Dualtree and doubledensity 2D wavelet transform
returns the wt
= dddtree2(typetree
,x
,level
,fdf
,df
)typetree
discrete wavelet transform of the 2D input image,
x
, down to level, level
. The wavelet transform
uses the decomposition (analysis) filters, fdf
, for the first level and
the analysis filters, df
, for subsequent levels. Supported wavelet
transforms are the critically sampled DWT, doubledensity, real oriented dualtree, complex
oriented dualtree, real oriented dualtree doubledensity, and complex oriented dualtree
doubledensity wavelet transform. The critically sampled DWT is a filter bank decomposition
in an orthogonal or biorthogonal basis (nonredundant). The other wavelet transforms are
oversampled filter banks with differing degrees of directional selectivity.
uses the filters specified in wt
= dddtree2(typetree
,x
,level
,fname1
,fname2
)fname1
for the first stage of the
dualtree wavelet transform and the filters specified in fname2
for
subsequent stages of the dualtree wavelet transform. Specifying different filters for stage
1 is valid and necessary only when typetree
is
'realdt'
, 'cplxdt'
, 'realdddt'
,
or 'cplxdddt'
.
Visualize the six directional wavelets of the real oriented dualtree wavelet transform.
Create the firststage Farras analysis filters for the two trees.
Faf{1} = [0 0 0.0884 0.0112 0.0884 0.0112 0.6959 0.0884 0.6959 0.0884 0.0884 0.6959 0.0884 0.6959 0.0112 0.0884 0.0112 0.0884 0 0]; Faf{2} = [ 0.0112 0 0.0112 0 0.0884 0.0884 0.0884 0.0884 0.6959 0.6959 0.6959 0.6959 0.0884 0.0884 0.0884 0.0884 0 0.0112 0 0.0112];
Create the 6tap Kingsbury Qshift analysis filters for subsequent stages of the multiresolution analysis.
af{1} = [ 0.0352 0 0 0 0.0883 0.1143 0.2339 0 0.7603 0.5875 0.5875 0.7603 0 0.2339 0.1143 0.0883 0 0 0 0.0352]; af{2} = [0 0.0352 0 0 0.1143 0.0883 0 0.2339 0.5875 0.7603 0.7603 0.5875 0.2339 0 0.0883 0.1143 0 0 0.0352 0];
To visualize the six directional wavelets, you will modify the wavelet coefficients of a four level real oriented dualtree wavelet transform of an image of zeros. Create an image of zeros whose size satisfies the following constraints:
The row and column dimensions are divisible by $${2}^{4}$$.
The minimum of the row and column size must be greater than or equal to the product of the maximum length of the analysis filters and $${2}^{3}$$.
J = 4; L = 3*2^(J+1); N = L/2^J; x = zeros(2*L,3*L); [numrows,numcols] = size(x)
numrows = 192
numcols = 288
Obtain the real oriented dualtree wavelet transform of the image of zeros down to level 4.
wt = dddtree2('realdt',x,J,Faf,af)
wt = struct with fields:
type: 'realdt'
level: 4
filters: [1x1 struct]
cfs: {1x5 cell}
sizes: [14x2 double]
The fourth element in wt.cfs
are the level 4 wavelet coefficients. Insert a 1 in one position of the six wavelet subbands (three orientations $$\times $$ two trees) at the coarsest scale, and invert the wavelet transform.
wt.cfs{4}(N/2,N/2+0*N,1,1) = 1; wt.cfs{4}(N/2,N/2+1*N,2,1) = 1; wt.cfs{4}(N/2,N/2+2*N,3,1) = 1; wt.cfs{4}(N/2+N,N/2+0*N,1,2) = 1; wt.cfs{4}(N/2+N,N/2+1*N,2,2) = 1; wt.cfs{4}(N/2+N,N/2+2*N,3,2) = 1; xrec = idddtree2(wt);
Visualize the six directional wavelets.
imagesc(xrec); colormap gray; axis off; title('Real Oriented DualTree Wavelets')
Obtain the doubledensity wavelet transform of an image.
