MATLAB Wavelet Toolbox Skill

Expert skill for 2D wavelet analysis in MATLAB. Focus areas: custom wavelet design via lifting schemes, medical image analysis (MRI/CT/ultrasound), and deep learning integration.

Read Before Coding

Always open the relevant knowledge card before writing code:

Task	Knowledge File
Mathematical foundations	knowledge/mathematical-foundations.md
Custom wavelet design	knowledge/cards/custom-wavelets.md
Learning wavelets from data	knowledge/cards/custom-wavelets.md
MRI/CT/ultrasound processing	knowledge/cards/medical-imaging.md
Deep learning + wavelets	knowledge/cards/deep-learning.md
Basic 2D decomposition	knowledge/cards/2d-transforms.md
Denoise images	knowledge/cards/denoising.md
Directional edge detection	knowledge/cards/dual-tree.md
Curvilinear features (vessels)	knowledge/cards/shearlets.md
Filter coefficients	knowledge/cards/filters.md

Critical Rules

1. Check level limit: wmaxlev(size(img), wname) before decomposition
2. Match transform pairs: dwt2↔idwt2, wavedec2↔waverec2, lwt2↔ilwt2
3. Verify wavelet: waveinfo('db') or wavemngr('read')
4. Boundary mode: Use 'symmetric' for medical images
5. For custom wavelets: Verify perfect reconstruction, vanishing moments

Transform Selection

Analysis type?
├── Standard 2D decomposition
│   ├── Critically sampled → wavedec2/waverec2
│   └── Shift-invariant → swt2/iswt2 (MODWT is 1D only)
├── Directional analysis
│   ├── 6 orientations → dualtree2/idualtree2
│   └── Curvilinear (vessels) → shearletSystem
├── Custom wavelet design
│   └── Lifting scheme → liftingScheme + lwt2/ilwt2
└── Time-frequency (signals)
    └── CWT → cwt/icwt

Wavelet Selection for Images

Image Type	Wavelet	Rationale
Medical (MRI, CT)	db4-db8	Good edge preservation
Smooth gradients	sym6-sym8	Higher vanishing moments
Sharp edges	db2-db4	Shorter filters
Compression	bior4.4, bior6.8	Symmetric (linear phase)

Quick Patterns

Multi-level Decomposition

[C, S] = wavedec2(img, 4, 'db4');
cA = appcoef2(C, S, 'db4');           % Approximation
cH = detcoef2('h', C, S, 1);          % Horizontal detail, level 1
imgRec = waverec2(C, S, 'db4');

Medical Image Denoising

% MRI (Rician noise)
mriClean = wdenoise2(mri, 'DenoisingMethod', 'Bayes', ...
    'Wavelet', 'sym4', 'Level', 4);
% Valid methods: 'UniversalThreshold', 'Minimax', 'SURE', 'Bayes', 'FDR'

% Ultrasound (multiplicative speckle)
logUS = log(1 + double(us));
denoised = wdenoise2(logUS, 'DenoisingMethod', 'Bayes');
usClean = exp(denoised) - 1;

Custom Wavelet via Lifting

ls = liftingScheme('Wavelet', 'haar');
ls = addlift(ls, liftingStep('Type', 'predict', ...
    'Coefficients', [-0.5, 0.5], 'MaxOrder', 1));
ls = addlift(ls, liftingStep('Type', 'update', ...
    'Coefficients', [0.25, 0.25], 'MaxOrder', 0));
[LL, LH, HL, HH] = lwt2(img, liftingScheme=ls);

Directional Analysis

[a, d] = dualtree2(img, Level=3);
% d{level}(:,:,dir) for 6 directions: ±15°, ±45°, ±75°

Deep Learning (R2025a+)

x = dlarray(img, 'SSCB');
[A, D] = dldwt(x, Wavelet='db4');  % A=approx, D=detail (concatenated)
% For 2D: D(:,:,1,:)=H, D(:,:,2,:)=V, D(:,:,3,:)=D subbands
xRec = dlidwt(A, D, Wavelet='db4');  % Inverse transform

Function Quick Reference

Function	Purpose
wavedec2/waverec2	Multi-level 2D DWT
lwt2/ilwt2	Lifting-based DWT (custom wavelets)
dualtree2/idualtree2	Dual-tree complex (6 orientations)
shearletSystem	Curvilinear feature detection
wdenoise2	Image denoising
liftingScheme	Create/modify lifting scheme
liftingStep	Define predict/update step
wfilters	Get filter coefficients
dldwt/dlidwt	Differentiable DWT (R2025a+)

Knowledge Base Summary

~1,740 lines of curated, mathematically rigorous content:

• mathematical-foundations.md: MRA theory, Daubechies construction, perfect reconstruction
• custom-wavelets.md: Lifting scheme, learnable wavelets, constraints
• medical-imaging.md: Noise models (Rician, Poisson, speckle), fusion, features
• deep-learning.md: Network layers, GPU acceleration, learnable filters
• Plus: 2d-transforms, denoising, dual-tree, shearlets, filters

See knowledge/INDEX.md for full navigation.