For practical application of x-ray in-line phase contrast imaging, a high-quality image is essential for object perceptibility and quantitative imaging. The existing approach to improve image quality is limited by high cost and physical limitations of the acquisition hardware. A useful image restoration algorithm based on fast wavelet transform is proposed. It takes advantage of degradation model and extends the modulation transform function (MTF) compensation algorithm from Fourier domain to wavelet domain. The modified algorithm is evaluated through comparison with the conventional MTF compensation algorithm. Its deblurring property is also characterized with the evaluation parameters of image quality. The results demonstrate that the modified algorithm is fast and robust, and it can effectively restore both the lost detail and edge information while ringing artifacts are reduced.
©2011 Optical Society of America
For its great potential of enhancing x-ray imaging sensitivity and tremendously reducing radiation dose to the sample, phase contrast imaging is a powerful nondestructive detection technology in medicine, biology, and materials science [1–4]. Among all phase contrast imaging techniques, x-ray in-line phase contrast imaging  is the simplest, since no additional complex optical components are needed. In this technique, the phase-shifted x-ray diffracts and generates dark-bright interference fringes at boundaries and interfaces of the sample. These fringes carry phase information, and it is essential for phase retrieval and quantitative imaging . However, fringe images are degraded by a practical imaging system, which impedes precise reconstruction and detail identification.
The degradation arises from partial coherence of the source, finite resolution of the detector, and unavoidable noise of the system. To obtain a higher quality image, great efforts have been made to improve source size and the imaging detector’s resolution and sensitivity [7–10]. Nevertheless, it is limited by high cost and physical limitations of the acquisition hardware. An alternative method is image restoration. Suffering from oversmoothing of edges [11–13], ringing artifacts [14–16], or being computationally intensive and complex [17–20], the classical algorithms (including frequency and spatial domain algorithms) are unable to be directly applied in phase-contrast image restoration. It is reported that the restoration algorithms extended from spatial domain to wavelet domain can deal with the oversmoothing of edge and ringing artifacts, since wavelet analysis has the properties of multiresolution and excellent time-frequency localization [21–24]. However, wavelet algorithms are still complex or slow in computation. Therefore, a special image restoration method for in-line phase contrast imaging is needed. We noted that the MTF (modulation transform function) compensation algorithm is a fast and simple tool for image restoration, although it will lead to ringing artifacts and noise amplification. If we extend the MTF compensation algorithm from Fourier domain to wavelet domain, it may provide a unique solution to phase-contrast image restoration.
In this paper, a modified algorithm combining MTF compensation with wavelet transform is proposed. Its procedures are summarized in the following. First, a series of denoising processing that preserve edges is implemented to obtain the image with high signal-to-noise ratio (SNR). Then the image is deblurred through MTF compensation in wavelet domain. Finally, the image denoising with wavelet thresholding is applied again to suppress the noise that is amplified by deblurring processing. The results indicate that the lost detail and edge information are restored effectively and artifacts are suppressed as well.
2. Degradation model of x-ray in-line phase contrast image and MTF compensation
To expound the effect of the degradation on an ideal image, the degradation model is described in a way that differs from an ideal imaging system to a practical one. The simplified schematic of x-ray in-line phase contrast imaging is shown in Fig. 1 . Consider a sample of light element material irradiated by a monochromatic x-ray of wavelength. In the ideal imaging system, the incident wave emitted from a point source interacts with an object forming a particular distribution of wave amplitude just after the object plane. Then the wave propagates through free space and is registered in a detector. The propagation process is accompanied by the interference of scattered and transmitted waves causing the fringe along the edge of fine structure in the sample. Here we assume that the detector is an ideal detector with infinite resolution. The Fourier transform of intensity distribution in the detector plane is calculated as follows :
In practice, an ideal interference pattern is always deteriorated owing to the corruption of blurring and noise. The blurring effect is caused by partial coherence of the source and spatial resolution of the x-ray detector. It can be described with MTF of the system, which is generally measured by blade-edge method . The noise includes additive and multiplicative noises. The additive noise is composed of dark current and thermal noises, while the multiplicative noise is arising from the nonuniform distribution of x-ray and difference in gain coefficient of pixels.
As a partial coherence imaging system, x-ray in-line phase contrast imaging is assumed to be a linear shift-invariant system for objects with weak absorption. In frequency domain, the degradation model, i.e., the relationship among observed imageoriginal image additive noise image and multiplicative noise image is expressed as 
On the basis of the degradation model, the lost information is recovered via denoising and deblurring. The additive and multiplicative nonstochastic noises are canceled respectively by subtraction and division in spatial domain. Using a logarithm, the stochastic multiplicative noise is converted to stochastic additive noise, which is erased by utilizing wavelet threshold denoise . These denoising procedures effectively preserve the edge information and tremendously improve the SNR. Then a fast deblurring algorithm, which is valid only for an image with a high SNR is employed. The basic principle of the algorithm is to measure the descend degree of the MTF of the system, then calculate the ascend degree of high frequency part of the true image. To avoid noise amplification, wavelet denoise is employed again for the deblurred image. In conclusion, the restored image using MTF compensation in Fourier domain is given by 
3. Wavelet transform theory and the modified algorithm
3.1 Wavelet transform based on FFT
Wavelet transform represents an image as a sum of wavelet functions (wavelet bases) with different locations and scales. It is implemented by a pair of analysis filters (low pass filter L and high pass filter H). Figure 2(a) shows the wavelet transform of cameraman image (256 × 256) up to the second scale, and Fig. 2(b) gives the configuration of subbands. Using the synthesis filters, the original image is exactly reconstructed from the coefficients of the bases.