Load the image and obtain the doubledensity wavelet transform using 6tap filters (see dtfilters
).
load xbox imagesc(xbox) colormap gray
wt = dddtree2('ddt',xbox,1,'filters1')
wt = struct with fields:
type: 'ddt'
level: 1
filters: [1x1 struct]
cfs: {[64x64x8 double] [64x64 double]}
sizes: [10x2 double]
In the critically sampled 2D discrete wavelet transform, there is one highpass filter. Filtering the rows and columns of the image with the highpass filter corresponds to extracting details in the diagonal orientation. In the doubledensity wavelet transform, there are two highpass filters, H1 and H2. Diagonally oriented details are extracted by filtering the image rows and columns with four combinations of the highpass filters. Visualize the diagonal details in the four wavelet highpasshighpass subbands.
H1H1 = wt.cfs{1}(:,:,4); H1H2 = wt.cfs{1}(:,:,5); H2H1 = wt.cfs{1}(:,:,7); H2H2 = wt.cfs{1}(:,:,8); subplot(2,2,1) imagesc(H1H1); title('H1 H1') colormap gray; subplot(2,2,2); imagesc(H1H2); title('H1 H2') subplot(2,2,3) imagesc(H2H1) title('H2 H1') subplot(2,2,4) imagesc(H2H2) title('H2 H2')
Obtain the complex dualtree wavelet transform of an image. Show that the complex dualtree wavelet transform can detect the two different diagonal directions.
Load the image and obtain the complex dualtree wavelet transform.
load xbox imagesc(xbox) colormap gray
wt = dddtree2('cplxdt',xbox,1,'FSfarras','qshift10')
wt = struct with fields:
type: 'cplxdt'
level: 1
filters: [1x1 struct]
cfs: {[5D double] [64x64x2x2 double]}
sizes: [5x2 double]
Obtain and display the diagonally oriented details from the two trees.
waveletcfs = wt.cfs{1}; subplot(2,2,1) imagesc(waveletcfs(:,:,3,1,1)) title('Diagonal  Tree 1  Real') colormap gray subplot(2,2,2) imagesc(waveletcfs(:,:,3,1,2)) title('Diagonal  Tree 1  Imaginary') subplot(2,2,3) imagesc(waveletcfs(:,:,3,2,1)) title('Diagonal  Tree 2  Real') subplot(2,2,4) imagesc(waveletcfs(:,:,3,2,2)) title('Diagonal  Tree 2  Imaginary')
typetree
— Type of wavelet decomposition'dwt'
 'ddt'
 'realdt'
 'cplxdt'
 'realdddt'
 'cplxdddt'
Type of wavelet decomposition, specified as one of 'dwt'
,
'ddt'
, 'realdt'
, 'cplxdt'
,
'realdddt'
, or 'cplxdddt'
. The type,
'dwt'
, produces a critically sampled (nonredundant) discrete
wavelet transform. The other decomposition types produce oversampled wavelet transforms.
'ddt'
produces a doubledensity wavelet transform with one scaling
and two wavelet filters for both row and column filtering. The doubledensity wavelet
transform uses the same filters at all stages. 'realdt'
and
'cplxdt'
produce oriented dualtree wavelet transforms consisting
of two and four separable wavelet transforms. 'realdddt'
and
'cplxdddt'
produce doubledensity dualtree wavelet transforms. The
dualtree wavelet transforms use different filters for the first stage (level).
x
— Input imageInput image, specified as a matrix with evenlength row and column dimensions. Both the row and column dimensions must be divisible by 2^{L}, where L is the level of the wavelet transform. Additionally, the minimum of the row and column dimensions of the image must be greater than or equal to the product of the maximum length of the decomposition (analysis) filters and 2^{(L1).}
Data Types: double
level
— Level of wavelet decompositionLevel of the wavelet decomposition, specified as a positive
integer. If L is the value of level
,
2^{L} must divide both
the row and column dimensions of x
. Additionally,
the minimum of the row and column dimensions of the image must be
greater than or equal to the product of the maximum length of the
decomposition (analysis) filters and 2^{(L1)}.
fdf
— Levelone analysis filtersThe levelone analysis filters, specified as a matrix or cell
array of matrices. Specify fdf
as a matrix when typetree
is 'dwt'
or 'ddt'
.
