idwt (Wavelet Toolbox)

Single-level inverse discrete 1-D wavelet transform

Syntax

X = idwt(cA,cD,'wname')
X = idwt(cA,cD,Lo_R,Hi_R)
X = idwt(cA,cD,'wname',L)
X = idwt(cA,cD,Lo_R,Hi_R,L)
X = idwt(...,'mode',MODE)

Description

The idwt command performs a single-level one-dimensional wavelet reconstruction with respect to either a particular wavelet ('wname', see wfilters for more information) or particular wavelet reconstruction filters (Lo_R and Hi_R) that you specify.

X = idwt(cA,cD,'wname') returns the single-level reconstructed approximation coefficients vector X based on approximation and detail coefficients vectors cA and cD, and using the wavelet 'wname'.

X = idwt(cA,cD,Lo_R,Hi_R) reconstructs as above using filters that you specify.

Lo_R is the reconstruction low-pass filter.
Hi_R is the reconstruction high-pass filter.

Lo_R and Hi_R must be the same length.

Let la be the length of cA (which also equals the length of cD) and lf the length of the filters Lo_R and Hi_R; then length(X) = LX where LX = 2*la if the DWT extension mode set to periodization. For the other extension modes LX = 2*la-lf+2.

For more information about the different Discrete Wavelet Transform extension modes, see dwtmode.

X = idwt(cA,cD,'wname',L) or X = idwt(cA,cD,Lo_R,Hi_R,L) returns the length-L central portion of the result obtained using idwt(cA,cD,'wname'). L must be less than LX.

X = idwt(...,'mode',MODE) computes the wavelet reconstruction using the specified extension mode MODE.

X = idwt(cA,[],...) returns the single-level reconstructed approximation coefficients vector X based on approximation coefficients vector cA.

X = idwt([],cD,...) returns the single-level reconstructed detail coefficients vector X based on detail coefficients vector cD.

idwt is the inverse function of dwt in the sense that the abstract statement idwt(dwt(X,'wname'),'wname') would give back X.

Examples

% The current extension mode is zero-padding (see dwtmode).

% Construct elementary one-dimensional signal s. 

randn('seed',531316785) 
s = 2 + kron(ones(1,8),[1 -1]) + ... 
    ((1:16).^2)/32 + 0.2*randn(1,16);

% Perform single-level dwt of s using db2.

[ca1,cd1] = dwt(s,'db2'); 
subplot(221); plot(ca1); 
title('Approx. coef. for db2'); 
subplot(222); plot(cd1); 
title('Detail coef. for db2');

% Perform single-level inverse discrete wavelet transform, 
% illustrating that idwt is the inverse function of dwt. 

ss = idwt(ca1,cd1,'db2'); 
err = norm(s-ss);     % Check reconstruction. 
subplot(212); plot([s;ss]'); 
title('Original and reconstructed signals'); 
xlabel(['Error norm = ',num2str(err)])

% For a given wavelet, compute the two associated
% reconstruction filters and inverse transform using 
% the filters directly.

[Lo_R,Hi_R] = wfilters('db2','r'); 
ss = idwt(ca1,cd1,Lo_R,Hi_R);

% Using some plotting commands,
% the following figure is generated.

Algorithm

Starting from the approximation and detail coefficients at level j, cA_j and cD_j, the inverse discrete wavelet transform reconstructs cA_j-1, inverting the decomposition step by inserting zeros and convolving the results with the reconstruction filters.

See Also
dwt, dwtmode, upwlev

References

Daubechies, I. (1992), Ten lectures on wavelets, CBMS-NSF conference series in applied mathematics. SIAM Ed.

Mallat, S. (1989), "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Pattern Anal. and Machine Intell., vol. 11, no. 7, pp. 674-693.

Meyer, Y. (1990), Ondelettes et opérateurs, Tome 1, Hermann Ed. (English translation: Wavelets and operators, Cambridge Univ. Press. 1993.)

get idwt2