In this work, we propose a no-reference method for automatic choice of the parameters of Perona-Malik image difusion algorithm for the problem of image denoising. The idea of the approach it to analyze and quantify the presence of structures in the diference image between the noisy image and the processed image as the mutual information value. We apply the proposed method to photographic images and to retinal images with modeled Gaussian noise with diferent parameters and analyze the efects of no-reference parameter choice compared to the optimal results. The proposed algorithm shows the efectiveness of no-reference parameter choice for the problem of image denoising.
One of the main challenges in image processing is
denoising, as images are often corrupted by noise
during acquisition, transmission or storage. The goal is to
restore the original image by removing all noise while
preserving the contents. Image denoising is usually
needed as a preparation step in other image
processing methods. There has been a great research efort
in that field, yet the problem remains unsolved. In
this paper, we will use non-linear difusion method
proposed in [
must be controlled by parameters entered by a user
or estimated automatically. A common approach for
automatic choice of the parameters is to estimate the
noise level and then choose the parameters according
to this noise level [
A less common approach is to analyze the
preservation of image contents after image restoration and
to pose the stopping criterion of anisotropic
difusion. For example, the work [
case the diference image must contain just random values without any structures from the original image.
of the parameters for Perona-Malik image difusion for
on non-linear difusion that considers the cleaned
image as the solution of the heat conduction. The
diffusion coefficient is chosen to reduce the difusivity
in locations, which have more likelihood to be edges.
Such methods allow to preserve edges while denoising
due to the right choice of coefficient. Koenderink [
∂u
= div(c∇u), (x, t) ∈ Ω × [0, T ], ∂t u(x, 0) = l0,
x ∈ Ω, ∂u
= 0, (x, t) ∈ ∂Ω × [0, T ], ∂⃗n where l0 is the input image defined in spatial domain Ω, c is the difusion coefficient, u(x, T ) is the result of heat distribution at moment T .
In linear difusion, the coefficient c is considered
to be constant and independent of the image. In
nonlinear difusion, the coefficient c is a function of image
gradient magnitude c = c(|∇u|), which controls the
blurring efect. Setting c to 1 in interior of each
region and 0 at the boundaries will encourage smoothing
within a region and stop it on the edge, so that the
boundaries remain sharp. In [
( ( s )2) c1(s) = exp − K (1) and c2(s) = (1 s )2 , (2) 1 +
K where K is the parameter of the method.
un+1 = un + tn · c(|∇u|)∆u, u0 = u(x, 0) = I0, ∑ tn = T. n
rameters for the Perona-Malik image difusion
algorithm for images of the following two classes:
• Photographic images from TID database [
white Gaussian noise with diferent levels σ in [
4. Fullreference parameter analysis
of (K, T ) parameters that maximizes PSNR and
image there is a set of (K, T ) values producing the results that are almost indistinguishable from the optimal result. The set is banana-shaped and lies perpendicular to the line passing though the zero point. Fig. 2 shows an example of optimal (K, T ) values for one of the images for diferent noise levels.
Noise = 3 Noise = 8 Fig. 2. A visualization of optimal (K, T ) parameters for an image with different noise levels in terms of
logarithmic scale. The vertical axis represents T value. Top-left corner is (0, 0) point. White regions corresponds to (K, T ) values that produce images with PSNR values close to the optimal value. Black regions correspond to PSNR values equal or less than
ratio K/T can be fixed, and the parameter optimization becomes one-dimensional, but for diferent noise level the optimal ratio K/T set is diferent.
In order to go from two-dimensional to onedimensional parameter optimization for any noise level, we have analyzed the behavior of optimal (K, T ) values and have found out that a set of optimal points (log K, √T ) lies along a line. Therefore, we introduce single-argument parameterization for (K, T ) values: p K = q1q2 ,
T = p2, (3) where the coefficients q1 and q2 are chosen experimentally by optimizing the full-reference metrics values.
q1 = 0.1 and optimized q2 value. The ranges of optimal values for TID images and for DRIVE images are diferent, but they intersects. We have chosen q2 = 4600 from the intersection. 5. Noreference parameter choice
We use the algorithm [
compared to noise removal.
We use the following scenario: an image denoising algorithm is executed with diferent parameters, then the mutual information µ value is calculated between the input image and each denoising result, and the image that minimizes the mutual information is chosen as the optimal result. In practice, there can be several local minima, and a special analysis should be performed in order to choose the optimal result.
single-parameter model (3), we find the optimal p value using both full-reference and no-reference approach based on calculating the mutual information coefficient.
It has been found that mutual information correlates well with PSNR and SSIM values for noise level σ > 2. An example is shown in Fig. 3. A argument where PSNR and/or SSIM reaches its maximum is close to a local minimum of µ(p) function. In the case of several local minima points, we find the one that maximizes the drop: popt = argp mp′<axp µ(p′) − µ(p). (4)
In the case of very low noise level (σ ≤ 2), the method has limited application. Non-linear difusion improves the image very little in the case of low noise.
TID, I04, noise 10 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 PSNR SSIM MU
DRIVE, I03, noise 6 1,2 1 0,8 0,6 0,4 0,2 0 1,2 1 0,8 0,6 0,4 0,2 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61
PSNR SSIM MU Fig. 3. Examples of the dependence of PSNR, SSIM and mutual information on the parameter p corresponding to denoising strength. The PSNR and
noising parameter choice are presented in table 1. The results are averaged for all the images with noise level σ > 2.
gorithm has worse PSNR and SSIM values than the optimal ones, the diference between the results of the proposed algorithm and the optimal results is practically indistinguishable, and the efectiveness o f image denoising is clearly visible.
The work was supported by Russian Science Foundation grant 17-11-01279. 0,1 0,09 0,08 0,07 0,06 0,05 0,04 0,03 0,02 0,01 0 0,06 0,05 0,04 0,03 0,02 0,01 0
TID