The size and structure of the matrix depend on the typetree
input
as follows:
'dwt'
— This is the critically
sampled discrete wavelet transform. In this case, fdf
is
a twocolumn matrix with the lowpass (scaling) filter in the first
column and the highpass (wavelet) filter in the second column.
'ddt'
— This is the doubledensity
wavelet transform. The doubledensity DWT is a threechannel perfect
reconstruction filter bank. fdf
is a threecolumn
matrix with the lowpass (scaling) filter in the first column and the
two highpass (wavelet) filters in the second and third columns. In
the doubledensity wavelet transform, the single lowpass and two highpass
filters constitute a threechannel perfect reconstruction filter bank.
This is equivalent to the three filters forming a tight frame. You
cannot arbitrarily choose the two wavelet filters in the doubledensity
DWT. The three filters together must form a tight frame.
Specify fdf
as a 1by2 cell array of matrices
when typetree
is a dualtree transform, 'realdt'
, 'cplxdt'
, 'realdddt'
,
or 'cplxdddt'
. The size and structure of the matrix
elements in the cell array depend on the typetree
input
as follows:
For the dualtree complex wavelet transforms, 'realdt'
and 'cplxdt'
, fdf{1}
is
an Nby2 matrix containing the lowpass (scaling)
and highpass (wavelet) filters for the first tree and fdf{2}
is
an Nby2 matrix containing the lowpass (scaling)
and highpass (wavelet) filters for the second tree.
For the doubledensity dualtree complex wavelet transforms, 'realdddt'
and 'cplxdddt'
, fdf{1}
is
an Nby3 matrix containing the lowpass (scaling)
and two highpass (wavelet) filters for the first tree and fdf{2}
is
an Nby3 matrix containing the lowpass (scaling)
and two highpass (wavelet) filters for the second tree.
df
— Analysis filters for levels > 1Analysis filters for levels > 1, specified as a matrix or
cell array of matrices. Specify df
as a matrix
when typetree
is 'dwt'
or 'ddt'
.
The size and structure of the matrix depend on the typetree
input
as follows:
'dwt'
— This is the critically
sampled discrete wavelet transform. In this case, df
is
a twocolumn matrix with the lowpass (scaling) filter in the first
column and the highpass (wavelet) filter in the second column. For
the critically sampled orthogonal or biorthogonal DWT, the filters
in df
and fdf
must be identical.
'ddt'
— This is the doubledensity
wavelet transform. The doubledensity DWT is a threechannel perfect
reconstruction filter bank. df
is a threecolumn
matrix with the lowpass (scaling) filter in the first column and the
two highpass (wavelet) filters in the second and third columns. In
the doubledensity wavelet transform, the single lowpass and two highpass
filters constitute a threechannel perfect reconstruction filter bank.
This is equivalent to the three filters forming a tight frame. For
the doubledensity DWT, the filters in df
and fdf
must
be identical.
Specify df
as a 1by2 cell array of matrices
when typetree
is a dualtree transform, 'realdt'
, 'cplxdt'
, 'realdddt'
,
or 'cplxdddt'
. For dualtree transforms, the filters
in fdf
and df
must be different.
The size and structure of the matrix elements in the cell array depend
on the typetree
input as follows:
For the dualtree wavelet transforms, 'realdt'
and 'cplxdt'
, df{1}
is
an Nby2 matrix containing the lowpass (scaling)
and highpass (wavelet) filters for the first tree and df{2}
is
an Nby2 matrix containing the lowpass (scaling)
and highpass (wavelet) filters for the second tree.
