Equally Sloped Tomography (EST)
(Free Matlab Source Code and User-Friendly Software)

I. Overview :

Tomography has made revolutionary impacts in a number of fields ranging from medical imaging to electron microscopy. Conventional tomography reconstructs a 3D object from a set of equal-angle 2D projections. Since the set of projections are in polar coordinates and the object in Cartesian coordinates, interpolation has to be used in the reconstruction process, which introduces artifacts in the reconstructed 3D object. In some application, there are two additional difficulties: a limited number of projections due to radiation damage to the specimen and the missing wedge problem (i.e., the specimen cannot usually be tilted beyond and the data in the remaining projections are missing). In 2005, Miao and collaborators developed equally sloped tomography (EST) to overcome or alleviate these difficulties [1]. Unlike conventional tomography, EST makes use of a set of equally sloped projections. In a combination of the pseudopolar fast Fourier transform (PPFFT) [2,3] and the oversampling method with an iterative algorithm, EST makes superior 3D reconstructions to conventional tomography, such as weighted back-projection (WBP), the algebraic reconstruction technique (ART) and the simultaneous algebraic reconstruction technique (SART), when there is a limited number of projections and a missing wedge [1,4,5].

II. Principle of EST :

When the projections of a tilt series use equal slope increments, it has been shown that a direct fast Fourier transform, the PPFFT, exists between a pseudopolar grid and a Cartesian grid [2,3]. Figure 1 shows a pseudopolar grid and the PPFFT. For an N * N Cartesian grid, the corresponding pseudopolar grid is defined by a set of 2N lines, each line consisting of 2N grid points mapped out on N concentric squares. The 2N lines are subdivided into a horizontal group (in blue) defined by y = sx, where s is the slope and |s|<=1, and a vertical group (in red) defined by x = sy, where |s|<=1; the horizontal and vertical groups are symmetric under the interchange of x and y, and s = 2/N. When these conditions are met, the PPFFT and its inverse algorithm are mathematically faithful [3]. Note that the PPFFT and its inverse algorithm were originally developed to interpolate tomographic projections from a polar to a Cartesian grid in reciprocal space. The idea of acquiring tomographic tilt-series at equal slope increments and then combining the PPFFT with iterative algorithms for 3D image reconstructions was first suggested by Miao in 2005 [1].

Figure 1. Geometrical relationship between a pseudopolar and a Cartesian grid. For an N N Cartesian grid, the corresponding pseudopolar grid is defined by a set of 2N lines, each line consisting of 2N grid points mapped out on N concentric squares (left) with N = 8 in this example. The 2N lines are subdivided into a horizontal group (in blue) defined by y = sx, where |s|<=1, and a vertical group (in red) defined by x = sy, where |s|<=1. The horizontal and vertical groups are symmetric under the interchange of x and y, and s=2/N. The dashed circle on the pseudopolar grid represents the resolution circle. The grid points outside of the resolution circle cannot be obtained by applying the Fourier transform to the experimental projections [1].

Compared to other data acquisition approaches such as the Saxton scheme [6], the EST data acquisition approach is different in that it acquires projections with equal slope increments in order to use the PPFFT. Although the PPFFT and its inverse provide an algebraically faithful way to do the fast Fourier transform between the Cartesian and pseudopolar grids, three difficulties limit its direct application to electron tomography. First, the tilt range has to be from -90 degree to +90 degree. Second, the number of projections in a tilt series needs to be 2N for an N*N object. Third, the grid points past the resolution circle (dashed circle in Fig. 1) cannot be experimentally determined. These limitations are overcome by combining the PPFFT with an iterative process [1,4,5,7-13]. Figure 2 shows the schematic layout of the iterative EST method. The 2D projections are first converted to Fourier slices in the pseudopolar grid. As illustrated in Fig. 1, the distance between the sampling points on the individual 2N lines of the pseudopolar grid varies from line to line. In order to calculate the Fourier slices from the projections, the fractional Fourier transform (FrFT) is used to vary the output sampling distance of the Fourier slices [14]. By applying the inverse PPFFT, a 3D image in real space is obtained. A 3D support is defined to separate the object from a zero region where the size of the zero region is proportional to the oversampling of the projections [15]. The negative-valued voxels inside the support and the voxel values outside the support are set to zero, and a new 3D image is obtained. The forward PPFFT is applied to the new image and a set of calculated Fourier slices is obtained. The corresponding calculated Fourier slices are then replaced with the measured ones, and the remaining slices are kept unchanged. The iterative process is then repeated with each iteration monitored by an Rrecip, defined as

Where, and represent the measured and calculated Fourier slices. The algorithm is terminated after reaching a maximum number of iterations. A more detailed description of the EST method can be found in literature [7,8].

