WO2000044104A1 - Procedes de filtrage numerique et de compression de donnees multidimensionnelles au moyen de la quadrature de farey, d'ondelettes arithmetiques, en eventail et modulaires - Google Patents

Procedes de filtrage numerique et de compression de donnees multidimensionnelles au moyen de la quadrature de farey, d'ondelettes arithmetiques, en eventail et modulaires Download PDF

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WO2000044104A1
WO2000044104A1 PCT/US1999/030584 US9930584W WO0044104A1 WO 2000044104 A1 WO2000044104 A1 WO 2000044104A1 US 9930584 W US9930584 W US 9930584W WO 0044104 A1 WO0044104 A1 WO 0044104A1
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cos
sin
tan
arithmetic
arrow
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PCT/US1999/030584
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English (en)
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Robert C. Penner
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Penner Robert C
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Priority to AU22065/00A priority Critical patent/AU2206500A/en
Priority to US09/869,640 priority patent/US7158569B1/en
Publication of WO2000044104A1 publication Critical patent/WO2000044104A1/fr

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    • G—PHYSICS
    • G10—MUSICAL INSTRUMENTS; ACOUSTICS
    • G10L—SPEECH ANALYSIS TECHNIQUES OR SPEECH SYNTHESIS; SPEECH RECOGNITION; SPEECH OR VOICE PROCESSING TECHNIQUES; SPEECH OR AUDIO CODING OR DECODING
    • G10L19/00—Speech or audio signals analysis-synthesis techniques for redundancy reduction, e.g. in vocoders; Coding or decoding of speech or audio signals, using source filter models or psychoacoustic analysis
    • G10L19/02—Speech or audio signals analysis-synthesis techniques for redundancy reduction, e.g. in vocoders; Coding or decoding of speech or audio signals, using source filter models or psychoacoustic analysis using spectral analysis, e.g. transform vocoders or subband vocoders
    • G10L19/0212—Speech or audio signals analysis-synthesis techniques for redundancy reduction, e.g. in vocoders; Coding or decoding of speech or audio signals, using source filter models or psychoacoustic analysis using spectral analysis, e.g. transform vocoders or subband vocoders using orthogonal transformation
    • G10L19/0216—Speech or audio signals analysis-synthesis techniques for redundancy reduction, e.g. in vocoders; Coding or decoding of speech or audio signals, using source filter models or psychoacoustic analysis using spectral analysis, e.g. transform vocoders or subband vocoders using orthogonal transformation using wavelet decomposition
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
    • G06F17/148—Wavelet transforms
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
    • H04N19/60—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using transform coding
    • H04N19/63—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using transform coding using sub-band based transform, e.g. wavelets
    • H04N19/635—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using transform coding using sub-band based transform, e.g. wavelets characterised by filter definition or implementation details