For the doubledensity dualtree complex wavelet transforms, 'realdddt'
and 'cplxdddt'
, df{1}
is
an Nby3 matrix containing the lowpass (scaling)
and two highpass (wavelet) filters for the first tree and df{2}
is
an Nby3 matrix containing the lowpass (scaling)
and two highpass (wavelet) filters for the second tree.
fname
— Filter namefname1
— Firststage filter nameFirststage filter name, specified as a character vector or string scalar. Specifying a
firstlevel filter that is different from the wavelet and scaling filters in
subsequent levels is valid and necessary only with the dualtree wavelet
transforms, 'realdt'
, 'cplxdt'
,
'realdddt'
, and 'cplxdddt'
.
fname2
— Filter name for stages > 1Filter name for stages > 1, specified as a character vector or string scalar. Specifying a
different filter for stages > 1 is valid and necessary only with the
dualtree wavelet transforms, 'realdt'
,
'cplxdt'
, 'realdddt'
, and
'cplxdddt'
.
wt
— Wavelet transformWavelet transform, returned as a structure with these fields:
type
— Type of wavelet decomposition (filter bank)'dwt'
 'ddt'
 'realdt'
 'cplxdt'
 'realdddt'
 'cplxdddt'
Type of wavelet decomposition used in the analysis returned
as one of 'dwt'
, 'ddt'
, 'realdt'
, 'cplxdt'
, 'realdddt'
,
or 'cplxdddt'
. 'dwt'
is the
critically sampled DWT. 'ddt'
produces a doubledensity
wavelet transform with one scaling and two wavelet filters for both
row and column filtering. 'realdt'
and 'cplxdt'
produce
oriented dualtree wavelet transforms consisting of 2 and 4 separable
wavelet transforms. 'realdddt'
and 'cplxdddt'
produce
doubledensity dualtree wavelet transforms consisting of two and
four separable wavelet transforms.
level
— Level of wavelet decompositionLevel of wavelet decomposition, returned as a positive integer.
filters
— Decomposition (analysis) and reconstruction (synthesis) filtersDecomposition (analysis) and reconstruction (synthesis) filters, returned as a structure with these fields:
Fdf
— Firststage analysis filtersFirststage analysis filters, returned as an Nby2 or Nby3 matrix for singletree wavelet transforms, or a 1by2 cell array of two Nby2 or Nby3 matrices for dualtree wavelet transforms. The matrices are Nby3 for the doubledensity wavelet transforms. For an Nby2 matrix, the first column of the matrix is the scaling (lowpass) filter and the second column is the wavelet (highpass) filter. For an Nby3 matrix, the first column of the matrix is the scaling (lowpass) filter and the second and third columns are the wavelet (highpass) filters. For the dualtree transforms, each element of the cell array contains the firststage analysis filters for the corresponding tree.
Df
— Analysis filters for levels > 1Analysis filters for levels > 1, returned as an Nby2 or Nby3 matrix for singletree wavelet transforms, or a 1by2 cell array of two Nby2 or Nby3 matrices for dualtree wavelet transforms. The matrices are Nby3 for the doubledensity wavelet transforms. For an Nby2 matrix, the first column of the matrix is the scaling (lowpass) filter and the second column is the wavelet (highpass) filter. For an Nby3 matrix, the first column of the matrix is the scaling (lowpass) filter and the second and third columns are the wavelet (highpass) filters. For the dualtree transforms, each element of the cell array contains the analysis filters for the corresponding tree.
Frf
— Firstlevel reconstruction filtersFirstlevel reconstruction filters, returned as an Nby2 or Nby3 matrix for singletree wavelet transforms, or a 1by2 cell array of two Nby2 or Nby3 matrices for dualtree wavelet transforms. The matrices are Nby3 for the doubledensity wavelet transforms. For an Nby2 matrix, the first column of the matrix is the scaling (lowpass) filter and the second column is the wavelet (highpass) filter. For an Nby3 matrix, the first column of the matrix is the scaling (lowpass) filter and the second and third columns are the wavelet (highpass) filters. For the dualtree transforms, each element of the cell array contains the firststage synthesis filters for the corresponding tree.
