ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 607 (2009) 208–211
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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Enhancement of spatial resolution of roentgenographic methods Frantisek Krejci a,b,, Jan Jakubek a, Jiri Dammer a, Daniel Vavrik a a b
Institute of Experimental and Applied Physics, Czech Technical University in Prague, Horska 3a/22, CZ 12800 Prague 2, Czech Republic Faculty of Biomedical Engineering, Czech Technical University in Prague, Nam. Sitna 3105, 272 01 Kladno, Czech Republic
a r t i c l e in f o
a b s t r a c t
Available online 25 March 2009
The spatial resolution of X-ray micro-radiography is usually the crucial parameter determining its performance and range of possible applications. The lower limit of the spatial resolution of a roentgenographic method is given by the size and shape of the so-called Point Spread Function (PSF) of the imaging system. We demonstrate two methods for resolution enhancement (high- and lowmagnification) based on the precise measurement of the PSF and the implementation of deconvolution. In the case of high geometrical magnification, the X-ray source shape becomes the most important factor for PSF determination. A technique for the precise full 2-D measurement of X-ray spot size has been developed. The methods have been successfully demonstrated on the real radiographic images. The spatial resolution has been thus improved up to three times. Another possibility for resolution enhancement is sub-pixel micro-shifting of the X-ray detector used. The successful demonstration of X-ray radiograph spatial resolution enhancement based on micro-shifting of the Medipix2 detector and measurement of its PSF is included. & 2009 Elsevier B.V. All rights reserved.
Keywords: X-ray imaging Spatial resolution Deconvolution Medipix
1. Introduction Combination of state-of-the-art hybrid pixel semiconductor detectors with newly available micro- and nano-focus X-ray sources opens many new possibilities in radiation imaging [1,2]. Semiconductor single particle counting pixel detectors (Medipix2, Timepix [3,4]) offer many advantages for X-ray imaging: energy discrimination, noiseless digital integration (counting), high frame rate, high detection efficiency at low energies (5–15 kV), and virtually unlimited dynamic range (i.e. the dynamic range is essentially unlimited and it is possible to reach almost any arbitrary signal to statistical noise ratio just by exposure time prolongation). All these properties allow achieving high-quality microradiographs even for very low contrast and low absorption objects. The spatial resolution of X-ray micro-radiography (similar to any other imaging method) is the crucial parameter and its improvement still remains an experimental and data-processing challenge. From theoretical principles, the signal on the detector is given by convolution of the PSF of the imaging system and the spatial transmission function of the object (see Fig. 1). The image on the detector has limited spatial resolution, which is theoretically given by the size of the PSF. Thus, deconvolution should lead to improvement of the spatial resolution beyond this
Corresponding author at: Institute of Experimental and Applied Physics, Czech
Technical University in Prague, Horska 3a/22, CZ 12800 Prague 2, Czech Republic. Tel.: +420 224 359 260; fax: +420 224 35 93 92. E-mail address:
[email protected] (F. Krejci). 0168-9002/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2009.03.153
Fig. 1. Image formation in ideally linear shift invariant systems. The image is given by convolution of the object under imaging and the system PSF.
limit. The imaged Object (its transmission function) can be reached by solving convolution Eq. (1.1) (e.g. easily in Fourier domain). Object PSF ¼ Image
(1.1)
2. Simulations In physical measurements the situation is usually closer to ðObject PSFÞ þ ¼ Image
(2.1)
where e is some type of noise that has entered the recorded signal. However, in this case it is not clear as to how this equation should be deterministically solved. A number of deconvolution algorithms have been developed that attend specifications of each task. Selection and implementation
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of the most suitable method for the case of micro-radiography was one part of our work. In our numerical simulations we have tested the ability of some deconvolution methods (namely Wiener deconvolution, regularized deconvolution, blind deconvolution [5], and Lucy– Richardson algorithm [6,7]) to improve spatial resolution by deconvolution of the image obtained on the detector with the deterministically known PSF. The general scenario of simulations is illustrated in Fig. 2. The object is blurred (i.e. convoluted) with the deterministically known PSF of Gaussian shape. Using the Poisson random number generator the corresponding image is computed, which should be obtained by an ideal photon counting detector. The real object is then restored using the deconvolution algorithm with the deterministically known PSF. The typical results of edge profile restoration are shown in Fig. 3. It can be seen that the deconvoluted image usually shows two types of errors: (i) resolution restoration, which is not optimal, and (ii) the artefacts. Fig. 4. Spatial resolution enhancement as a function of photon statistics. Resolution is given as the width of the edge response measured between 10% and 90% of the relative signal.
Fig. 2. Illustration of principle of deconvolution scenario used in simulations.
Fig. 5. Artefacts in deconvoluted image as a function of photon statistics. The measure of artefact effects is evaluated in the sense of mean square error.
However, in our simulations it has been demonstrated that both types of errors (i, ii) are sufficiently reduced by increasing photodetection statistics. With the simulation it can be determined that the sufficient number of detected photons is about 105–106 per pixel as illustrated in Fig. 4 (resolution improvement as a function of photon statistics) and Fig. 5 (artefacts in deconvoluted image as a function of photon statistics).
3. PSF measurement and resolution enhancement 3.1. High geometrical magnification
Fig. 3. Typical results of edge profile restoration. Two errors are evident: artefacts and imperfect resolution restoration.
