### 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:... ..."

### Table 1: Change of Fourier coefficients after operations in the continuous spatial domain.

1999

Cited by 10

### Table 1: Change of Fourier coefficients after operations in the continuous spatial domain.

1999

Cited by 10

### 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 1: Four types of signals and their properties in the time domain. Type Time Domain Frequency Domain Transform

"... In PAGE 1: ... INTRODUCTION There are four types of signals often used in signal processing, and to analyze these signals, there are four types of Fourier trans- forms. In Table1 we list the four types of signals along with their properties and the appropriate Fourier transform. Since the Fourier transform is linear, the discrete Fourier transforms are samples of the continuous Fourier transform under the appropriate sampling conditions.... In PAGE 1: ... The Cohen class of time-frequency distributions [1, 2] was originally formulated for type I signals. Recently, this class has been extended to the three types of discrete signals in Table1 us- ing both an axiomatic approach [3, 4] and an operator theory ap- proach [5, 6, 7].... ..."

### Table 3: The average mean square errors between the projections derived from the Fourier Projection Theorem and those from spatial domain summing, for various viewing angles. The color values are normalized to the range between 0 and 255.

"... In PAGE 6: ... All the reported measurementsbelow are 2D mean square errors (MSE) from the head data set. Table3 shows the average mean square errors between the projection sums derived from the Fourier Projection Theorem and those from spatial domain summing, for subcubesof different sizes from different viewing angles. The pro- jection angles are specified in the second row in terms of multi- ples of #19.... ..."

### Table 1: Fast Fourier and other transform factorizations.

1997

"... In PAGE 2: ... 2 The Language of Factorizations A fast transform algorithm can be seen as a sparse factorization of the transform matrix. Table1 displays di erent factorizations for the DFT matrix as well as other transforms that arise in signal processing. The abbreviation Bn;m = (Fn Im)Dn;m is used in the FFT, with D a diagonal matrix of weights, Pn;p is the stride permutation matrix.... In PAGE 10: ... In general, the same transformation strategy can han- dle all problems of the same dimension, di erent transformation strategies are needed for multidimensional problems. bbb We successfully transform all fast Fourier and other transform and transposition algo- rithms shown on Table1 using the above rules for determining the loop structure and high level assignments. We generated the radix code, the straight-line code that computes an FFT in the loop nest by symbolically capturing the assignments and operations of an FFT calculation of appropriate length using the prime factor FFTs as the base cases.... ..."

Cited by 4

### Table 1. Radial basis functions and Fourier transforms

1999

"... In PAGE 2: ...2) is symmetric and positive de nite on (Pd m)?. Table1 shows some conditionally positive de nite functions with their minimal orders m. Any functional 2 (Pd m)? of the form (2.... In PAGE 9: ...5) that makes the integral well{de ned near zero. Table1 shows the functions ^ for various choices of . As a referee correctly pointed out, the assumption (4.... ..."

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### Table 4: Basic Properties of the Discrete Fourier Transform

1995

"... In PAGE 13: ... (We extend the de nition to non-periodic sequences of the same length N, by regarding them as periodic, as de ned earlier in this section). Table4 lists the discrete counterparts of the properties given in Table 3 for the continuous case. We assume that we are dealing only with well de ned combinations.... ..."

Cited by 9

### Table 4: Basic Properties of the Discrete Fourier Transform

1995

"... In PAGE 14: ... (We extend the de nition to non-periodic sequences of the same length N, by regarding them as periodic, as de ned earlier in this section). Table4 lists the discrete counterparts of the properties given in Table 3 for the continuous case. We assume that we are dealing only with well de ned combinations.... ..."

Cited by 9