Rf
— Reconstruction filters for levels > 1Reconstruction filters for levels > 1, returned as an Nby2 or Nby3 matrix for singletree wavelet transforms, or a 1by2 cell array of two Nby2 or Nby3 matrices for dualtree wavelet transforms. The matrices are Nby3 for the doubledensity wavelet transforms. For an Nby2 matrix, the first column of the matrix is the scaling (lowpass) filter and the second column is the wavelet (highpass) filter. For an Nby3 matrix, the first column of the matrix is the scaling (lowpass) filter and the second and third columns are the wavelet (highpass) filters. For the dualtree transforms, each element of the cell array contains the firststage analysis filters for the corresponding tree.
cfs
— Wavelet transform coefficientsWavelet transform coefficients, specified as a
1by(level
+1) cell array of matrices. The size and
structure of the matrix elements of the cell array depend on the type of wavelet
transform, typetree
as follows:
'dwt'
— cfs{j}(:,:,d)
j = 1,2,... level
is the level.
d = 1,2,3 is the orientation.
cfs{level+1}(:,:)
are the lowpass, or scaling,
coefficients.
'ddt'
—
cfs{j}(:,:,d)
j = 1,2,... level
is the level.
d = 1,2,3,4,5,6,7,8 is the orientation.
cfs{level+1}(:,:)
are the lowpass, or scaling,
coefficients.
'realdt'
—
cfs{j}(:,:,d,k)
j = 1,2,... level
is the level.
d = 1,2,3 is the orientation.
k = 1,2 is the wavelet transform tree.
cfs{level+1}(:,:,k)
are the lowpass, or scaling,
coefficients.
'cplxdt'
—
cfs{j}(:,:,d,k,m)
j = 1,2,... level
is the level.
d = 1,2,3 is the orientation.
k = 1,2 is the wavelet transform tree.
m = 1,2 are the real and imaginary parts.
cfs{level+1}(:,:,k,m)
are the lowpass, or
scaling, coefficients.
'realdddt'
—
cfs{j}(:,:,d,k)
j = 1,2,... level
is the level.
d = 1,2,3,4,5,6,7,8 is the orientation.
k = 1,2 is the wavelet transform tree.
cfs{level+1}(:,:,k)
are the lowpass, or scaling,
coefficients.
'cplxdddt'
—
cfs{j}(:,:,d,k,m)
j = 1,2,... level
is the level.
d = 1,2,3,4,5,6,7,8 is the orientation.
k = 1,2 is the wavelet transform tree.
m = 1,2 are the real and imaginary parts.
cfs{level+1}(:,:,k,m)
are the lowpass, or
scaling, coefficients.
Each orientation corresponds to a particular subband. The doubledensity
transforms 'ddt'
, 'realddt'
, and
'cplxdddt'
generate wavelet coefficients of eight
orientations. The other transforms, 'dwt'
,
'realdt'
, and 'cplxdt'
generate wavelet
coefficients of three orientations. The correspondence to subbands is as
follows.
typetree  Orientations 

'dwt' , 'realdt' ,
'cplxdt' 

'ddt' , 'realdddt' ,
'cplxdddt' 

sizes
— Sizes of componentsSizes of components in cfs
, returned as an Nby2
integervalued matrix. The value of N depends on the level of wavelet
decomposition and the type of wavelet decomposition: N = 2 +
level
× (number of orientations).
cfs(1,:)
= dimensions of input image.
cfs(2+level,:)
= dimensions of scaling
coefficients.
cfs(1+no×(i–1)+(1:no),:)
= dimensions of level
i
detail coefficients, where no
is
the number of orientations.
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