dtfilters

Analysis and synthesis filters for oversampled wavelet filter banks

Syntax

df = dtfilters(name)

[df,rf] = dtfilters(name)

Description

df = dtfilters(name) returns the decomposition (analysis) filters corresponding to name. These filters are used most often as input arguments to dddtree and dddtree2.

[df,rf] = dtfilters(name) returns the reconstruction (synthesis) filters corresponding to name.

Examples

Filters for Complex Dual-Tree Wavelet Transform

Obtain valid filters for the complex dual-tree wavelet transform. The transform uses Farras nearly symmetric filters for the first stage and Kingsbury Q-shift filters with 10 taps for subsequent stages.

Load the noisy Doppler signal. Obtain the filters for the first and subsequent stages of the complex dual-tree wavelet transform. Demonstrate perfect reconstruction using the complex dual-tree wavelet transform.

load noisdopp;
df = dtfilters("dtf2");
dt = dddtree("cplxdt",noisdopp,5,df{1},df{2});
xrec = idddtree(dt);
max(abs(noisdopp-xrec))

ans = 
1.3056e-13

Filters for Double-Density Wavelet Transform

Obtain valid filters for the double-density wavelet transform.

Load the noisy Doppler signal. Obtain the filters for the double-density wavelet transform. The double-density wavelet transform uses the same filters at all stages. Demonstrate perfect reconstruction using the double-density wavelet transform.

df = dtfilters("filters1");
load noisdopp;
dt = dddtree("ddt",noisdopp,5,df,df);
xrec = idddtree(dt);
max(abs(noisdopp-xrec))

ans = 
2.3892e-13

Dual-Tree and Double-Density Wavelet Transforms Using Filter Names and Filters

Load a 1-D signal.

load noisdopp
x = noisdopp;

"dwt" - Critically sampled discrete wavelet transform

The critically sampled discrete wavelet transform can be applied to 1-D and 2-D data. The filter can be any valid orthogonal or biorthogonal wavelet name, or "farras".

Specify a valid filter name. Use dtfilters to obtain the corresponding decomposition filters. Confirm the decomposition filters are returned as a two-column matrix.

fname = "db4";
df = dtfilters(fname);
df

df = 8×2

   -0.0106   -0.2304
    0.0329    0.7148
    0.0308   -0.6309
   -0.1870   -0.0280
   -0.0280    0.1870
    0.6309    0.0308
    0.7148   -0.0329
    0.2304   -0.0106

Use dddtree to obtain two wavelet decompositions of the 1-D signal. Use the filter name for the first decomposition, and the filters for the second decomposition.

wtA = dddtree("dwt",x,3,fname);
wtB = dddtree("dwt",x,3,df,df);

Confirm the wavelet coefficients in the decompositions are equal.

for k=1:length(wtA.cfs)
    t = max(abs(wtA.cfs{k}(:)-wtB.cfs{k}(:)));
    fprintf("level %d maximum difference: %f\n",k,t)
end

level 1 maximum difference: 0.000000
level 2 maximum difference: 0.000000
level 3 maximum difference: 0.000000
level 4 maximum difference: 0.000000

Confirm the filters in both decompositions are equal.

max(abs(wtA.filters.FDf(:)-wtB.filters.FDf(:)))

ans = 
0

max(abs(wtA.filters.Df(:)-wtB.filters.Df(:)))

ans = 
0

max(abs(wtA.filters.FRf(:)-wtB.filters.FRf(:)))

ans = 
0

max(abs(wtA.filters.Rf(:)-wtB.filters.Rf(:)))

ans = 
0

"ddt" - Double-density wavelet transform

The double-density wavelet transform can be applied to 1-D and 2-D data. Valid filter names for the double-density wavelet transform are "filters1", "filters2", and "doubledualfilt".