Definitions

  • the initial form of the data may be digital samples of analog data, and the intermediate form of the data may be the coefficients of the Fourier transform; the initial form of the data may be the coefficients of the Fourier transform, and the intermediate form of the data may be the coefficients in some wavelet or other expansion; the initial form of the data may be an analog signal, and the intermediate form of the data may be some digital representation of it; the initial form of the data may be digital samples of an analog signal, and the intermediate form of the data may be a quantization of the digital samples.
  • transform coding methods for data compression in signal or data processing are realized in practice as transform of data/ quantization, storage or transmission, de-quantization/reconstruction.
  • Another class of examples consists of finite-impulse response digital filters where the input data is filtered using the indirect calculation of convolution given by Fourier transform/multiplication/inverse Fourier transform. For analog data, sampling and filtering together determine fidelity and speed of manipulation.
  • the invention disclosed here provides a new method for sampling analog or digital input data which may be used to calculate new wavelet transforms as well as their inverse reconstruction algorithms. This immediately provides novel and efficient methods for data compression based on this new method of non-linear transform coding. In combination with these wavelet filters, the invention also provides new methods for calculating various classical transforms including the Fourier transform and its inverse. This immediately provides novel and efficient methods for digital filtering.
  • practical applications of the methods include non- speech audio compression, speech compression, speech recognition, speech synthesis, voice printing, audio filtering for hearing aids, still two-dimensional image compression, moving video compression, video compression for purposes of telephony, precision Fourier analysis, precision trigonometry, denoising, interpolation, medical imaging, geological imaging for recovery of oil or other resources, and other emission/detection apparatus.
  • transforms whose efficient digital implementations are important and to which the invention is relevant include the transforms of Hubert, Haar, Laplace, Bessel, Laguerre, Hermite, Chebyshev, Hotelling, Mersenne, and Fermat; see, for instance, US Patents 3,891,443 by Lynch et al. and 4,093,994 by Nussbaumer.
  • Each filter depends upon a new quadrature on the circle, called the Farey quadrature, which relies on a novel method of non-equally-spaced sampling.
  • the output of each wavelet filter plays the role in the invention of an intermediate representation between input data and any one of a number of useful output transformations of data which may be computed from the intermediate quantity.
  • the inverse wavelet transform or reconstruction algorithm requires calculating spatial data from wavelet data and admits an especially convenient implementation: the values of the spatial function at certain non-equally-spaced grids of arbitrarily fine spacing may be computed exactly using a specified finite set of wavelet coefficients.
  • This reconstruction algorithm combines with the wavelet filter into a binary cascade giving an efficient procedure for data compression which is especially well-suited to discontinuous input data and to iterative refinement of output data.
  • the methods extend directly to procedures for compression of multi-dimensional data.
  • Computer coding for the reconstruction algorithm itself is sufficiently abbreviated and low-level that it might be transmitted together with compressed data, for instance, for down-loading from satellite to home computer.
  • FIG la illustrates the standard circle C of radius one in the complex plane, where C bounds the standard disk D of radius one.
  • the complex numbers +1 and —1 are also indicated, and the chord of C with these endpoints is labeled by the matrix
  • the three chords determine a triangle inscribed in C, and the vertices of this triangle are indicated as complex numbers.
  • FIG 2 depicts a classical figure in mathematics called “the Farey tesselation in Klein's model of hyperbolic geometry” . It is constructed beginning from the chord in Figure la by recursively taking descendants as in Figure lb and Figure lc, respectively, on the top and bottom semi-circle of C. The figure itself is classical, but the sampling of input data on C at the endpoints of the chords of the Farey tesselation in increasing generation is a novel aspect of the invention disclosed here.
  • FIG 3a illustrates the circle C, the chord labeled / and the two triangles in the Farey tesselation on either side of this chord.
  • the vertices of these triangles are indicated inside the circle as complex numbers. Outside the circle are given four explicit combinations of cosine and sine functions, one such function next to each circular arc determined by the vertices, where ⁇ is the usual angular coordinate on the circle.
