Multiresolutio codig ad wavelets Predictive (closed-loop) pyramids Ope-loop ( Laplacia ) pyramids Discrete Wavelet Trasform (DWT) Quadrature mirror filters ad cojugate quadrature filters Liftig ad reversible wavelet trasform Wavelet theory Embedded zero-tree wavelet (EZW) codig Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. Iterpolatio error codig, I Iput picture Q - Recostructed picture Subsamplig Iterpolator - Q Coder icludes Decoder Subsamplig Iterpolator Sample ecoded i curret stage Previously coded sample Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o.
Iterpolatio error codig, II origial image sigals to be ecoded Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 3 Predictive pyramid, I Iput picture Q - Recostructed picture Filterig Iterpolator Subsamplig - Q Coder icludes Decoder Filterig Subsamplig Iterpolator Sample ecoded i curret stage Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 4
Predictive pyramid, II Number of samples to be ecoded = N N... = N x umber of origial image samples N subsamplig factor Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 5 Predictive pyramid, III origial image sigals to be ecoded Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 6 3
Compariso: iterpolatio error codig vs. pyramid Resolutio layer #, iterpolated to origial size for display Iterpolatio Error Codig Pyramid Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 7 Compariso: iterpolatio error codig vs. pyramid Resolutio layer #, iterpolated to origial size for display Iterpolatio Error Codig Pyramid Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 8 4
Compariso: iterpolatio error codig vs. pyramid Resolutio layer #, iterpolated to origial size for display Iterpolatio Error Codig Pyramid Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 9 Compariso: iterpolatio error codig vs. pyramid Resolutio layer #3 Iterpolatio Error Codig Pyramid = (origial) Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 5
Ope-loop pyramid (Laplacia pyramid) Iput picture - Q Recostructed picture Trasmitter Filterig Iterpolator Iterpolator Receiver Subsamplig - Q Filterig Subsamplig Iterpolator Iterpolator [Burt, Adelso, 983] Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. Whe multiresolutio codig was a ew idea... This mauscript is okay if compared to some of the weaker papers. [...] however, I doubt that ayoe will ever use this algorithm agai. Aoymous reviewer of Burt ad Adelso s origial paper, ca. 98 Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 6
Cascaded aalysis / sythesis filterbaks h g h g h g h g Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 3 Discrete Wavelet Trasform Recursive applicatio of a two-bad filter bak to the lowpass bad of the previous stage yields octave bad splittig: frequecy Same cocept ca be derived from wavelet theory: Discrete Wavelet Trasform (DWT) Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 4 7
-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: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 5 -d Discrete Wavelet Trasform example Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 6 8
-d Discrete Wavelet Trasform example Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 7 -d Discrete Wavelet Trasform example Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 8 9
-d Discrete Wavelet Trasform example Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 9 -d Discrete Wavelet Trasform example Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o.
Two-chael filterbak x( z ) ˆx ( z) h g h g xz ˆ( ) = [ h( zg ) ( z) h( zg ) ( z) ] xz ( ) Aliasig cacellatio if : [ h ( z) g ( z) h( z) g ( z) ] x( z) g ( z) = h( z) g ( z) = h ( z) Aliasig Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 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: EE398A Image Commuicatio I Multiresolutio & Wavelets o.