Figure 2 Schematic layout of the iterative EST method. The measured projections are first converted to the Fourier slices by the fractional Fourier transform (FrFT) [14]. The algorithm iterates back and forth between real and reciprocal space using the PPFFT and its inversion (Fig. 1). Recent work has shown that the inverse PPFFT can be replaced by the adjoint PPFFT, allowing for faster convergence without compromising the accuracy [11]. In real space, the negative-valued voxels inside the support and the voxel values outside the support are set to zero (that is, constraints are applied). In reciprocal space, the corresponding calculated slices are updated with the measured ones (in blue) and the remaining slices (in green) are unchanged. The algorithm is terminated after reaching a maximum number of iterations [7,8].

III. Application of EST :

EST has found application in a variety of fields, including atomic resolution electron tomography [7,8], phase contrast X-ray imaging [12,13], coherent diffraction imaging (CDI) [9,10], medical CT [5] and cryo-electron microscopy [4]. Below is a brief summary of the EST application.

III.1 Atomic resolution electron tomography

In a combination of annular dark field scanning transmission electron microscopy (ADF-STEM) with the center of mass and EST methods, electron tomography has been demonstrated for 3D imaging of a ~10 nm gold nanoparticle at resolution [7]. Individual atoms are observed in some regions of the particle and several grains are identified at three dimensions. The 3D surface morphology and internal lattice structure revealed are consistent with a distorted icosahedral multiply-twinned particle. More recently, the combination of 3D Fourier filtering and EST with high angle annular dark field (HAADF)-STEM has allowed the observation of nearly all the atoms in a multiply-twinned Pt nanoparticle [8]. The existence of atomic steps at 3D twin boundaries of the Pt nanoparticle is visualized, and the 3D core structure of edge and screw dislocations in the nanoparticle is imaged at atomic resolution for the first time. These dislocations and the atomic steps at the twin boundaries are hidden in conventional 2D projections, and appear to be a significant stress-relief mechanism. The ability to image 3D disordered structures such as dislocations at atomic resolution is expected to find application in materials sciences, nanoscience, solid state physics and chemistry.

III.2 High resolution, low dose phase contrast X-ray tomography

EST has been applied to achieve high resolution, low dose phase contrast X-ray tomographic imaging [12,13]. By combining phase contrast x-ray imaging with EST, we and collaborators have recently imaged a human breast in three dimensions and identified a malignant cancer with a pixel size of 92 m and a radiation dose less than that of dual-view mammography [13]. According to a blind evaluation by five independent radiologists, our method can reduce the radiation dose and acquisition time by ~74% relative to conventional phase contrast x-ray tomography, while maintaining high image resolution and image contrast. These results demonstrate that high resolution 3D diagnostic imaging of human breast cancers can, in principle, be performed at clinical compatible doses.

III.3 Three-dimensional coherent diffraction imaging (CDI)

EST has also been combined with CDI for 3D structural determination of materials science and biological samples [9,10]. In a combination of direct phase retrieval of coherent X-ray diffraction patterns [16] with EST, we carried out quantitative 3D imaging of a heat-treated GaN particle with each voxel corresponding to 17*17*17 [9]. We observed the platelet structure of GaN and the formation of small islands on the surface of the platelets, and successfully captured the internal core shell structure in three dimensions. We have also reported quantitative 3D imaging of a whole, unstained cell at a resolution of 50-60 nm, and identified the 3D morphology and structure of cellular organelles including cell wall, vacuole, endoplasmic reticulum, mitochondria, granules, nucleus and nucleolus inside a yeast spore cell [10].

III.4 Radiation dose reduction in medical CT

EST has been implemented with advanced mathematical regularization to investigate radiation dose reduction in medical CT [5]. The EST method was experimentally implemented on fan-beam CT and evaluated as a function of imaging dose on a series of image quality phantoms and anonymous pediatric patient data sets. Based on the phantom and pediatric patient fan-beam CT data, it has demonstrated that EST reconstructions with the 39 mAs flux setting produced comparable image quality, resolution and contrast relative to FBP with the 140mAs flux setting. Compared to ART and the expectation maximization statistical reconstruction algorithm, a significant reduction in computation time has been achieved in EST. Finally, numerical experiments on helical cone beam CT data suggest that the combination of EST and the advanced single-slice rebinning (ASSR) method produces reconstructions with higher image quality and lower noise than the Feldkamp Davis and Kress (FDK) method and the conventional ASSR approach.

III.5 Radiation dose reduction and image enhancement in biological imaging

Cryo-electron tomography is currently the highest resolution imaging modality available to study the 3-D structures of pleomorphic macromolecular assemblies, viruses, organelles and cells. Unfortunately, the resolution is currently limited to 3-5 nm by several factors including the dose tolerance of biological specimens and the inaccessibility of certain tilt angles. In 2008, we and collaborators reported the experimental demonstration of EST to alleviate these problems [4]. As a proof of principle, we applied EST to reconstructing frozen-hydrated keyhole limpet hemocyanin molecules from a tilt-series taken with constant slope increments. In comparison with WBP, ART and SART, EST reconstructions exhibited higher contrast, less peripheral noise, more easily detectable molecular boundaries and reduced missing wedge effects. More surprisingly, EST reconstructions including only two-thirds the original images appeared to have the same resolution as full WBP reconstructions, suggesting that EST can either reduce the dose required to reach a given resolution or allow higher resolutions to be achieved with a given dose. EST was also applied to reconstructing a frozen-hydrated bacterial cell from a tilt-series taken with constant angular increments. The results further confirmed the benefits of EST, even when applied to standard tilt-series.

