### Table 2: Comparison of row BAS Function and its inverse Fourier Transform with various T values.

"... In PAGE 8: ... The plot shows the rapid decay of the spectral coefficients which becomes zero after n=60. Table2... ..."

Cited by 1

### Table 4.2: Timing results in seconds on the CM-5 for di erent implementations of the Pasciak algorithm. The three major steps in the algorithm are shown, were the ltering step is combined with that of the two- dimensional inverse Fourier transform. The last column shows the total time needed for reconstructing.

### Table 2: Timing results in seconds on the CM-5 for di erent implementations of the Pasciak algorithm. The three major steps in the algorithm are shown, were the ltering step is combined with that of the two-dimensional inverse Fourier transform. The last column shows the total time needed for reconstruc- tion.

"... In PAGE 7: ... The parameter N has the same meaning. The results are shown in Table2 for di erent values of N. In the interpolation step there is some communica- tion between processors when at a certain point the actual interpolation is computed.... In PAGE 8: ... The last column shows the total time needed for reconstruc- tion. When the timing results in Table2 are compared to the timings for ltered backprojection (Table 1), it can be concluded that the Pasciak method is indeed faster than ltered backprojection. 5 Conclusions Of the reconstruction algorithms considered, the l- tered backprojection algorithm turns out to be the most di cult to parallelise.... ..."

### Table 1: A list of three-dimensional Fourier transforms of various integrable functions used in meshless methods. with the exception of that of the Gaussian, the inverse multiquadric, and the Sobolev spline [28, Theorems 6.10, 6.13, and Page 133], the Fourier transform of each function was computed using (6).

"... In PAGE 6: ...Examples of integrable radial basis functions are given in Table1 . Non-integrable radial basis functions and polynomial terms are sometimes used, examples of which include the multiquadric and the thin-plate spline.... ..."

### Table I Table I contains the inverse Fourier and wavelet transformation characteristics. In Column 1 we list the examined datasets. Column 2 gives the number of data in the various time series. In column 3 we indicate the processing techniques used, FP stands for Fourier Processing, whereas WP means Wavelet Processing. Column 4 provides the percentage of Fourier and wavelet coe cients above the threshold. In column 5, is the cross-correlation coe cient between the `models apos; and the original averaged data. data number of points signal nb coe cients

### Table 3: Radix-2r twiddle factor storage as a function of input order. 4.3 Summary of twiddle factor storage requirements The preferred combinations of data allocation and FFT type is summarized in Table 3. For multi-dimensional FFT each axis has its set of twiddle factors. The twiddle factors for an axis is a subset of the twiddle factors for the longest axis. With axes of length P1 P2 : : : Pk the minimum number of twiddle factors is max`(R ? 1)P` R . With separate storage of the twiddle factors for each axis the total storage is P`(R ? 1)P` R , which is still less than the storage required for a one-dimensional FFT of size `P`. The Inverse Discrete Fourier Transform can be computed as a Discrete Fourier Transform by using conjugate twiddle factors.

1992

### Table1. Some Properties of the Fourier Transformation Signal Fourier Transform

"... In PAGE 2: ... The discrete Fourier trans- formation (FT) F(u; v) of a 2D discrete image f[x; y] 2 IRN N is de ned as F(u; v) = 1 N N?1 X j=0 N?1 X k=0 f[j; k] exp(?2 i(uj + vk) N ) (1) with i = p?1 and u; v = 0; 1; :::; N ? 1. Using the FT properties shown in Table1 , the following characteristics of the amplitude spectrum A of F(u; v) can be derived: it is invariant with respect to translation, inverse-variant with respect to scaling and variant with respect to rotation. Thus, features based on the amplitude spectrum of an image are translation invariant.... ..."

### Table 3: The parallel execution time (in sec.) of the optimized code for images of size 512x512.

1997

"... In PAGE 18: ... For example, using the above parameters and p = 4, the shortest path in Figure 9 was found to pass through the block partition node for the histogram equalization operation, the row partition node for the masking operation, the row partition node for the Fourier transform, and the row partition node for the inverse Fourier transform. The parallel time was found to be 3:62sec: Table3 shows the breakdown of the parallel execution time of the optimized codes using 4, 16, 24, and 32 processors. Note that some DR columns, which show the data redistribution time in between two adjacent stages, are... ..."

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### Table 3: The parallel execution time (in sec.) of the optimized code for images of size 512x512. HIST DR MED DR FFT DR IFFT TOTAL

1997

"... In PAGE 26: ... For example, using the above parameters and p = 4, the shortest path in Figure 9 was found to pass through the block partition node for the histogram equalization operation, the row partition node for the masking operation, the row partition node for the Fourier transform, and the row partition node for the inverse Fourier transform. The parallel time was found to be 3:62sec: Table3 shows the breakdown of the parallel execution time of the optimized codes using 4, 16, 24, and 32 processors. Note that some DR columns, which show the data redistribution time in between two adjacent stages, are empty because there was no data redistribution in between the stages.... ..."

Cited by 4

### Table 1: Some properties of the Fourier transform

"... In PAGE 16: ...imension of u is 1=time, i.e., frequency). In the case of image processing, the signal is a function of space rather than time, and in that case the domain of the Fourier transform is called spatial frequency. Table1 summarizes some important properties of the one-dimensional Fourier transform. The symbol is used to denote convolution:... ..."