  • a function ⁇ ( ⁇ ) is uniquely determined by the figure, where on each such circular arc, ⁇ ) takes the values of the nearby combination of cosine and sine functions.
  • FIG 3b illustrates th ⁇ lord in the top semi-circle of C labeled th the matrix A, where a > c. As before, the figure determines a function on the circle denoted "0A (&) .
  • FIG 3d illustrates the chord in the bottom semi-circle of C labeled with the matrix A, where ⁇ > — c. As before, the figure determines a function on the circle denoted X ⁇ A ⁇ ) .
  • FIG 5 provides a flow chart for the main driving routine in calculating the Fourier transform.
  • FIG 6 provides a flow chart for the main recursive routine in calculating the Fourier transform.
  • FIG 7 provides a flow chart for the procedure of updating coefficients in calculating the Fourier transform.
  • FIG 8 provides a flow chart for the procedure of processing terminal arrows of the recursion in calculating the Fourier transform.
  • FIG 9 provides a flow chart for the main driving routine in calculating the inverse Fourier transform.
  • FIG 10 provides a flow chart for the main recursive routine in calculating the inverse Fourier transform.
  • FIG 11 illustrates the region U consisting of all complex numbers with positive imaginary part. Map the disk D to U via the function z f ⁇ i(z + l)/(z — 1); if two points in C are the endpoints of a chord in Klein's model of the Farey tesselation as in Figure 2, then draw the semi-circle in U passing through the corresponding two points which is perpendicular to the real axis. This produces another classical figure in mathematics, called the "Farey tesselation in the upper half-space model of hyperbolic geometry" , which is indicated in Figure 11.
  • the doe which is illustrated in Figure la, is both a top arithmetic arrow and a bottom arithmetic arrow.
  • the sampling of it at the points Q C. C in this order of increasing generation will be referred to as the Farey quadrature.
  • the Farey quadrature In practice, one might interpolate given ta as required at the sampling points of tl ⁇ arey quadrature or restrict the Farey quadrature to a circular arc in C. (In the subsequent discussion of mathematical basis, the sample points of the Farey quadrature will be seen to correspond to the rational points in the real line enumerated in increasing order, where the generation is the length of a corresponding continued fraction expansion.)
  • Figure 3a depicts the doe itself with A — I
  • Figure 3b and 3c depict top arrows with a > c and c > a, respectively
  • Figure 3d and 3e depict bottom arrows with a > —c and — c > ⁇ , respectively.
  • the notation inside the circles indicates the endpoints of the chords as complex numbers, and the notation outside the circles in each case of Figures 3a-3e is yet to be explained.
  • C is decomposed into a finite number of non-overlapping circular arcs, and on each such arc, / takes the values of a trigonometric function.
  • any two- by-two matrix A I , 1 determines a self-mapping MA ' ⁇ C — C of the circle C as follows:
  • the arithmetic wavelets enjoy an important renormalization property, namely,
  • f( ⁇ ) may be expressed in its representation as a Fourier series / ⁇ ⁇ _ n c n e ⁇ n ⁇ ' , so too it may be expressed as a series / « ⁇ CA ⁇ A ( ⁇ ) of arithmetic wavelets, which together form a linearly independent set. The sum is over all arithmetic arrows, and the coefficients CA are called the arithmetic wavelet coefficients.
  • the calculation of arithmetic wavelet c -icients from the input data / is called th rithmetic wavelet filter.
  • the arithmetic wavelet coefficients are not quite uniquely determined by / as will be explained.
  • the reconstruction algorithm or inverse wavelet transform provides a method for calculating function values at points of the circle from arithmetic wavelet coefficients.
  • ⁇ A( ⁇ ) ⁇ U- IA( ⁇ ) - ⁇ A( ⁇ )
  • ⁇ SA( ⁇ ) ⁇ U-ISA( ⁇ ) - ⁇ SA( ⁇ )
  • the fan wavelets also arise from ⁇ j in the analogous manner.
  • Both fan and modular wavelets enjoy precisely the same finiteness property as arithmetic wavelets. Furthermore, both fan and modular wavelets enjoy renormalization properties, namely,
  • Source Codes is an implementation in the computer language C of a preferred embodiment of the method described in this section employing arithmetic wavelets.
  • Stepl Choose a finite set S of arithmetic arrows and approximate / « _2A _ S e A A calculating the coefficients e ⁇ in a manner to be described in terms of the values of / at specified points using the Farey quadrature.
  • Step 2 Substitute into the expression in Step 1 the known Fourier expansions ⁇ A ⁇ ) ⁇ _ n c n e%n ⁇ m order to derive the approximation c n — / _, C-A C n .
  • the coefficients require n Step 2 are known theoretically and for n 0, ⁇ 1 by
  • ⁇ c ⁇ ⁇ h (c 2 -d 2 ⁇ 2icd) + ⁇ r [(b - d) 2 - (a - c) 2 + 2i(b - d)(a - c)]/2 + ⁇ t (a 2 -b 2 ⁇ 2iab) + ⁇ t [(b + d) 2 - ( ⁇ + c) 2 + 2i(b + d)(a + c)]/2,
  • ⁇ c A 2 ⁇ h (c 2 + d 2 ) - ⁇ r [(b - d) 2 + ( ⁇ - c) 2 ] + 20 f ( ⁇ 2 + ⁇ > 2 ) - ⁇ [(6 + d) 2 + ( ⁇ + c) 2 ] where the angles are
  • Steps 1 and 2 are merged into a single binary cascade which keeps track of an ongoing approximation to the Fourier coefficients as follows.
