WTHaarFwd, WTHaarInv

Syntax

Parameters

pSrc	Pointer to the source vector for forward transform.
lenSrc	Number of elements in the source vector pSrc.
pDstLow	Pointer to the array with the coarse “low frequency” components of the output for forward transform.
pDstHigh	Pointer to the array with the detail “high frequency” components of the output for forward transform.
pSrcLow	Pointer to the array with the coarse “low frequency” components of the input for inverse transform.
pSrcHigh	Pointer to the array with the detail “high frequency” components of the input for inverse transform.
pDst	Pointer to the array with the output signal for inverse transform.
lenDst	Number of elements in the destination vector pDst
scaleFactor	Scale factor, refer to Integer Scaling.

Description

The functions ippsWTHaarFwd and ippsWTHaarInv are declared in the ipps.h file. These functions perform forward and inverse single-level discrete Haar transforms. These transforms are orthogonal and reconstruct the original signal perfectly.

The forward transform can be considered as wavelet signal decomposition with lowpass decimation filter coefficients {1/2, 1/2} and highpass decimation filter coefficients {1/2, -1/2}.

The inverse transform is represented as wavelet signal reconstruction with lowpass interpolation filter coefficients {1, 1} and highpass interpolation filter coefficients {-1, 1}.

The decomposition filter coefficients are frequency response normalized to provide the same value range for both input and output signals. Thus, the amplitude of the low pass filter frequency response is 1 for zero-valued frequency, and the amplitude of the high pass filter frequency response is also 1 for the frequency value near to 0.5.

As the absolute values of the interpolation filter coefficients are equal to 1, the reconstruction of the signal requires few operations. It is well suited for usage in data compression applications. As the decomposition filter coefficients are powers of 2, the integer functions perform lossless decomposition with the scaleFactor value equal to -1. To avoid saturation, use higher-precision data types.

Note that the filter coefficients can be power spectral response normalized, see [Strang96] for more information. Thus, the decomposition filter coefficients are {2^-1/2, 2^-1/2} and {2^-1/2, -2^-1/2}; accordingly; the reconstruction filter coefficients are {2^-1/2, 2^-1/2} and {-2^-1/2, 2^-1/2}.

In the following definition of the forward single-level discrete Haar transform, N = lenSrc. The coarse “low-frequency” component c(k) is pDstLow[k] and the detail “high-frequency” component d(k) is pDstHigh[k]; also x(2k) and x(2k+1) are even and odd values of the input signal pSrc, respectively.

c(k) = (x (2k) + x(2k+1 ))/2

d(k) = (x(2k+1) - x(2k))/2

In the inverse direction, N= lenDst. The coarse “low-frequency” component c(k) is pSrcLow[k] and the detail “high-frequency” component d(k) is pSrcHigh[k]; also y(2i) and y(2i+1) are even and odd values of the output signal pDst, respectively.

y(2i) = c(i) - d(i)

y(2i+1) = c(i) + d(i)

For even length N, 0 ≤ k < N/2 and 0 ≤ i < N/2. Also, “low-frequency” and “high-frequency” components are of size N/2 for both original and reconstructed signals. The total length of components is equal to the signal length N.

In case of odd length N, the vector is considered as a vector of the extended length N+1 whose two last elements are equal to each other x[N] = x[N - 1]. The last elements of the coarse and detail components of the decomposed signal are defined as follows:

c((N+1 )/2 - 1) = x(N - 1)

d((N + 1)/2 - 1) = 0

Correspondingly, the last element of the reconstructed signal is defined as:

y(N) = y(N - 1) = c((N+ 1)/2 - 1)

For odd length N, 0 ≤ k < N - 1)/2 and 0 ≤ i < N - 1)/2 , assuming that c((N+ 1)/2 - 1) = x(N - 1) and y(N - 1) = c((N + 1)/2 - 1). The “low-frequency” component is of size (N + 1)/2. The “high-frequency” component is of size (N - 1)/2, because the last element d((N + 1)/2 - 1) is always equal to 0. The total length of components is also N.

Such an approach applies continuation of boundaries for filters having the symmetry properties, see [Bris94].

When performing block mode transforms, take into consideration that for decomposition and reconstruction of even-length signals no extrapolations at the boundaries is used. In case of odd-length signals, a symmetric continuation of the signal boundary with the last point replica is applied.

When it is necessary to have a continuous set of output blocks, all the input blocks are to be of even length, besides the last one (which can be either of odd or even length). Thus, if the whole amount of elements is odd, only the last block can be of odd length.

ippsWTHaarFwd. Thise function performs the forward single-level discrete Haar transform of a lenSrc-length signal pSrc and stores the decomposed coarse “low-frequency” components in pDstLow, and the detail “high-frequency” components in pDstHigh.

ippsWTHaarInv. This function performs the inverse single-level discrete Haar transform of the coarse “low-frequency” components pSrcLow and detail “high-frequency” components pSrcHigh, and stores the reconstructed signal in the lenDst-length vector pDst.

For more information on wavelet transforms see [Strang96] and [Bris94].

Example below illustrates the use of the function ippsWTHaarFwd_32f.

Return Values

ippStsNoErr	Indicates no error.
ippStsNullPtrErr	Indicates an error when the pDst or pSrc pointer is NULL.
ippStsSizeErr	Indicates an error when len is less than 4 for the function ippsWinBlackmanOpt and less than 3 for all other functions of the family.

Using the ippsWTHaarFwd Function

IppStatus wthaar(void) {

      Ipp32f x[8], lo[4], hi[4];

      IppStatus status;

      ippsSet_32f(7, x, 8);  --x[4];

      status = ippsWTHaarFwd_32f(x, 8, lo, hi);

      printf_32f(“WT Haar low  =”, lo, 4, status);

      printf_32f(“WT Haar high =”, hi, 4, status);

      return status;

}

Output:
    WT Haar low  =  7.000000 7.000000 6.500000 7.000000

    WT Haar high =  0.000000 0.000000 0.500000 0.000000