Image deblurring or deconvolution using computational techniques have made a remarkable impact in fluorescence imaging by overcoming the limitations of optical microscopy which requires staining of biological samples. There are many deconvolution techniques available, but most of the techniques do not perform well in presence of noise.
The all pervasiveness of noise and improper estimation of point spread function degrade the resolution posing a major challenge to recover the true image in fluorescence microscopy. The motivation of this research proposal is to carry out comparative study and find optimal and robust deconvolution technique for various noise levels.
The research methodology will be divided in two stages: The initial phase of research will focus on understanding the noise models and study its impact on the imaging system which is characterized by point spread function. Since in fluorescence imaging noise is mostly due to photon counting, the Poisson noise model will be considered, and second stage will be investigation of the non-iterative and iterative algorithms for solving this optimization problem.
The non-iterative Wiener (mean square error) filter approach and the iterative approach will be analyzed. Results of the algorithm will be analyzed in terms quantitative metrics like signal to noise ratio and mean square error for both experimental and simulated data.
The proposed research contributions can be summarized as follows:
A comparative study of the system performances using the two different algorithms for Poisson and Gaussian noise model.
Identify pros and cons associated with each technique using test object and experimental samples.
Fluoresence imaging: widefield and confocal system
A fluorescence microscopy system is based on the phenomenon of fluorescence and phosphorescence. where 3-D information from the object is recorded in a set of 2-D images of different in-focus planes of the object.
Deconvolution in fluorescence image why is it problem application?
Image deblurring is still a major challenge in all fields especially in biological and astronomical field where low intensity signals are present. The presence of noise further aggravates the deblurring problem. The imaging system is in general, modeled by the impulse response of the system that is the point spread function .
Mathematically for linear shift invariant system in 3-D space, the observed image ‘I’ is obtained by the convolution of real object ‘O’ with the Point Spread function ‘H’ and ‘N’ is the noise shown as Eq. (1). .
i(x,y,z)=N(o(x,y,z) ⨂_3 h(x,y,z)) (1)
Here,⨂_3 refers to 3D convolution. Fourier Transform of equation (1) is given by
O ̂(u,v,w)= (I(u,v,w))/(H(u,v,w)) (3)
If noise is ignored in Eq. (1) then Fourier transformation of Eq. (1) is given by Eq. (2) where convolution operator becomes multiplication., the simple approach to find the true object is to divide the Fourier transform of the observed image by Fourier transform of PSF which is known as Optical transfer function (OTF). This method is referred as direct inverse filtering. However, it does not give stable result because of zeros in OTF.
Also, in presence of additive noise this becomes a serious problem at values where the Fourier transform of PSF (OTF) is very small or zero, which will amplify the noise and degrade the restoration by amplifying noise. The above problem in equation (1) for attaining real object ‘O’ by taking inverse is an ill posed problem as the solution is unstable and not unique .
Literature Review on solutions to solve ill posed problem:
Several algorithms have been developed to address the problem of inverse imaging [1, 2, 3, and 4]. The objective of all these algorithms was to estimate true object degraded by the process of image formation and acquisition system given by equation (1).
Wiener filter (Mean Square Error) restoration technique involves estimating O ̂, such that HO ̂, approximates I in the least squares sense . The issue with spectral zeros obtained by direct inverse filtering is resolved and has faster convergence. The Wiener filter criterion for 3D space is given below as
O ̂(u,v,w)=(H^* (u,v,w))/(|H(u,v,w)|^2+k) I(u,v,w) (4)
Here, k is the constant and is defined as the power spectra of the noise and the object, and H^* (u,v,w) is complex conjugate of H(u,v,w) .
From Eq. (4), it is noted that if 𝑘 = 0 then it becomes 1/H(u,v,w) which is an inverse filter. The Wiener filter is also termed as linear method as it requires single operation to estimate true object.
The limitations of the above approach are that it works for additive Gaussian noise. Since in fluorescence imaging noise is generally due to photon counting, it is Poisson noise model in general. Also, the characteristics of noise must be known to implement the above technique . Another noteworthy drawback is this method gives rise to ringing artifacts and the edge errors .
Janson -van Cittert’s iterative approach:
Van Cittert’s iterative image restoration is a classical method to recover the original image o (x, y, z) which calculates a new estimate of the object as a function of the previous estimate. Van Cittert’s algorithm was the first iterative method for constrained deconvolution . A positivity constraint on the restoration used to reduce the ringing artifact.