Use dtfilters to obtain the filters corresponding to "filters1". Inspect the filters. Confirm the decomposition filters are returned as a three-column matrix.

fname = "filters1";
df = dtfilters(fname);
df

df = 6×3

    0.1430   -0.0185   -0.0460
    0.5174   -0.0669   -0.1666
    0.6396   -0.0739    0.0031
    0.2443    0.0004    0.6776
   -0.0755    0.5811   -0.4681
   -0.0546   -0.4222         0

Use dddtree to obtain two wavelet decompositions of the 1-D signal. Use the filter name for the first decomposition, and the filters for the second decomposition.

wtA = dddtree("ddt",x,3,fname);
wtB = dddtree("ddt",x,3,df,df);

Confirm the filters in both decompositions are equal.

max(abs(wtA.filters.FDf(:)-wtB.filters.FDf(:)))

ans = 
0

max(abs(wtA.filters.Df(:)-wtB.filters.Df(:)))

ans = 
0

max(abs(wtA.filters.FRf(:)-wtB.filters.FRf(:)))

ans = 
0

max(abs(wtA.filters.Rf(:)-wtB.filters.Rf(:)))

ans = 
0

Use dtfilters to obtain the filters corresponding to "doubledualfilt". Inspect the filters. Confirm the decomposition filters are returned as 1-by-2 cell array consisting of three-column matrices.

fname = "doubledualfilt";
df = dtfilters(fname);
df

df=1×2 cell array
    {10x3 double}    {10x3 double}

Use dddtree to obtain two wavelet decompositions of the 1-D signal. Use the filter name for the first decomposition, and the filters for the second decomposition.

wtA = dddtree("ddt",x,3,fname);
wtB = dddtree("ddt",x,3,df{1},df{2});

Confirm the filters in both decompositions are equal.

max(abs(wtA.filters.FDf(:)-wtB.filters.FDf(:)))

ans = 
0

max(abs(wtA.filters.Df(:)-wtB.filters.Df(:)))

ans = 
0

max(abs(wtA.filters.FRf(:)-wtB.filters.FRf(:)))

ans = 
0

max(abs(wtA.filters.Rf(:)-wtB.filters.Rf(:)))

ans = 
0

"realdt" - Real oriented dual-tree wavelet transform

The real oriented dual-tree wavelet transform can only be applied to 2-D data. Valid filter names are:

• Any orthogonal or biorthogonal wavelet name, but only as a first-stage filter.

• "dtfP", where P can equal 1, 2, 3, 4, or 5.

• "FSfarras", but only as a first-stage filter.

• "qshiftN", where N can equal 6, 10, 14, 16, or 18, for stages > 1.

Obtain a 2-D image.

x2 = x'*x;

Use dtfilters to obtain the decomposition filters corresponding to "dtf1". Confirm the filters are returned as 1-by-2 cell array consisting of 1-by-2 cell arrays.

dtf = dtfilters("dtf1")

dtf=1×2 cell array
    {1x2 cell}    {1x2 cell}

Obtain the filters corresponding to "FSfarras" and "qshift6". Confirm the filters are returned as 1-by-2 cell array consisting of two-column matrices.

fs = dtfilters("FSfarras")

fs=1×2 cell array
    {10x2 double}    {10x2 double}

qs = dtfilters("qshift6")

qs=1×2 cell array
    {10x2 double}    {10x2 double}

Confirm the dtf filters are equal to the fs and qs filters.

max(abs(dtf{1}{1}(:) - fs{1}(:)))

ans = 
0

max(abs(dtf{1}{2}(:) - fs{2}(:)))

ans = 
0

max(abs(dtf{2}{1}(:) - qs{1}(:)))

ans = 
0

max(abs(dtf{2}{2}(:) - qs{2}(:)))

ans = 
0

Use dddtree2 to obtain two realdt decompositions of the image. Use the filter name "dtf1" for the first decomposition, and the filters fs and qs for the second decomposition. Confirm the wavelet coefficients in both decompositions are equal.

wtA = dddtree2("realdt",x2,4,"dtf1");
wtB = dddtree2("realdt",x2,4,fs,qs);
for k=1:length(wtA.cfs)
    t = max(abs(wtB.cfs{k}(:)-wtA.cfs{k}(:)))
end

t = 
0

t = 
0

t = 
0

t = 
0

t = 
0

"cplxdt" - Complex oriented dual-tree wavelet transform

The complex oriented dual-tree wavelet transform can be applied to 1-D and 2-D data. Valid filter names are:

• Any orthogonal or biorthogonal wavelet name, but only as a first-stage filter.

