Multiresolutio image processig Laplacia pyramids Discrete Wavelet Trasform (DWT) Quadrature mirror filters ad cojugate quadrature filters Liftig ad reversible wavelet trasform Wavelet theory Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. Image pyramid example origial image Gaussia pyramid Laplacia pyramid Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o.
Image pyramids [Burt, Adelso, 983] Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 3 Overcomplete represetatio Number of samples i Laplacia or Gaussia pyramid = 4 + + +... + x umber of origial image samples P 4 4 4 3 Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 4
Image processig with Laplacia pyramid Iput picture + Processed picture - Aalysis Filterig Iterpolator Iterpolator Sythesis Subsamplig + - Filterig Subsamplig Iterpolator Iterpolator Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 5 Expaded Laplacia pyramid Iput picture + Processed picture - Aalysis Filterig Sythesis Filterig + - Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 6
Mosaicig i the image domai Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 7 Mosaicig by bledig Laplacia pyramids Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 8
Expaded Laplacia pyramids Origial: begoia Gaussia level Laplacia level Origial: dahlia Gaussia level Laplacia level Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 9 Bledig Laplacia pyramids Level Level Level 4 Level 6 Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o.
Example: multiresolutio oise reductio Origial Noisy Noise reductio: 3-level Haar trasform o subsamplig w/ soft thresholdig Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. Image aalysis with Laplacia pyramid Iput picture + - Aalysis Filterig Iterpolator Subsamplig Filterig + - Recogitio/ detectio/ segmetatio result Subsamplig Iterpolator Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o.
Multiscale edge detectio Zero-crossigs of Laplacia images of differet scales Spurious edges removed Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 3 Multiscale face detectio Iput image pyramid Extracted widow ( by pixels) Correct lightig Histogram equalizatio Receptive fields Hidde uits subsamplig Network Iput Output by pixels Preprocessig Neural etwork [Rowley, Baluja, Kaade, 995] Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 4
-d Discrete Wavelet Trasform Recursive applicatio of a two-bad filter bak to the lowpass bad of the previous stage yields octave bad splittig: frequecy Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 5 Cascaded aalysis / sythesis filterbaks h g h g h g h g Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 6
-d Discrete Wavelet Trasform ω y ω y ω y ω x ω x ω x ω y ω y ω x ω x ω y ω y...etc ω x ω x ω y ω y ω x ω x Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 7 -d Discrete Wavelet Trasform example Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 8
-d Discrete Wavelet Trasform example Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 9 -d Discrete Wavelet Trasform example Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o.
-d Discrete Wavelet Trasform example Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. -d Discrete Wavelet Trasform example Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o.
Review: Z-trasform ad subsamplig Geeralizatio of the discrete-time Fourier trasform [ ] ; C ; x z = x z z r < z < r = + Fourier trasform o uit circle: substitute z = j e ω Dowsamplig ad upsamplig by factor x( z ) x z + x z Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 3 Two-chael filterbak x( z ) ˆx ( z) h g h g xˆ( z) = g( z) [ h( zxz ) + h( zx ) ( z) ] + g( z) [ h( zxz ) + h( zx ) ( z) ] = + + + Aliasig [ h ( z) g ( z) h( z) g ( z) ] x( z) [ h ( z) g ( z) h( z) g ( z) ] x( z) Aliasig cacellatio if : g ( z) = h( z) g ( z) = h ( z) Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 4
Example: two-chael filter bak with perfect recostructio Impulse resposes, aalysis filters: Lowpass highpass 3,,,, 4 4,, 4 4 Impulse resposes, sythesis filters Lowpass highpass,, 4 4 3,,,, 4 4 Biorthogoal 5/3 filters LeGall filters Madatory i JPEG Frequecy resposes: Frequecy respose h g π Frequecy g h π Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 5 Classical quadrature mirror filters (QMF) QMFs achieve aliasig cacellatio by choosig Example: 6-tap QMF filterbak h( z) = h ( z) = g ( z) = g ( z) [Croisier, Esteba, Galad, 976] Highpass bad is the mirror image of the lowpass bad i the frequecy domai Need to desig oly oe prototype filter frequecy ω Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 6
