red line: HL+Mask ).
Fig.4. The HeLa cell dataset results using 4
different methods. (a): The ground truth
data. (b): The blurred images. (c): Result
with TM regularization . (d): Result with
TV regularization. (e): Result with the
proposed HL. (f): Result with HL+Mask.
The second figure shows the trend of the
RMSE during iterations ( Blue line: TM,
Green line: TV, Black line: HL, Red line:
HL+Mask ).
Fig.5. The Zebrafish data results using 4
different methods. The central xy-section and
central xz-section of the z-stack images are
shown in each block. (a): Central frame of the
Recorded images. (b): TM. (c): TV. (d):
Proposed HL. (e): Proposed HL+Mask.
To observe the results in detail, we magnify
the results shown in the second figure: Central
xy-section with a yellow square. (b): Zoom-in
of the yellow square region. (c): TM. (d): TV.
(e): Proposed HL. (f): Proposed HL+Mask.
Reference
[1] Q. Shan, J. Jia, and A. Agarwala, “High-quality motion deblurring from a single image,”
ACM Transactions on Graphics, vol. 27, no. 3, p. 73, 2008.
[2] S. Hiware, P. Porwal, R. Velmurugan, and S. Chaudhuri, “Modeling of PSF for refractive
index variation in fluorescence microscopy,” in IEEE Conference on Image Processing, 2011,
pp. 2037–2040.
7. ISBI 2014 challenge Data
1. Abstract
In this work, we propose a Bayesian Maximum a-Posteriori (MAP) method with
the sparse image priors to solve three-dimensional (3D) deconvolution problem for
Wide Field (WF) fluorescence microscopy images from zebrafish embryos. The
novel sparse image priors include a global Hyper-Laplacian model and a local
smooth region mask. These two kinds of prior are deployed for preserving sharp
edges and suppressing ringing artifacts, respectively.
Fig.1. (a) is one frame of a zebrafish
embryo z-stack. (b): The blue curve
describes the gradient distribution (5xf) of
the image. The Hyper-Laplacian with α =
1/3 is a more appropriate model to describe
the log-gradient distribution (blue),
compared with the Gaussian (black),
Laplacian (yellow) or Hyper-Laplacian
with α= 2/3 (green) log-distribution.
Fig.2. Applying the triangle threshold
algorithm to a WF fluorescent zebrafish
image. (a) Original frame. (b) Binary
mask produced by the triangle
threshold algorithm. The white area
indicates the texture region. (c) The
ringing map is visualized by computing
the difference between the result
using the local prior and the result
without using the local prior, which is
visualized between the intensity range
[0, 50].
Fig.6. The results of testing the HL+Mask method on the dataset from the
’Second International Challenge on 3D Deconvolution Microscopy’.
Maximum-intensity projections of the data (a), (c), (e), (f) and
corresponding results (b), (d), (f), (h) are shown in the figure. All the
results are displayed as the maximum intensity value projection along the
vertical (z) direction.
Fig.7. The results of testing the TM, TV, and
HL+Mask methods on the dataset from the
’Second International Challenge on 3D
Deconvolution Microscopy’. All the results
are displayed as the maximum intensity value
projection along the vertical (z) direction.
(a) Original (b) TM
(c) TV (d) HL+Mask