• "dtfP", where P can equal 1, 2, 3, 4, or 5.

• "FSfarras", but only as a first-stage filter.

• "qshiftN", where N can equal 6, 10, 14, 16, or 18, for stages > 1.

Use dtfilters to obtain the decompositions filters corresponding to the db4 orthogonal wavelet and the Kingsbury Q-shift filter with 14 taps.

wf = dtfilters("db2")

wf = 4×2

   -0.1294   -0.4830
    0.2241    0.8365
    0.8365   -0.2241
    0.4830   -0.1294

qf = dtfilters("qshift14")

qf=1×2 cell array
    {14x2 double}    {14x2 double}

Use dddtree and the filters to obtain the complex oriented dual-tree wavelet decomposition of the 1-D signal.

wtA = dddtree("cplxdt",x,4,{wf,wf},qf);

Demonstrate perfect reconstruction.

xrec = idddtree(wtA);
max(abs(xrec(:)-x(:)))

ans = 
3.4159e-12

"realdddt" - Real double-density dual-tree wavelet transform

The real double-density dual-tree wavelet transform can only be applied to 2-D data. Valid filter names are:

• "dddtf1"

• "self1"

• "self2"

Use dtfilters to obtain the decomposition filters corresponding to "dddtf1". Confirm the filters are returned as 1-by-2 cell array consisting of 1-by-2 cell arrays.

fname = "dddtf1";
df = dtfilters(fname)

df=1×2 cell array
    {1x2 cell}    {1x2 cell}

Use dddtree2 to obtain two wavelet decompositions of the image. Use the filter name for the first decomposition, and the filters for the second decomposition. Confirm the decompositions are equal.

wtA = dddtree2("realdddt",x2,4,fname);
wtB = dddtree2("realdddt",x2,4,df{1},df{2});
for k=1:length(wtA.cfs)
    t = max(abs(wtB.cfs{k}(:)-wtA.cfs{k}(:)))
end

t = 
0

t = 
0

t = 
0

t = 
0

t = 
0

"cplxdddt" - Complex double-density dual-tree wavelet transform

The complex double-density dual-tree wavelet transform can be applied to 1-D and 2-D data. Valid filter names are:

• "dddtf1"

• "self1"

• "self2"

Use dtfilters to obtain the decomposition filters corresponding to "self1". Confirm the filters are returned as 1-by-2 cell array consisting of 1-by-2 cell arrays.

fname = "self1";
df = dtfilters(fname)

df=1×2 cell array
    {1x2 cell}    {1x2 cell}

Use dddtree to obtain two wavelet decompositions of the 1-D signal. Use the filter name for the first decomposition, and the filters for the second decomposition. Confirm the decompositions are equal.

wtA = dddtree("cplxdddt",x,4,fname);
wtB = dddtree("cplxdddt",x,4,df{1},df{2});
for k=1:length(wtA.cfs)
    t = max(abs(wtB.cfs{k}(:)-wtA.cfs{k}(:)))
end

t = 
0

t = 
0

t = 
0

t = 
0

t = 
0

Input Arguments

name — Filter name

"dtf1" | "dddtf1" | "self1" | "self2" | ...

Filter name, specified as a character vector or string scalar. Valid entries for name are:

• Any valid orthogonal or biorthogonal wavelet name. See wfilters for details. An orthogonal or biorthogonal wavelet is only valid when the filter bank type is "dwt", or when you use the filter as the first stage in a complex dual-tree transform, "realdt" or "cplxdt".

  Note

  An orthogonal or biorthogonal wavelet filter is not a valid filter if you have a double-density, "ddt" or dual-tree double-density, "realdddt" or "cplxdddt", filter bank. An orthogonal or biorthogonal wavelet filter is not a valid filter for complex dual-tree filter banks for stages greater than 1.

• "dtfP" — With P equal to 1, 2, 3, 4, or 5 returns the first-stage Farras filters ("FSfarras") and Kingsbury Q-shift filters ("qshiftN") for subsequent stages. This input is only valid for a dual-tree transform, "realdt" or "cplxdt". Setting P = 1, 2, 3, 4, or 5 specifies the Kingsbury Q-shift filters with N = 6, 10, 14, 16, or 18 taps, respectively.