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: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 3 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 F ( ω) F( ω π) ± = Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 4
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: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 5 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 built ito liftig structure Powerful for both implemetatio ad filter/wavelet desig Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 6 3
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: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 7 Reversible subbad 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 [ ] Used i JPEG as part of 5/3 biorthogoal wavelet trasform K high bad y Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 8 4
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: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 9 Multi-resolutio aalysis Nested subspaces ( ) ( ) ( ) ( ) ( ) ( ) V V V V V L Upward completeess Dowward complete ess m Z m Z () ( m) ( m) ( ) {} ( ) m Self-similari t y xt V iff x t V Traslatio ivariace { ϕ } () V V = L = ( m) ( ) ( ) ( ) ( ) xt V iff xt 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: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 3 5
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: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 3 Sice V Relatio to subbad filters ( ) ( ) V Orthoormality ( () [ ] ) () ( ) ( ) [ ] =,, recursive defiitio of scalig fuctio liear combiatio ( of scalig fuctios i V ) [ ] ( ) ϕ t = g ϕ t = g ϕ t = = δ ϕ ϕ ( [] ) ( ) = g i ϕi () t g[ j] ϕj () t dt i j ( ) ( ) [] [ ] ϕi, ϕj [] [ ] = g i g j = g i g i i, j i g [ k ] uit orm ad orthogoal to its -traslates: correspods to sythesis lowpass filter of orthoormal subbad trasform Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 3 6
Wavelets from scalig fuctios ( p) ( p) ( p ) W is orthogoal complemet of V i V ( p) ( p) ( p) ( p) ( p ) W V ad W V = V Orthoormal wavelet basis ψ ( () t g [ ] ϕ ) () t = = liear combiatio of scalig fuctios ( i V ) [ ] = ( ) ( ) ( ) ( ) ( ) { ψ } for W V [ ] ϕ ( ) = g t Usig cojugate quadrature high-pass sythesis filter g g ( ) ( ) ( ) { ψ } { ϕ } V Z Z The mutually orthoormal fuctios, ad, together spa. Easy to exted to dilated versios of ψ t = () ( m) { ψ } L ( ) m, Z for. to costruct orthoormal wavelet basis Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 33 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 () () 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 Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 34 7
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: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 35 Differet wavelets Haar / coeffs. Daubechies 8/8 Symlets 8/8 Cohe- Daubechies- Feauveau 7/ [Gozalez, Woods, ] Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 36 8
Daubechies orthoormal 8-tap filters [Gozalez, Woods, ] Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 37 8-tap Symlets [Gozalez, Woods, ] Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 38 9
Biorthogoal Cohe-Daubechies-Feauveau 7/ wavelets [Gozalez, Woods, ] Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 39 Wavelet compressio results.74 bpp.48 bpp Origial 5x5 8bpp Error images elarged [Gozalez, Woods, ] Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 4
Embedded zero-tree wavelet algorithm X X X X X X X X X X X X X X X Paret Childre Descedats Idea: Coditioal codig of all descedats (icl. childre) Coefficiet magitude > threshold: sigificat coefficiets Four cases ZTR: zero-tree, coefficiet ad all descedats are ot sigificat IZ: coefficiet is ot sigificat, but some descedats are sigificat POS: positive sigificat NEG: egative sigificat Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 4 Embedded zero-tree wavelet algorithm (cot.) For the highest bads, ZTR ad IZ symbols are merged ito oe symbol Z Successive approximatio quatizatio ad ecodig Iitial domiat pass Set iitial threshold T, determie sigificat coefficiets Arithmetic codig of symbols ZTR, IZ, POS, NEG Subordiate pass Refie magitude of all coefficiets foud sigificat so far by oe bit (subdivide magitude bi by two) Arithmetic codig of sequece of zeros ad oes. Repeat domiat pass Omit previously foud sigificat coefficiets Decrease threshold by factor of, determie ew sigificat coefficiets Arithmetic codig of symbols ZTR, IZ, POS, NEG Repeat subordiate ad domiate passes, util bit budget is exhausted. Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 4
Embedded zero-tree wavelet algorithm (cot.) Decodig: bitstream ca be trucated to yield a coarser approximatio: embedded represetatio Further details: J. M. Shapiro, Embedded image codig usig zerotrees of wavelet coefficiets, IEEE Trasactios o Sigal Processig, vol. 4, o., pp. 3445-346, December 993. Ehacemet SPIHT coder: A. Said, A., W. A. Pearlma, A ew, fast, ad efficiet image codec based o set partitioig i hierarchical trees, IEEE Trasactios o Circuits ad Systems for Video Techology, vol. 63, pp. 43-5, Jue 996. Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 43 Summary: multiresolutio ad subbad codig Resolutio pyramids with subsamplig : horizotally ad vertically Predictive pyramids: quatizatio error feedback ( closed loop ) Trasform pyramids: o quatizatio error feedback ( ope loop ) Pyramids: overcomplete represetatio of the image Critically sampled subbad decompositio: umber of samples ot icreased Discrete Wavelet Trasform = cascaded dyadic subbad splits Quadrature mirror filters ad cojugate quadrature filters: aliasig cacellatio Liftig: powerful for implemetatio ad wavelet costructio Liftig allows reversible wavelet trasform Zero-trees: exploit statistical depedecies across subbads Berd Girod: EE398A Image Commuicatio I Multiresolutio & Wavelets o. 44