  • a cascade element is called an arrow- structure and is defined to consist of the specification of
  • the coefficients , ⁇ , ⁇ in an arrow-structure keep a lagged/updated running tally of the overall effect of what has come before it in the cascade; as a result of this technique, the method requires essentially no memory other than the storage of the running approximation to Fourier coefficients and the stack required for the recursive computation in the cascade of arrow-structures.
  • the final technical point involves the stopping criteria and terminal processing for the cascade of arrow-structures.
  • the basic stopping parameter is NVAN, where a branch of the cascade terminates whenever there have been NVAN consecutive generations of offspring whose contributions to all coefficients c n in the bandwidth N have been of magnitude at most EPS.
  • NVAN NVAN
  • terminal arrow-structures could be stored for restart or iterative refinement capabilities.
  • One favorable aspect of the method is its advantageous mix of floating-point and integer operation types. Moreover, except for the coefficients c 0 , c ⁇ i, the implementation of the algorithm is purely algebraic, that is, requires only addition and multiplication. Furthermore, by its very nature as a cascade, the algorithm is amenable to parallelization and efficient hardware implementation.
  • Step 1 the calculation of the modular wavelet transform where the input function is approximated as f( ⁇ ) ⁇ __ A 9A ⁇ A ( ⁇ ) > and a basic Step 2, substitution of known expressions for the Fourier coefficients of the modular wavelets; these two steps are again conveniently merged into a binary cascade of arrow-structures in practice.
  • Step 1 for modular wavelets refer again to Figure 4 for the notation near the arithmetic arrow labeled by the matrix A.
  • the single value of the input function f ⁇ o) together with the ongoing calculation of the updated trigonometric function cos ⁇ + ⁇ sin ⁇ + 7 in the arrow-structure this time uniquely determines the modular wavelet coefficient
  • the method of calculating Fourier transforms using modular wavelets is easily derived from the method using arithmetic wavelets.
  • Step 1 In the same way for fan wavelets, there is again a Step 1 and Step 2, which are merged into a binary cascade of arrow-structures.
  • Step 1 for fan wavelets again requires a regularization scheme.
  • FIG. 5 is presented a flow chart for the main driving routine.
  • the input data is normalized as indicated in program segment 1.
  • program segment 1 There are separate recursions established in program segments 2 and 3 for the top and bottom of the circle respectively.
  • Each recursion is estab- .ed with a call to the subroutine gener, w ' h then calls itself in turn.
  • the Fourier coefficients are stored internally with an overall suppressed factor of ⁇ , and output data normalization of multiplying by - is accomplished in program segment 4.
  • Output Fourier coefficients are finally displayed before exiting in program segment 5.
  • Program segments 2 and 3 are entirely independent and could be performed in parallel; more generally, each of program segments 2 and 3 could be decomposed further into multiple parallel procedures.
  • a flow chart for the main recursive routine gener is given in Figure 6. The procedure starts with a test in program segment 6 to determine if:
  • the procedure passes to program segment 7, where the descendant arrows in the cascade are determined using the least- squares fit to the next generation of data as described before to compute the regularized wavelet coefficients of the descendants. These are combined with integer calculations to update the lagged trigonometric functions.
  • the procedure then passes to program segment 8, where the ongoing approximations to Fourier coefficients are updated to include contributions from the two descendant arrows with a call of the subroutine charles for each descendant.
  • the procedure continues with a test in program segment 9. If the contribution calculated in the subroutine charles for either descendant to any Fourier coefficient in the bandwidth was non-negligible, then the recursive argument envy is set to zero in program segment 10. In the contrary case that both contributions to all Fourier coefficients in the bandwidth were negligible, the control parameter envy is increased by one in program segment 13.
  • program segment 14 establishes the recursion by calling gener once for each of the descendant arrows with the updated control parameter envy and an incremented generation.
  • Figure 7 is presented a flow chart for the subroutine charles.
  • Calling the subroutine charles has the effect of updating the ongoing approximations to the Fourier coefficients for a single argument arrow and returning a flag which keeps track of whether any such contribution has been non-negligible.
  • the flag is cleared in program segment 16, and several preliminary calculations are accomplished in program segment 17.
  • the procedure passes to program segment 18, where it is determined whether to calculate the 0,+l,-l Fourier coefficients. If these are to be calculated, then the procedure passes to program segment 19, which calls a subroutine to update these three Fourier coefficients.