• "dddtf1" — Returns the filters for the first and subsequent stages of the double-density dual-tree transform. This input is only valid for the double-density dual-tree transforms, "realdddt" and "cplxdddt".

• "self1" — Returns 10-tap filters for the double-density wavelet transform. This option is only valid for double-density wavelet transforms, "realdddt", and "cplxdddt".

• "self2" — Returns 16-tap filters for the double-density wavelet transform. This option is only valid for double-density wavelet transforms, "realdddt", and "cplxdddt".

• "filters1" — Returns 6-tap filters for the double-density wavelet transform, "ddt".

• "filters2" — Returns 12-tap filters for the double-density wavelet transform, "ddt".

• "farras" — Farras nearly symmetric filters for a two-channel perfect reconstruction filter bank. This option is meant to be used for one-tree transforms and is valid only for an orthogonal critically sampled wavelet transform, "dwt". The output of dtfilters is a two-column matrix. The first column of the matrix is a scaling (lowpass) filter, and the second column is a wavelet (highpass) filter.

• "FSfarras" — Farras nearly symmetric first-stage filters intended for a dual-tree wavelet transform. With this option, the output of dtfilters is a cell array with two elements, one for each tree. Each element is a two-column matrix. The first column of the matrix is a scaling (lowpass) filter, and the second column is a wavelet (highpass) filter.

• "qshiftN" — Kingsbury Q-shift N-tap filters with N = 6, 10, 14, 16, or 18. The Kingsbury Q-shift filters are used most commonly in dual-tree wavelet transforms for stages greater than 1.

• "doubledualfilt" — Filters for one stage of the double-density dual-tree wavelet transforms, "realdddt" or "cplxdddt". This option can also be used in the double-density wavelet transform, "ddt".

This table can help you decide which filter to choose:

Type of Wavelet Decomposition	Valid Filters
"dwt" — Critically sampled (nonredundant) discrete wavelet transform (1-D and 2-D)	Any valid orthogonal or biorthogonal wavelet name "farras"
"ddt" — Double-density wavelet transform (1-D and 2-D)	"filters1" "filters2" "doubledualfilt"
"realdt" — Real oriented dual-tree wavelet transform (2-D only) "cplxdt" — Complex oriented dual-tree wavelet transform (1-D and 2-D)	Any valid orthogonal or biorthogonal wavelet name (only as first stage) "dtfP" "FSfarras" (only as first stage) "qshiftN" (only for stages > 1)
"realdddt" — Real double-density dual-tree wavelet transform (2-D only) "cplxdddt" — Complex double-density dual-tree wavelet transform (1-D and 2-D)	"dddtf1" "self1" "self2" "doubledualfilt" (for one stage of the double-density dual-tree wavelet transform)

Output Arguments

df — Decomposition (analysis) filters

matrix | cell array

Decomposition (analysis) filters, returned as a matrix or cell array of matrices.

rf — Reconstruction (synthesis) filters

matrix | cell array

Reconstruction (synthesis) filters, returned as a matrix or cell array of matrices.

References

[1] Abdelnour, A. F., and I. W. Selesnick. “Design of 2-Band Orthogonal near-Symmetric CQF.” In 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221), 6:3693–96. Salt Lake City, UT, USA: IEEE, 2001. https://doi.org/10.1109/ICASSP.2001.940644.

[2] Kingsbury, Nick. “Complex Wavelets for Shift Invariant Analysis and Filtering of Signals.” Applied and Computational Harmonic Analysis 10, no. 3 (May 2001): 234–53. https://doi.org/10.1006/acha.2000.0343.

[3] Selesnick, Ivan W., and A. Farras Abdelnour. “Symmetric Wavelet Tight Frames with Two Generators.” Applied and Computational Harmonic Analysis 17, no. 2 (September 2004): 211–25. https://doi.org/10.1016/j.acha.2004.05.003.

[4] Selesnick, I.W. “The Double-Density Dual-Tree DWT.” IEEE Transactions on Signal Processing 52, no. 5 (May 2004): 1304–14. https://doi.org/10.1109/TSP.2004.826174.

Version History

Introduced in R2013b