Cojugate quadrature filters Achieve aliasig cacelatio by Prototype filter Impulse resposes h k = g k = f k Orthoormal subbad trasform! Perfect recostructio: fid power complemetary prototype filter h z = g z f z h z = g z = zf z [Smith, Barwell, 986] [ ] [ ] [ ] k + [ ] = [ ] = ( ) ( + ) h k g k f k j j ( ω ) F e ω± π F e + = Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 7 Liftig Aalysis filters [ ] eve samples x K low bad y λ λ λl λ L odd samples [ ] x + K high bad y L liftig steps [Sweldes 996] First step ca be iterpreted as predictio of odd samples from the eve samples Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 8
Liftig (cot.) Sythesis filters [ ] eve samples x odd samples [ ] x + λ λ λl - - - - λ L K K low bad y high bad y Perfect recostructio (biorthogoality) is directly build ito liftig structure Powerful for both implemetatio ad filter/wavelet desig Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 9 Example: liftig implemetatio of 5/3 filters [ ] eve samples x low bad y ( z) + + z 4 odd samples [ ] x + / high bad y Verify by cosiderig respose to uit impulse i eve ad odd iput chael. Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 3
Iteger-to-iteger wavelet trasform Observatio: liftig operators ca be oliear. Icorporate the ecessary roudig ito liftig operator: [ ] eve samples x K low bad y λ λ λl λ L odd samples [ ] x + K high bad y Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 3 Wavelet bases x = x( t) Cosider Hilbert space of fiite-eergy fuctios. Wavelet basis for L that spa ψ L : family of liearly idepedet fuctios ( m ) m m () t = ψ ( t ) L. Hece ay sigal x L x = m= = y ( m ) [ ] ψ ( m) mother wavelet ca be writte as Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 3
Multi-resolutio aalysis Nested subspaces ( ) ( ) K V V V V V K L Upward completeess Dowward complete ess U I m Z m Z () ( m) ( m) {} m Self-similari t y x t V iff x t V Traslatio ivariace { ϕ } () V V = L = ( m) x t V iff x t V for all Z () t () t = ( t ) There exists a "scalig fuctio" ϕ with iteger traslates ϕ ϕ - such that forms a orthoormal basis for V Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 33 Multiresolutio Fourier aalysis spa ( p 3) ( p 3) { ϕ } = V ( p ) ( p ) spa{ ϕ } = V ( p ) ( p ) spa{ ψ } = W ( p ) ( p ) spa{ ϕ } = V ( p ) ( p ) spa{ ψ } = W ( p) ( p) spa{ ϕ } = V ( p) ( p) spa{ ψ } = W spa ( p 3) ( p 3) { ψ } = W ω Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 34
Sice V Relatio to subbad filters ( ) Orthoormality [ ] V, recursive defiitio of scalig fuctio () [ ] ( ) () = liear combiatio ( of scalig fuctios i V ) [ ] ϕ t = g ϕ t = g ϕ t = 4 44443 δ = ϕ, ϕ = g[] i ϕ( t i) g[ j] ϕ( ( t ) j) dt i j [] [ ] ( ) ( ) ϕi ϕ j i [] [ ] = g i g j, = g i g i i, j 4 44443 g [ k ] uit orm ad orthogoal to its -traslates: correspods to sythesis lowpass filter of orthoormal subbad trasform Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 35 Wavelets from scalig fuctios ( p) ( p) ( p ) W is orthoormal complemet of V i V ( p) ( p) ( p) ( p) ( p ) W V ad W V = V Orthoormal wavelet basis ψ () t = g [ ] ϕ t = 4 44443 ( ) { ψ } for W V ( ) () liear combiatio of scalig fuctios ( i V ) + [ ] = ( ) ( ) [ ] ϕ = g t Usig cojugate quadrature high-pass sythesis filter g g The mutually orthoormal fuctios, Easy to exted to dilated versios of ψ t = ( ) { ψ } ad { ϕ }, together spa V Z Z () ( m) { ψ } L m, Z for. to costruct orthoormal wavelet basis. Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 36
Calculatig wavelet coefficiets for a cotiuous sigal Sigal sythesis by discrete filter bak Suppose cotiuous sigal Write as superpositio of x t = y t = y ϕ V Z Z () () () () x () t V ad w () t W ( ) () () () () () = [] ϕ + [ ] ψ i Z j Z 4443 44443 () () x () t V () w t W x t y i y j [ ] ϕ Sigal aalysis by aalysis filters h [k], h [k] Discrete wavelet trasform () () ( ) [ ] y [ ] ( ) () = ϕ y [ ] g[ i] + y [ j] g[ i] Z i Z j Z 44444444444444443 Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 37 Discrete Wavelet Trasform () [ ] y h ( ) [ ] y g () [ ] y h ( ) [ ] y g x( t) Samplig ( ) [ ] y h g ( ) [ ] y Iterpolatio ϕ ( t) x( t) h ( ) [ ] y g Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 38
Differet wavelets Haar / coeffs. Daubechies 8/8 Symlets 8/8 Cohe- Daubechies- Feauveau 7/ [Gozalez, Woods, ] Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 39 Daubechies orthoormal 8-tap filters [Gozalez, Woods, ] Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 4
8-tap Symlets [Gozalez, Woods, ] Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 4 Biorthogoal Cohe-Daubechies-Feauveau 7/ wavelets [Gozalez, Woods, ] Berd Girod: EE368 Digital Image Processig Multiresolutio & Wavelets o. 4