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Figure from:Fast Algorithms for Discrete and Continuous

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Fig. 7. “‘A trous’’ structure as derived by Holschneider ef al. (b) Basic computational cell used for computing CWT coefficients octave and octave. (c) Connection of the cells used in this paper and corresponding location of the wavelet coefficients in the time-scale plane.

Figure 7 “‘A trous’’ structure as derived by Holschneider ef al. (b) Basic computational cell used for computing CWT coefficients octave and octave. (c) Connection of the cells used in this paper and corresponding location of the wavelet coefficients in the time-scale plane.

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Fig. 1. Basic computational cell of (a) the DWT and (b) the inverse DWT. (c) Overall organization gives wavelet coefficients that correspond to a dyadic grid in the time-scale plane. Signal is reconstructed using the transposed scheme (b).
Fig. 1. Basic computational cell of (a) the DWT and (b) the inverse DWT. (c) Overall organization gives wavelet coefficients that correspond to a dyadic grid in the time-scale plane. Signal is reconstructed using the transposed scheme (b).
Fig. 2. Sampling of the time-scale plane corresponding to 3 voices pet octave in a WST. The imbrication of the computation is shown using points labelled by circles, squares, and crosses, which can be computed separately using octave-by-octave DWT algorithms.
Fig. 2. Sampling of the time-scale plane corresponding to 3 voices pet octave in a WST. The imbrication of the computation is shown using points labelled by circles, squares, and crosses, which can be computed separately using octave-by-octave DWT algorithms.
Fig. 4. Rearrangement of the DWT cell of Fig. 1(a) that avoids subsam- pling and, hence, allows the application of fast filtering techniques. fore some time processing must be done in order to avoid wrap-around effects. There are two well-known methods for this, called the overlap-add and overlap-save methods [24]. One is the transposed form of the other and both require exactly the same complexity. For one filter of length L /2, with input block length B/2, wrap-around effects are avoided if the FFT-length AN satisfies the inequality
Fig. 4. Rearrangement of the DWT cell of Fig. 1(a) that avoids subsam- pling and, hence, allows the application of fast filtering techniques. fore some time processing must be done in order to avoid wrap-around effects. There are two well-known methods for this, called the overlap-add and overlap-save methods [24]. One is the transposed form of the other and both require exactly the same complexity. For one filter of length L /2, with input block length B/2, wrap-around effects are avoided if the FFT-length AN satisfies the inequality
Fig. 5. FFT-based implementation of the DWT cell of Fig. 4. Overlap-adc (or overlap-save) procedure is not explicitly shown. A more precise comparison with (31) can be done for large filter lengths by minimizing the criterion (mults + adds) of (42)
Fig. 5. FFT-based implementation of the DWT cell of Fig. 4. Overlap-adc (or overlap-save) procedure is not explicitly shown. A more precise comparison with (31) can be done for large filter lengths by minimizing the criterion (mults + adds) of (42)
*Each entry gives the number of operations per input or output point in the form mults + adds, and either the FFT length or the type of fast running FIR algorithm used (see text). Complexities should be multiplied by 2(1 — 277) for a computation of the DWT on J octaves. ARITHMETIC COMPLEXITY PER POINT AND PER CELL* FOR VARIOUS DWT ALGcoriTHMs
*Each entry gives the number of operations per input or output point in the form mults + adds, and either the FFT length or the type of fast running FIR algorithm used (see text). Complexities should be multiplied by 2(1 — 277) for a computation of the DWT on J octaves. ARITHMETIC COMPLEXITY PER POINT AND PER CELL* FOR VARIOUS DWT ALGcoriTHMs
FFT-Basep DWT ALcoriTHMS: ARITHMETIC COMPLEXITY PER PoInT AND PER Octave” *Each entry gives the number of operations per input or output point in the form mults + adds, and the corresponding initial FFT length. Complexities should be multiplied by 2(1 — 2~7) for a computation of the DWT on J octaves. TABLE I
FFT-Basep DWT ALcoriTHMS: ARITHMETIC COMPLEXITY PER PoInT AND PER Octave” *Each entry gives the number of operations per input or output point in the form mults + adds, and the corresponding initial FFT length. Complexities should be multiplied by 2(1 — 2~7) for a computation of the DWT on J octaves. TABLE I
Fig. 6. Simple example of fast running FIR filtering algorithm with decimation ratio R = 2 [23]. Subscripts 0 and 1 indicate biphase decomposi- tion.
Fig. 6. Simple example of fast running FIR filtering algorithm with decimation ratio R = 2 [23]. Subscripts 0 and 1 indicate biphase decomposi- tion.
ARITHMETIC COMPLEXITY PER COMPUTED Pont* ror Various CWT ALGORITHMS *Each entry gives the number of operations per computed wavelet coefficient (i.e., per input point per octave) in the forn mults + adds, and either the FFT length or the type of fast running FIR algorithm used (see text). TABLE I
ARITHMETIC COMPLEXITY PER COMPUTED Pont* ror Various CWT ALGORITHMS *Each entry gives the number of operations per computed wavelet coefficient (i.e., per input point per octave) in the forn mults + adds, and either the FFT length or the type of fast running FIR algorithm used (see text). TABLE I