  • Program segments 20 and 21 set the flag as required depending upon whether these three contributions are non-negligible. In any case, the proce re passes to program segment 22.
  • FIG 8 is presented a flow chart for the subroutine prune, which modifies the array of Fourier coefficients when terminating the cascade.
  • the final update of lagged trigonometric functions to produce ⁇ , T 2 is performed in program segment 28. There is one such function for each possible descendant of the argument arrow.
  • the procedure passes to program segment 29, where further input data is sampled in order to compute two trigonometric extrapolations ⁇ , ⁇ , one such extrapolation for each possible descendant of the argument arrow.
  • Each function ⁇ , — Tj, for i — 1, 2 is truncated as described before, and the Fourier coefficients of the truncated functions are added to the ongoing approximations of Fourier coefficients in program segment 30.
  • Program segments 31 and 32 implement the calculation of 0, +1,-1 Fourier coefficients if desired, and the subroutine prune is terminated with the return in program segment 33.
  • Source Codes is an implementation in the computer language C of a preferred embodiment of the method described in this section employing arithmetic wavelets.
  • the input data includes the specification of a collection of Fourier coefficients c n in a given bandwidth N.
  • Stepl Calculate arithmetic wavelet coefficients e from the given Fourier coefficients c perpetual.
  • Step 2 Use the wavelet coefficients from Step 1 and the reconstruction algorithm to output values of the function ⁇ _ A CA ⁇ A ⁇ ) at the Farey quadrature points on the circle.
  • Step 2 in the method for calculating inverse Fourier transforms depends upon the reconstruction algorithm to output function values on the circle taken by the linear combination f( ⁇ ) m c' 0 + c e % ⁇ + d_ l e ⁇ 'l ⁇ + _ A e A ⁇ A ⁇ ).
  • the finiteness property of the reconstruction algorithm allows the exact calculation of these function values at the sample points Q . C in their ordering determined by the Farey quadrature.
  • the two steps are again conveniently merged into a single binary cascade as follows. There is only one con 1 parameter, which is called SCALE, v. ;e the method is required to determine at least one output value in each circular arc subtending an angle SCALE.
  • Step 1 the calculation of wavelet coefficients from Fourier coefficients, and a Step 2, the reconstruction algorithm; in each case of fan or modular wavelets, these two steps are again conveniently merged into a binary cascade of arrow-structures in practice.
  • Step 2 The finiteness condition on fan or modular wavelets mentioned before renders Step 2 entirely analogous to that for arithmetic wavelets.
  • the method of calculating inverse Fourier transforms using fan or modular wavelets is easily derived from the method using arithmetic wavelets.
  • Figure 9 is given a flow chart for the main driving routine.
  • Modified Fourier coefficients c 0 ' , c' l , c'_ 1 are computed in program segment 34.
  • the formula given in Step 1 for the inverse Fourier transform is applied in program segment 35 to calculate the arithmetic wavelet coefficients in terms of Fourier coefficients for a particular family of nine arrows. These nine arithmetic wavelet coefficients are required to initialize the trigonometric functions for recursions established in program segment 36 with calls to the recursive subroutine gener.
  • the procedure passes to program segment 37 to perform an overall correction of the function values using the modified Fourier coefficients c' Q , c' 1 , c_ 1 .
  • Output data is displayed and the procedure terminated in program segment 38.
  • Figure 10 is given a flow chart for the main recursive routine gener.
  • the two descendant arrows of the argument arrow are generated in program segment 39 employing Step 1 of the inverse Fourier transform method to calculate the wavelet coefficients of the descendants.
  • These expressions are then used to update the lagged trigonometric functions in program segment 39.
  • the recursion is established in program segment 41 with two calls to gener, where the arguments are given by the two current descendant arrows.
  • the procedure performs the final update of the lagged trigonometric functions in program segment 42, then stuffs the output array with the new function values in program segment 43, and finally returns in program segment 44.
  • the wavelet filter has already been fully disclosed as Step 1 for the calculation of the Fourier transform, and the wavelet inverse filter or reconstruction algorithm has likewise already been described as Step 2 for the calculation of the inverse Fourier transform.
  • Application-specific quantization is done according to psychovisual or psychoacoustic thresholds. Quantization and storage are furthermore merged with wavelet filtering into a single binary cascade as before, and retrieval or transmission are likewise merged with reconstruction into another single binary cascade.
  • the reconstruction algorithm is exact.
