Basically dwt2() applies 2-D discreet wavelet transformation on the image and provides the approximation and detail coefficients matrices directly. Whereas, wavedec2() only provides the wavelet decomposition vector and the bookkeeping matrix, which contains the number of coefficients by level and orientation. And you can to get approximation and detail coefficients matrices from the wavelet decomposition vector and the bookkeeping matrix using the detcoef2() and appcoef2() function.
Exact same approximation and detail coefficients matrices can be generated using any of those (dwt2 or wavedec2) functions. See the example below
close all; clc;
% Decomposition using wavedec2 and appcoef2
load woman;
[c, s] = wavedec2(X, 1, 'db1');
A1 = appcoef2(c, s, 'db1', 1);
figure; imshow(A1, []); title('Using wavedec2 and appcoef2');
% Decomposition using dwt2
load woman;
[A2, ~, ~, ~] = dwt2(X, 'db1');
figure; imshow(A2, []); title('Using dwt2');
% Difference image between A1 and A2
diffImage = imabsdiff(A1, A2);
figure; imshow(diffImage, []); title('Difference Image');
Now, here is an example how you can display different sub-bands at different wavelet decomposition levels.
clc; close all;
load woman;
[c, s] = wavedec2(X, 3, 'haar');
% Extract the level 1 approximation and detail coefficients.
[H1, V1, D1] = detcoef2('all', c, s, 1);
A1 = appcoef2(c, s, 'haar', 1);
% Display images
figure; subplot(2, 2, 1); imshow(A1, []); title('Approximation Coef. of Level 1');
subplot(2, 2, 2); imshow(H1, []); title('Horizontal Detail Coef. of Level 1');
subplot(2, 2, 3); imshow(V1, []); title('Vertical Detail Coef. of Level 1');
subplot(2, 2, 4); imshow(D1, []); title('Diagonal Detail Coef. of Level 1');
% Extract the level 2 approximation and detail coefficients.
[H2, V2, D2] = detcoef2('all', c, s, 2);
A2 = appcoef2(c, s, 'haar', 2);
% Display images
figure; subplot(2, 2, 1); imshow(A2, []); title('Approximation Coef. of Level 2');
subplot(2, 2, 2); imshow(H2, []); title('Horizontal Detail Coef. of Level 2');
subplot(2, 2, 3); imshow(V2, []); title('Vertical Detail Coef. of Level 2');
subplot(2, 2, 4); imshow(D2, []); title('Diagonal Detail Coef. of Level 2');
% Extract the level 3 approximation and detail coefficients.
[H3, V3, D3] = detcoef2('all', c, s, 3);
A3 = appcoef2(c, s, 'haar', 3);
% Display images
figure; subplot(2, 2, 1); imshow(A3, []); title('Approximation Coef. of Level 3');
subplot(2, 2, 2); imshow(H3, []); title('Horizontal Detail Coef. of Level 3');
subplot(2, 2, 3); imshow(V3, []); title('Vertical Detail Coef. of Level 3');
subplot(2, 2, 4); imshow(D3, []); title('Diagonal Detail Coef. of Level 3');
Hope this helps!
