Comparison of maximum entropy and filtered back-projection methods to reconstruct rapid-scan EPR images

Abstract

Reconstruction of two-dimensional images by filtered back-projection (FBP) and by the maximum entropy method (MEM) was compared for spectral-spatial EPR images with differing signal-to-noise ratios. Experimental projections were recorded using direct-detected rapid scans in the presence of a series of magnetic field gradients. The slow-scan absorption lineshapes were calculated by Fourier deconvolution. A Hamming filter was used in conjunction with FBP, but not for MEM. Imperfections in real experimental data, as well as random noise, contribute to discrepancies between the reconstructed image and experimental projections, which may make it impossible to achieve the customary MEM criterion for convergence. The Cambridge MEM algorithm, with allowance for imperfections in experimental data, produced images with more linear intensity scales and more accurate linewidths for weak signals than was obtained with another MEM method. The more effective elimination of noise in baseline regions by MEM made it possible to detect weak trityl (13)C trityl hyperfine lines that could not be distinguished from noise in images reconstructed by FBP. Another advantage of MEM is that projections do not need to be equally spaced. FBP has the advantages that computational time is less, the amplitude scale is linear, and there is less noise superimposed on peaks in images. It is useful to reconstruct images by both methods and compare results. Our observations indicate that FBP works well when the number of projections is large enough that the star effect is negligible. When there is a smaller number of projections, projections are unequally spaced, and/or signal-to-noise is lower MEM is advantageous.

Figure 1

Rapid scan projections in the presence of magnetic field gradients, after Fourier deconvolution to recover the slow-scan lineshapes. a) Gradient = 2.2 G/cm. b) Gradient = 0.01 G/cm. For each pair of scans the upper and lower traces were obtained by averaging 5,000 or 50,000 scans, respectively.

Figure 2

2-D spectral-spatial EPR image of a small tube containing solid LiPc and a larger tube containing an aqueous solution of trityl-CD₃. The centers of tubes were separated by 11 mm. Each of the 60 projection was averaged 5,000 times with scan frequencies of 1 to 8 kHz and a scan rate of 13.9 kG/s. A Hamming filter was used in conjunction with FBP. No filtering was used for reconstruction by CMEM algorithm. Statistics for the images are shown in Tables 1–3.

Figure 3

Spectral slices through the 2-D spectral-spatial image in Fig. 2 for (a) the LiPc signal and for (b) the trityl-CD₃ signal at the positions for maximum signal intensity. The red (upper) and blue (lower) lines are from the images reconstructed by CMEM and FBP, respectively, and calculated by summing three slices. The green lines (middle) are the corresponding slow-scan lineshapes obtained by Fourier deconvolution of non-gradient rapid scans recorded with 500 Hz modulation frequency and a 0.6 G sweep width.

Figure 4

Spatial profiles of the images shown in Fig. 2 obtained by the summation of numerical 2D image matrix along spectral axis. Red lines show MEM profile; blue lines show FBP profile.

Figure 5

2-D image of the same object as shown in Fig. 2 with the same experimental parameters, except that each of the 30 projections was averaged 50,000 times which results in better S/N. Statistics for the images are shown in Tables 1–3.

Figure 6

Spectral slices, analogous to those in Fig. 3, through the image in Fig. 5. The red (upper) and blue (lower) lines are from the images reconstructed by CMEM and FBP, respectively, and calculated by summing three slices. The green lines (middle) are the corresponding slow-scan lineshapes obtained by Fourier deconvolution of non-gradient rapid scans recorded with 500 Hz modulation frequency and a 0.6 G sweep width.

Figure 7

Spatial profiles of the images shown in Fig. 6 obtained by the summation of numerical 2D image matrix along spectral axis. Red lines show MEM profile; blue lines show FBP profile.