  • source code for it may be transmitted along with compressed data since the coding is sufficiently abbreviated and low-level.
  • a top arrow of generation g labeled by the two-by-two integral matrix A is specified by g bits, namely, by writing A uniquely as the matrix product of factors U or T.
  • the matrix A ( _ _ ⁇ factors as
  • the method is also especially well-suited to progressive picture build-up or other iterative refinement as follows.
  • the Farey quadrature determines a linear ordering on the set Q of quadrature points in the circle C as before.
  • the ordering from the Farey quadrature on Q ⁇ — i, ⁇ 1 ⁇ determines a corresponding ordering on the set of all arithmetic wavelets.
  • Storage, retrieval, transmission, and reconstruction of wavelet coefficients is accomplished at a specified input or output data resolution in this canonical ordering since energy compacts to the lesser wavelet coefficients.
  • the method can be further improved by the manipulation of previously computed wavelet coefficients in parallel with the calculation of subsequent ones.
  • the input function F must be normalized.
  • F( ⁇ ) F( ⁇ ) — v( ⁇ )
  • F is zero at each point of ⁇ — i, ⁇ 1 ⁇ M .
  • Each arithmetic wavelet is once-continuously differentiable on the circle, compactly supported, and localized in space.
  • Arithmetic wavelets are not compactly supported in frequency, but instead, the frequency profile of an arithmetic wavelet is given algebraically by the formula for c A used in Step 2 in the calculation of Fourier transforms; a non-compactly supported localization in frequency follows directly from this.
  • the asserted formula for c A can be derived without much difficulty but in several cases directly from Figures 3a-3e integrating twice by parts the usual expression for Fourier coefficients using the fact ⁇ A is once-continuously differentiable.
  • fan wavelets are only continuous (they are not differentiable), and modular wavelets are not even continuous.
  • Source code is presented for each of the following two implementations.
  • • awft is the implementation of a preferred embodiment of the method for computing the Fourier transform of real-valued input data defined on the circle.
  • the algorithm depends upon a bandwidth N, tolerance EPS, generation cut-off NVAN, and minimum generation MING as described before.
  • • awift is the implementation of a preferred embodiment of the method for computing the inverse Fourier transform of specified complex-valued Fourier coefficients defined in a specified bandwidth N.
  • the comparison of the subroutines tgener in awft and in awift illustrates the easy transition between source codes for real and for complex data.
  • the corresponding complex-awft and real-awift are therefore easily derived from the included source codes.
  • the met J for compression of one-dimensional inj data amounts to the first step of awft, then quantization/de-quantization, then the second step of awift.
  • the included source codes together thus also implement the method for compression of one-dimensional data since source code for it may be extracted from the included source codes.
  • it is straight-forward to write the further driving routine required for multi-dimensional data compression so the source codes included with this patent application are sufficient to readily implement the method for multi-dimensional data compression as well.
  • each of awft and awift is appended source code to replace the subroutines tgener and bgener by corresponding subroutines which do employ renormalization.
  • NVAN 1 //terminate the recursion when NVAN consecutive
  • double fbar(int p, int q) ⁇ //input p,q to fbar produces the nornalized value struct complex temp; //of the input data at the point (p-iq)/(p+iq) struct complex makecpx(int.int); double f (int.int); temp makecpx(p.q); return(f(p,q) - (abar*temp.x + bbar*temp.y+cbar));
  • tripl mob( //calculate sigma for first descendant makecpx(-(3*b+d),3*a+c), makecpx(-(2*b+d),2*a+c), makecpx(-(3*b+2*d),3*a+2*c), fbar(-(3*b+d),3*a+c), fbar(-(2*b+d),2*a+c), fbar(-(3*b+--*d),3*a+2*c)
  • trip2 mob( //calculate sigma for second descendant makecpx(-(2*b+3*d),2*a+3*c), makecpx(-(b+2*d),a+2*c), makecpx(-(b+3*d),a+3*c), fbar(-(2*b+3*d),2*a+3*c), fbar(-(b+2*d),a+2*c), fbar(-(b+3*d),a+3*c)
  • tripl mob( //arguments of mob differ from those in tprune makecpx(3*a+c,3*b+d), makecpx(2*a+c,2*b+d), makecpx(3*a4-2*c,3*b+2*d), fbar(3*a+c,3*b+d), fbar(2*a+c,2*b+d), fbar(3*a+2*c,3*b+2*d)
  • trip2 mob( //arguments of mob differ from those in tprune makecpx(2*a+3*c,2*b+3*d), makecpx(a+2*c,b+2*d), makecpx(a+3*c,b+3*d), fbar(2*a+3*c,2*b+3*d), f ar(a+2*c,b+2*d), fbar(a+3*c,b+3*d)
  • double fbar(int.int); int a,b,c,d,bpd,apc; double e,alp,bet,gam; double axl,ayl,ax2,ay2; double bxl,byl,bx2,by2; double cxl,cyl,cx2,cy2; double sigl,taul,sig2,tau2,chi; double vlc,vlu,vlt,v2c,v2u,v2t; double deriv(int,int,int,int,int); double dc,dn,dnp; double eu,et,fn,fnp; struct complex temp; struct edge *pnp; extern int count; pnp pn; pnp++;
  • cmon.x,cmon.y,czer.x,czer.y,cone.x,cone.y); for (i 0;i ⁇ outcount;i++) printf("(%-. -.d) %e+(i)%e ⁇ n", graf[i].p,graf[i].q,graf[i].x,graf[i].y);
  • nn (p->a)*(p->c)-t-(p->b)*(p->d); if(nn*nn ⁇ SCAT) retum(O); return(l);

