Digital Image Processing 3 November 6 Dr. ir. Aleksandra Pizurica Prof. Dr. Ir. Wilfried Philips Aleksandra.Pizurica @telin.ugent.be Tel: 9/64.345 UNIVERSITEIT GENT Telecommunicatie en Informatieverwerking Image transforms
version: 3//6 A. Pizurica, Universiteit Gent, 6 Introduction to image transforms transform inverse transform pixel data coefficients pixel data basis images Image transform decomposes an image into basis images. Coefficients can be seen as weighting factors for those different image components Important for compression, restoration and analysis. For example: Compression: define image transform such to pack the essential information about the image into as few big coefficients as possible Restoration: define image transform such that biggest image coefficients indicate image edges Other requirements can be defined for specific applications/problems 3a.3 version: 3//6 A. Pizurica, Universiteit Gent, 6 Projections onto basis images MN (, ) = b x y a p ( x, y) i= i i x =... M, y =... N a = B, i P i B, P = M N m= n b( m, n) p*( m, n) inner product The transform coefficient a i is simply the inner product of the i-th basis image with the given image. It is also called projection of the image on the i-th basis image 3a.4
version: 3//6 A. Pizurica, Universiteit Gent, 6 Discrete Cosine Transform The cosine transform has excellent energy compaction for highly correlated data Often used in compression (JPEG) 3a.5 version: 3//6 A. Pizurica, Universiteit Gent, 6 Application of DCT im image compression DCT transform 3a.6
Wavelet transform version: 3//6 A. Pizurica, Universiteit Gent, 6 Wavelets - localized waves a, b ( t) = t b a a.5.5 Mexican hat wavelets Wavelet family: shifts and dilations of the mother wavelet (t) -.5 - -6-4 - 4 6 8 Continuous wavelet transform: correlate signal with wavelets to reveal structures of different sizes Main idea: analyze according to scale! (see the forest and the trees!) 3a.8
version: 3//6 A. Pizurica, Universiteit Gent, 6 Wavelet analysis versus Fourier analysis Fourier transform Gabor transform Wavelet transform frequency time time time 3a.9 version: 3//6 A. Pizurica, Universiteit Gent, 6 Discrete Wavelet Transform (DWT) DWT algorithm: a filter bank iterated on the lowpass output highpass g s j h lowpass wavelet coefficients w j+ s j+ g h scaling coefficients s j+ w j+ g h w j+3 s j+3 3a.
version: 3//6 A. Pizurica, Universiteit Gent, 6 Choosing a wavelet: N v, support size K, symmetry ( = k N v - number of vanishing moments: t t) dt, A tradeoff: db db.5.5 K N v - Daubechies wavelets dbn v :.5.5 K= -.5.5.5 K=3 ϕ ϕ.5.5 -.5 - -.5 -.5 - -.5.5 sym8 k N Symmlets (Daubechies).5.5.5 -.5 -.8.6.4-5 5 ϕ.5 v -.5 5 5 Biorthogonal wavelets ϕ.5 -. -.5 db8 -.5 3.5.5 K=5 ϕ.5 - -.5.5-5 - ϕ ~ ~ 5-4 - 4 -.5 - - -.5 - -.5 5 5-5 5-5 5-4 - 4 3a. version: 3//6 A. Pizurica, Universiteit Gent, 6 Two dimensional DWT highpass g g h HH j+ HL j+ LL 3 HL 3 LH 3 HH 3 LH HL HH HL LL j lowpass h horizontal filtering g h vertical filtering LH j+ LL j+ LH HH 3a.
version: 3//6 A. Pizurica, Universiteit Gent, 6 Two dimensional DWT APPROXIMATION scaling coefficients Wavelet coefficient values DETAIL IMAGES wavelet coefficients Peaks indicate image edges 3a.3 version: 3//6 A. Pizurica, Universiteit Gent, 6 Curvelet transform Curvelets: specific tiling of the frequency plane: localized + directional 3a.4
Excersizes Excersize Wavelet domain image denoising
version: 3//6 A. Pizurica, Universiteit Gent, 6 Wavelet domain denosing Denoised output Noisy input scaling coefficients Wavelet transform s L w w Remove noise Inverse wavelet transform w L wavelet coefficients Estimated noise-free wavelet coefficients 3a.7 version: 3//6 A. Pizurica, Universiteit Gent, 6 Noise variance estimation Often the value of the input noise is unknown Noise has to be estimated from the observed noisy signal eliminating the influence of the actual signal A median measurement is highly insensitive to outliers Meadian Absolute Deviation (MAD) estimator HH ˆ σ = Median( w ) /.6745 3a.8
version: 3//6 A. Pizurica, Universiteit Gent, 6 Denoising by wavelet thresholding w = y + n w noisy coefficient; y noise-free coefficient; n i.i.d. Gaussian noise y, w < T ˆ = ht w, w T y, w < T ˆ = st sgn( w)( w T ), w T Hard thresholding keep or kill thresholded Soft thresholding shrink or kill thresholded T -T T -T T input input 3a.9 Excersize Wavelet domain image fusion
version: 3//6 A. Pizurica, Universiteit Gent, 6 Applications in image fusion Visible camera image MRI image CT image Infrared image 3a. version: 3//6 A. Pizurica, Universiteit Gent, 6 Applications in image fusion 3a.