IV. Free EST Matlab Source Code and User-Friendly Software :

If you use these codes for your publications and/or presentations, we request you at least cite the following papers:

IV.1 Free EST Matlab source code

IV.2 Free user-friendly EST software

If you are not an experienced Matlab user, you can freely install and use the user-friendly EST software, which does not require a Matlab license. To install this software package, you need a 64-bit machine with Windows OS (we have tested it with Windows 7 and Windows XP SP3). Below are the steps to install the user-friendly software.

References :

1. J. Miao, and O. Levi, Equally Sloped Tomography with Oversampling Reconstruction, Phys. Rev. B. 72, 052103 (2005).PDF]

2. R. M. Mersereau and A. V. Oppenheim, "Digital reconstruction of multidimensional signals from their projections," Proc. IEEE. 62, 1319-1338 (1974).

3. A. Averbuch, R. R. Coifman, D. L. Donoho, M. Israeli, Y. A. Shkolnisky, Y. A framework for discrete integral transformations I-the pseudopolar Fourier Transform. SIAM J. Sci. Comput. 30, 785-803 (2008).

4. E. Lee, B.P. Fahimian, C.V. Iancu, C. Suloway, G.E. Murphy, E.R. Wright, G.J. Jensen and J. Miao, Radiation Dose Reduction and Image Enhancement in Biological Imaging through Equally Sloped Tomography, J. Struct. Biol. 164, 221-227 (2008).[PDF]

5. B. P. Fahimian, Y. Zhao, Z. Huang, R. Fung, Y. Mao, C. Zhu, M. Khatonabadi, J. J. DeMarco, S. J. Osher, M. F. McNitt-Gray and J. Miao, "Radiation dose reduction in medical x-ray CT via Fourier-based iterative reconstruction",Med. Phys., 40, 031914(2013). [PDF]

6. W. O. Saxton, W. Baumeister, M. Hahn, "Three-dimensional reconstruction of imperfect two-dimensional crystals". Ultramicroscopy. 13, 57-71 (1984).

7. M. C. Scott, C. -Chun Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B. C. Regan & J. Miao, Electron tomography at 2.4-angstrom resolution, Nature, 483, 444-447 (2012). [PDF]

8. C.-C. Chen, C. Zhu, E. R. White, C.-Y. Chiu, M. C. Scott, B. C. Regan, L. D. Marks, Y. Huang and J. Miao, "Three-dimensional imaging of dislocations in a nanoparticle at atomic resolution" Nature, 496, 74-77(2013).PDF]

9. J. Miao, C. -C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T.K. Lee and S.H. Risbud, Three-Dimensional GaN-Ga2O3 Core Shell Structure Revealed by X-ray Diffraction Microscopy, Phys. Rev. Lett. 97, 215503 (2006).PDF]

10. H. Jiang, C. Song, C.-C. Chen, R. Xu, K. S. Raines, B. P. Fahimian, C.-H.Lu, T. K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, and J. Miao, Quantitative 3D Imaging of Whole, Unstained Cells by Using X-ray Diffraction Microscopy, Proc. Natl. Acad. Sci. USA 107, 11234-11239 (2010).[PDF]

11. Y. Mao, B. P. Fahimian, S. J. Osher, and J. Miao, Development and Optimization of Regularized Tomographic Reconstruction Algorithms Utilizing Equally-Sloped Tomography, IEEE Trans. Image Processing 19, 1259-1268 (2010). [PDF]

12. B. P. Fahimian, Y. Mao, P. Cloetens, and J. Miao, Low Dose X-ray Phase-Contrast and Absorption CT Using Equally-Sloped Tomograph, Phys. Med. Bio. 55, 5383-5400 (2010).[PDF]

13. Y. Zhao, E. Brun, P. Coan, Z. Huang, A. Sztrokay, P. C. Diemoz, S. Liebhardt, A. Mittone, S. Gasilov, J. Miao and A. Bravin, "High resolution, low dose phase contrast x-ray tomography for 3D diagnosis of human breast cancers,"Proc. Natl. Acad. Sci. USA 109, 18290-18294 (2012). [PDF]

14. D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier transform and applications". SIAM Rev. 33, 389-404 (1991).

15. J. Miao, D. Sayre and H. N. Chapman, Phase Retrieval from the Magnitude of the Fourier transform of Non-periodic Objects, J. Opt. Soc. Am. A. A 15, 1662-1669 (1998).[PDF]

16. J. Miao, P. Charalambous, J. Kirz and D. Sayre, Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens, Nature 400, 342-344 (1999).[PDF]