The fast wavelet transform is conducted with the Mallat algorithm , which makes use of the coefficients to decompose or reconstruct the image without knowing the concrete form of scaling and wavelet functions. Although halving the computation burden, the algorithm based on the convolution or matrix vector is still computationally intensive for dealing with large size images . To reduce calculation load, we can employ fast Fourier transform (FFT) to implement wavelet transform.
Let us assume that the Jth level image has a size of, where J = 1, 2, 3... is the level of the decomposition and is the size of origin image. The procedures of image wavelet decomposition based on FFT include the following:Eq. (4), and denote the horizontal and vertical unit vectors, respectively. Note that L and H are orthogonal mirror filters and n × 1 matrix. The Fourier transform (FT) of Eq. (4) is expressed as
Taking inverse FT of Eq. (9), we obtain
Steps (1)–(4) are repeated until the required level decomposition can be obtained.
3.2 The modified algorithm for image restoration
The decomposition mentioned above can transform any single-channel linear space-invariant filtering problem into a multichannel one. Transforming Eq. (3) into wavelet yields27]. We get the deblurred image by inverse wavelet transform for these subimages. In Eq. (12), the assumed stationary length in this image model is reduced from to. It means that the assumption of global space-invariant is relaxed , leading to some “traditional” artifacts in stationary image restoration, such as ringing artifacts and oversmoothing of edges, are reduced. Certainly, too much decomposition will make the computation become complex. Thus, it is necessary to choose a proper decomposition level. The procedures of the approach are given in Fig. 3 .
4. Experiment and image quality assessment
Our modified algorithm was tested by the experimental image of a cockroach’s leg. The image acquisition system includes a micro-focus x-ray source L7901-01 of Hamamatsu photonics and an x-ray FDI cooled scientific grade detector with pixel size of 6.5 μm × 6.5 μm. The sample was located at a distance of 200 mm from the detector, the imaging magnification M was 2, and the exposure time was 30 s. The experiment was conducted with tube currents of 50 μA and tube voltage of 80 kV. Figure 4(a) illustrates the x-ray image of an edge device (1384 × 1032) made of lead, and Fig. 4(b) shows the wavelet transform of the image whose approximation subimage is adopted to compute the MTF curve. The results of the edge spread function and MTF are given in Fig. 5 .
Figure 6(a) shows the x-ray in-line phase contrast image of the sample (1032 × 1032) acquired under the same experimental parameters. Obviously, the details and fringes nearly cannot be distinguished owing to the corruption of blur and noise. The denoising image is given in Fig. 6(b), where the details become cognizable. Utilizing the conventional MTF compensation algorithm, we obtain a deblurred image in which the lost fringe and detail information are restored. However, the deblurred image still suffers from ringing artifacts (Fig. 6(c)). From Fig. 6(d), it can be observed clearly that the lost edge and detail information are recovered without the oscillation along the edges, which demonstrates the validity of the modified MTF compensation algorithm.
The deblurring performance of the modified algorithm was also assessed with the parameters of image quality evaluation [29,30] by comparison between denoised and restored images, and the results are shown in Table 1 . The contrast of the image is the difference in visual properties that makes an object distinguishable from the background, and it is determined by , whereis the gray level co-occurrence matrix and i and j are the values of gray level. The contrast of restored image increases 2.17 times (from 0.06 to 0.19) compared with that of denoised image, which implies that the visibility of the image is enhanced remarkably. The energy describes the uniformity of gray level distribution within the image and is expressed as. Usually the image with high energy has a coarse and blurred texture. The decrease of the energy of the restored image (from 0.37 to 0.33) indicates that the texture becomes sharp. The entropy of image is a measure of information content and is defined as. The increase of the entropy of the restored image (from 1.48 to 1.87) results in an image with richer information contents. In other words, the modified algorithm can restore the lost information effectively. The rich degree of detail signal of an image is characterized by the local energy, which is expressed as 
In conclusion, we presented a useful image restoration method for an x-ray in-line phase contrast imaging system. It extends the MTF compensation algorithm from Fourier domain to wavelet domain and is tested by an experimental image. The deblurring performance of the modified algorithm is characterized in terms of parameters of image quality evaluation. The results indicate that the contrast, edge energy, and local energy increase remarkably, which demonstrates the effectiveness of our modified algorithm. The modified algorithm will be an effective preprocessing approach for quantitative phase-contrast imaging and tomography owing to its advantages of suppressing artifacts, preserving edge information, and having low computation complexity. It can also be applied in x-ray absorption contrast imaging.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 10875085, 91022002, and 10974143) and the Innovation Program of Shanghai Municipal Education Commission (Grant No. 11ZZ29).
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