Mathematically, the Van Cittert’s spatial domain iteration is given as:
f ̂_(i+1) (n_1, n_2)= f ̂_i (n_1, n_2)+β( 〖g(n_1, n_2) -d(n_1, n_2 )*f ̂〗_i (n_1, n_2)) (5)
Here, difference of the current restoration result f ̂_i (n_1,n_1) is compared to the recorded image g(n_1, n_1) . The error image is multiplied by β a relaxation factor, which is added to the initial guess to yield the first iteration estimate, f ̂_1 of the true image. This is to prevent negative intensities. Initial guess of the true image f ̂_0 is assumed to be identical to zero or g(n_1, n_1). This is repeated until the difference is under a threshold.
However, this method is iterative and require more computational time and becomes unstable after few number of iterations .
The noise is introduced into the system during the image acquisition process which will affect the restoration. The prior knowledge of noise model is important for choosing a good restoration algorithm.
Poisson Noise Model:
In fluorescence imaging the noise is majorly due to effect of photons and is mostly single dependent Poisson Noise. The noise follows the Poisson probability distribution function and Poisson noise model is given as below:
og(x,y,z)=P(o[f(x,y,z) ⨂_3 h(x,y,z)+b(x,y,z)] (6)
Here, P is a Poisson process, and b(x,y,z) is the background noise, og(x,y,z) number of measured photons and o is inverse of photon conversion factor and f(x,y,z) is 3D image’
Gaussian Noise Model:
This is given as in Eq. (7) where N is an additive gaussian noise caused by charged coupled device (CCD) camera in the fluorescence microscope system.
i(x,y,z)=o(x,y,z) ⨂_3 h(x,y,z)+N(x,y,z) (7)
This study requires analysis and implementation of the Poisson and Gaussian noise model and will determine the system performances for various noise levels. The study will provide optimal solution for various noise levels.
Simulation of model:
Three different imaging conditions will be simulated of 6-µm and 15-µm spherical beads -first, the ideal case of noiseless image and two noisy cases based on Poisson and Gaussian noise model. The investigation for noisy condition will be repeated for various noise levels introduced into the simulated data. The various noise levels will be quantified by Signal to Noise ratio (SNR).
The restoration results will be quantified and compared based on measure of image quality. The measure commonly used measures are mean-Square error and signal to noise ratio .
Signal to Noise ratio:
The various noise levels will be quantified by using Signal to Noise ratio (SNR). SNR measures the signal strength relative to the noise and is given
Mean Square Error:
As the original image will be available, the performance of algorithm will be measured using the mean square error (MSE), defined as average squared difference between original image and a distorted image .
Mathematically MSE is given by:〖||o-f_k ||〗^2
Where, undistorted image is denoted by o and the k is the current iteration.
The percentage error will be computed to study the percentage errors versus the number of iterations. This will aid to determine how many iterations are needed to reach stable solution.
The frequency response of the system also known as optical transfer function (OTF) is mainly characterized by axial and lateral cut off frequency.
The frequency response of the Widefield microscope is a torus like region or donut shaped as shown in Figure. 1; the ‘‘hole’’ of the torus is the “missing cone” of information around w axis as seen in Figure 2. The w direction in frequency corresponds to the axial (z) direction in real space .
The resolution of the system is estimated by〖 u〗_c, where u_c is the cutoff frequency of the system and is equal to 1/d of the system where is d is the resolvable distance between 2 points . Abbe’s diffraction for lateral (i.e. xy) resolution and axial (z) resolution  is given by:
Abbe’s diffraction for axial (i.e. Z)  resolution is given by:
d=2 λ/NA^2 (3)
Where, λ is wavelength and NA is numerical aperture
Rayleigh Criteria: The Rayleigh Criterion is a slightly refined formula based on Abbe’s diffraction limits and that 2 adjacent points are resolvable . Lateral Resolution (XY) is given by:
Here, u_c is the cutoff frequency.
Optical sectioning (OS) capability of the system is defined as to get thin sections or slices of thick sample along the Z plan. The integrated intensity is constant for Widefield systems as seen in Fig 7. Hence, WFM doesn’t have optical sectioning capability .
The research project requires understanding of concepts of Signal processing. This research will be done in Computational Imaging Research Laboratory and requires sound knowledge of various image deconvolution algorithms and will be implemented in MATLAB.
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- Lagendijk, R. L., & Biemond, J. (2009). Basic methods for image restoration and identification. In The essential guide to image processing (pp. 323-348). (van cttert equation)
- Bertero, M., et al. “Image Deblurring with Poisson Data: From Cells to Galaxies.” Inverse Problems, vol. 25, no. 12, 2009, pp. 1–26, doi:10.1088/0266-5611/25/12/123006.