From theoretical principles it is clear that in the case of high geometrical magnification the lower limit of the spatial resolution of a radiographic method is given by the spot size of the imaging X-ray source (e.g., pinhole collimated X-ray source, micro- or nano-focus X-ray tube radiation, divergence of the synchrotron or free electron laser X-ray beam). The finite X-ray source size causes
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the object under imaging to be irradiated from more than one point. This results in geometrical blurring, which linearly increases with geometrical magnification (see Fig. 6). The effective X-ray spot shape (i.e. shape of the spot in the detector plane) is in the view of linear theory the PSF, which can be used for spatial resolution enhancement via the deconvolution. For the effective operation of deconvolution algorithms with real radiographs it is necessary to measure the real PSF. Even a very small inaccuracy in PSF determination leads to huge artefacts and reconstruction is practically impossible. The measurement of PSF for the case of high geometrical magnification was done according to the scenario depicted in Fig. 7. The precise pinhole (Oxfordlasers: 100 mm diameter, stainless steel 50 mm width) was imaged with high geometrical magnification (100 ). The PSF (see Fig. 8) was then obtained as a result of deconvolution (Lucy–Richardson algorithm) of this image and an ideal circle aperture transmittance function.
Fig. 9. Spatial resolution enhancement using the Lucy–Richardson algorithm. The comparison of the measured image (after flat field correction) and the image after deconvolution application is illustrated.
Row 65 measured image deconvolution
1.2
Relative signal
1 90% 0.8 3 x resolution enhancement
0.6 0.4 0.2
10% 0 125 Fig. 6. Geometrical blurring in imaging caused by finite source size.
130
135
140
145 Pixel
150
155
Fig. 10. Quantitative results of resolution enhancement. The profile is taken from Fig. 9.
In Fig. 8 it can be seen that the measured PSF (which is in fact the spot intensity distribution of the X-ray tube FXE-160.50 from Feinfocus used during measurement) can be perfectly approximated by a 2D Gaussian. Using such an approximation the spatial resolution of the real X-ray radiographs has been improved for high-contrast objects up to three-fold (see Figs. 9 and 10). 3.2. Low geometrical magnification
Fig. 7. Deconvolution scenario used for the 2-D PSF measurement.
Fig. 8. Measured PSF for the case of high geometrical magnification. Measured data can be perfectly fitted by 2-D Gaussian.
The effect of geometrical magnification is commonly employed for spatial resolution enhancement using divergent X-ray source beams. However, due to the limited detector size, the dimension of the observed area decreases with increasing geometrical magnification. Another possibility for resolution enhancement is sub-pixel micro-shifting of the X-ray detector used. Although this approach is quite common for optical imaging, it has not been fully investigated for X-ray imaging. In view of the theoretical linearity, the spatial resolution can be enhanced using the deconvolution of oversampled image and the detector PSF. For this purpose we have measured the Medipix2 detector PSF (see Fig. 11) using imaging a perfect straight edge with low geometrical magnification. The measurement has been done with sub-pixel precision. For details see [8]. The principle of oversampling is demonstrated in Fig. 12. In our experiments with oversampling, the deconvolution problem (i.e. resolution enhancement problem) has been solved in 1-D. The results of the Lucy–Richardson deconvolution technique applied on the oversampled image of a straight edge of the metal sheet (three 18 mm micro-shifts of the Medipix2 used) are shown in Fig. 13.
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Fig. 13. Results of spatial resolution enhancement using the suppixel micro-shifts of Medipix2 detector and Lucy–Richardson algorithm. The comparison of oversampled image of the metal sheet and the reconstructed image is presented.
Fig. 11. Point Spread function of the Medipix2 detector: function was obtained as a derivative of step response function.
on real radiographic images. The spatial resolution has been improved for high-contrast objects up to three-fold. This means that, using our X-ray setup equipped with a state-of-the-art nanofocus X-ray source, the proposed method enables an imaging resolution below 250 nm. One possible task in the future will be testing this approach for real biological and material structure investigations using this resolution. We have successfully illustrated that the approach using detector micro-shifts can be also used for resolution enhancement in X-ray radiography. For this purpose we have measured the Medipix2 detector PSF. Software development for radiograms sequence processing, design and realization of the precise micro-scanning system for the full 2-D case is under way.
Acknowledgments This work has been carried out in frame of the Medipix Collaboration and has been supported by Projects LC06041 and MSMT 6840770040 of the Ministry of Education, Youth and Sports of the Czech Republic. References [1] J. Jakubek, Nucl. Instr. and Meth. Phys. Res. A 576 (2007) 223. [2] J. Jakubek, et al., Nucl. Instr. and Meth. Phys. Res. A 563 (2006) 278. [3] X. Llopard, M. Campbell, R. Dinapoli, D. San Segundo, E. Pernigotti, IEEE Trans. Nucl. Sci. NS-49 (2001) 2279. [4] X. Llopart, R. Ballabriga, M. Campbell, L. Tlustos, W. Wong, Nucl. Instr. and Meth. Phys. Res. A 581 (2007) 485. [5] R. Neelamani, Inverse problems in image processing, Ph.D. Thesis, Rice University, 2003. [6] L.B. Lucy, Astron. J. 79 (6) (1974) 745–754. [7] W.H. Richardson, J. Opt. Soc. Am. 62 (1) (1972) 55–59. [8] J. Jakubek, et al., Nucl. Instr. and Meth. Phys. Res. A 509 (2003) 294.
Fig. 12. Principle of sub-pixel micro-shifting. Every sub-pixel shift brings new information about the imaged object. If the shifts are equidistant the composite image is put together as described by the row numbers.
4. Conclusion We have successfully measured in 2-D the X-ray spot shape. Image restoration using deconvolution has been demonstrated