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Abstract

L'invention concerne des procédés de calcul de filtres d'ondelettes et leurs inverses, basés sur un nouveau procédé d'échantillonnage (UA, TA, A) de données numériques ou analogiques. Lesdits procédés sont combinés et étendus de sorte que de nouvelles procédures de compression de données multidimensionnelle non réversibles soient fournies. Pour des applications sélectionnées, ladite procédure permet l'amélioration des facteurs de compression pouvant être obtenus, par un à trois relations d'ordres décimales, et convient bien à la construction d'image ou à d'autres raffinements itératifs. La combinaison de ces filtres à ondelettes et de leurs inverses à un travail théorique antérieur permet la création de nouvelles méthodes pour le calcul de la transformée de Fourier et d'autre transformées. Dans un mode de réalisation préféré utilisé pour le calcul de la transformée de Fourier (1) et son inverse (34) appliqué aux données d'entrée numériques, ledit procédé remplace la transformation de Fourier rapide et son inverse et permet l'amélioration de la précision pouvant être obtenue. Le nouveau procédé d'échantillonnage est essentiellement du type multi-échelle, et grâce aux procédés de l'invention, la contrainte de Nyquist sur la largeur de bande significative en termes de nombre d'échantillons est supprimée. L'invention porte également sur une nouvelle interface analogique-numérique et numérique analogique efficace.
PCT/US1999/030584 1999-01-19 1999-12-21 Procedes de filtrage numerique et de compression de donnees multidimensionnelles au moyen de la quadrature de farey, d'ondelettes arithmetiques, en eventail et modulaires WO2000044104A1 (fr)

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AU22065/00A AU2206500A (en) 1999-01-19 1999-12-21 Methods of digital filtering and multi-dimensional data compression using the farey quadrature and arithmetic, fan, and modular wavelets
US09/869,640 US7158569B1 (en) 1999-01-19 1999-12-21 Methods of digital filtering and multi-dimensional data compression using the farey quadrature and arithmetic, fan, and modular wavelets

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US11654099P 1999-01-19 1999-01-19
US60/116,540 1999-01-19

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CN109288649A (zh) * 2018-10-19 2019-02-01 广州源贸易有限公司 一种智能语音控制按摩椅
CN113438014A (zh) * 2021-07-05 2021-09-24 重庆邮电大学 基于星间通信的低能耗路由方法

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US5802481A (en) * 1997-03-20 1998-09-01 Motorola, Inc. Adaptive filtering for use with data compression and signal reconstruction
US5819215A (en) * 1995-10-13 1998-10-06 Dobson; Kurt Method and apparatus for wavelet based data compression having adaptive bit rate control for compression of digital audio or other sensory data
US5974181A (en) * 1997-03-20 1999-10-26 Motorola, Inc. Data compression system, method, and apparatus
US5991454A (en) * 1997-10-06 1999-11-23 Lockheed Martin Coporation Data compression for TDOA/DD location system

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US5819215A (en) * 1995-10-13 1998-10-06 Dobson; Kurt Method and apparatus for wavelet based data compression having adaptive bit rate control for compression of digital audio or other sensory data
US5802481A (en) * 1997-03-20 1998-09-01 Motorola, Inc. Adaptive filtering for use with data compression and signal reconstruction
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN109288649A (zh) * 2018-10-19 2019-02-01 广州源贸易有限公司 一种智能语音控制按摩椅
CN113438014A (zh) * 2021-07-05 2021-09-24 重庆邮电大学 基于星间通信的低能耗路由方法
CN113438014B (zh) * 2021-07-05 2022-04-22 重庆邮电大学 基于星间通信的低能耗路由方法

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