Image deconvolution refers to the operation of reconstructing a given blur image into a sharp image. In general, the relationship is assumed as the following model
where ⊗ denotes finite discrete convolution operator, h is a psf (point spread function) blur kernel, x is an unknown true sharp 2D image, and e is a noise in which elements are independently followed by Gaussian distribution
When
However, many blind deconvolution techniques, like in Fergus
For non-blind deconvoution, the most popular algorithm is RL (Richardson, 1972 ; Lucy, 1974). This technique is based on the maximum likelihood estimation by the EM algorithm under the Poisson model. However, this technique requires a lot of time to obtain the desired result. Moreover, as shown in the experiment in Section 5, it contains undesirable artifacts especially when the iteration is not sufficient.
In this paper, we provide non-blind deconvolution using the EM algorithm under the Gaussian model. However, due to the inherent slow convergence speed of the EM algorithm, the processing time is impractical when the image size is large. Therefore, in this study, an immediate restored image of the EM algorithm at any iterates is presented so that an iterative process is not required.
The following section presents terms and concepts necessary for this paper. Section 3 presents an iterative EM algorithm. In Section 4, an immediate solution of the iterative EM algorithm is derived. And based on this result, we examine the statistical properties of the restored image in the middle of iteration. In Section 5, we demonstrates the effectiveness of the proposed method through a simple experiment. Finally, in Section 6 conclusions and discussions are presented.
where
where
The unbiased least squared estimator which maximizes the log-likelihood,
is a good choice if the noise
To alleviate this problem, for a positive definite matrix
By maximizing the
It is well-known that
However, the optimal solution
A natural setting for understanding our problem as a missing/observed data structure is to look at it as follows,
where
Noting that the density
where
In M-step, substituting
where
where
At hence, at (
where
As an alternative, by applying the Green (1990)’s One-Step-Late (OSL) method, one can also use
By the essential property of EM algorithm, proceeding iteration monotonically increases the penalized log-likelihood
If the image
where
As mentioned previously, however, algorithm (
We can rewrite the
saying that the current estimates
and for the
For an initial estimates
where
Note that if 0 < |
Accordingly, our immediate EM solution after
In our case, it is straightforward to show that |
From (
which is just the solution (
which is a mixture of the initial estimates and the limit estimates where the proportion rate is the “weight matrix”
It is worth investigating the bias and variance of
Since the limit estimator
From
Consider an initial estimator where the bias is greater than
From these opposite properties, it can be expected to obtain a visually more satisfactory restored image than the
Meanwhile, the Fourier version of
where
It should be further noted that our
In this section, we use real images to show that immediate EM works properly. In Figure 1, a true image x, a blur kernel h, and the observed blur image y are provided. The x is a 512 × 512 RGB image where pixels are scaled between [0, 1]. The h a 49 × 49 uniform blur kernel with hight 0.0016 and its sum is 1. And the y is an observed image obtained by adding Gaussian noise with a standard deviation of
Our goal is to reconstruct the distorted image
In this experiment, the penalty term in
where
And the initial estimates of immediate EM is
Figure 2 demonstrates the log-likelihood of the EM algorithm over iteration, where it monotonically increases in all cases. In particular, the original EM of
The first row of Figure 3 provides the RL deconvolution results for 50, 1500, and 2500 iterations (using function
In the second row of Figure 3, Wiener deconvolution results are provided (using function
which shows better result. If the correct
In the third row of Figure 3, our immediate EM deconvolution results are provided using
The PC used in the experiment in this chapter is Intel(R) i7-8700 CPU (3.20GHz), and the program is implemented in python.
In this paper, a non-blind deconvolution by iterative EM algorithm under Gaussian model was provided. Then an immediate EM without iterative processing that provides the same result of iterative EM was developed. This significantly reduces the processing time of image deconvolution with EM algorithm, thereby increasing the practicality. The result makes it possible to reveal the statistical properties of bias and variance for the restored image. This has hitherto been vaguely recognized but not explicitly known.
The Richardson-Lucy method is an EM algorithm based on Poisson model, and it is necessary to study whether an immediate solution can be obtained for this method as well.
In this study, very limited experiments were provided and an excessively simple penalty term was used. The use of various initial images and edge preserving penaty is expected to provide more interesting results.