Transform Coding. coefficient vectors u = As. vectors u into decoded source vectors s = Bu. 2D Transform: Rotation by ϕ = 45 A = Transform Coding
|
|
- Rosanna Black
- 5 years ago
- Views:
Transcription
1 Transfrm Cding Transfrm Cding Anther cncept fr partially expliting the memry gain f vectr quantizatin Used in virtually all lssy image and vide cding applicatins Samples f surce s are gruped int vectrs s f adjacent samples Transfrm cding cnsists f the fllwing steps 1 Linear analysis transfrm A, cnverting surce vectrs s int transfrm cefficient vectrs u = As Scalar quantizatin f the transfrm cefficients u u 3 Linear synthesis transfrm B, cnverting quantized transfrm cefficient vectrs u int decded surce vectrs s = Bu Adjacent Samples 4 S 1 A = U D Transfrm: Rtatin by ϕ = 45 [ ] sin ϕ cs ϕ cs ϕ sin ϕ = Transfrm Cefficients S U 4 January 3, 13 1 / 56
2 Transfrm Cding Overview Structure f Transfrm Cding Systems Mtivatin f Transfrm Cding Orthgnal Blck Transfrms Bit Allcatin fr Transfrm Cefficients Karhunen Léve Transfrm (KLT) Signal-Independent Transfrms Walsh-Hadamard Transfrm Discrete Furier Transfrm (DFT) Discrete Csine Transfrm (DCT) -d Transfrms in Image and Vide Cding Entrpy cding f transfrm cefficients Distributin f transfrm cefficients fr Images Mdified Discrete Csine Transfrm (MDCT) Summary January 3, 13 / 56
3 Transfrm Cding Structure f Transfrm Cding Systems s u u 1 Q Q 1 u u 1 A B s u N 1 Q N 1 u N 1 analysis transfrm quantizers synthesis transfrm Synthesis transfrm is typically inverse f analysis transfrm Separate scalar quantizer Q n fr each transfrm cefficient u n Vectr quantizatin f all bands r sme f them is als pssible, but Transfrms are designed t have a decrrelating effect = Memry gain f VQ is reduced Shape gain can be btained by ECSQ Space-filling gain is left as a pssible additinal gain fr VQ Cmbinatin f decrrelating transfrmatin, scalar quantizatin and entrpy cding is highly efficient - in terms f rate-distrtin perfrmance and cmplexity January 3, 13 3 / 56
4 Transfrm Cding Mtivatin f Transfrm Cding Explitatin f statistical dependencies Transfrm are typically designed in a way that, fr typical input signals, the signal energy is cncentrated in a few transfrm cefficients Cding f a few cefficients and many zer-valued cefficients can be very efficient (e.g., using arithmetic cding, run-length cding) Scalar quantizatin is mre effective in transfrm dmain Efficient trade-ff between cding efficiency & cmplexity Vectr Quantizatin: searching thrugh cdebk fr best matching vectr Cmbinatin f transfrm and scalar quantizatin typically results in a substantial reductin in cmputatinal cmplexity Suitable fr quantizatin using perceptual criteria In image & vide cding, quantizatin in transfrm dmain typically leads t an imprvement in subjective quality In speech & audi cding, frequency bands might be used t simulate prcessing f human ear Reduce perceptually irrelevant cntent January 3, 13 4 / 56
5 Transfrm Cding Transfrm Encder and Decder encder u α i s A u 1 α 1 i 1 γ b u N 1 α N 1 i N 1 analysis transfrm encder mapping entrpy cder decder b i i 1 β β 1 u u 1 γ 1 B s i N 1 β N 1 u N 1 entrpy decder decder mapping synthesis transfrm January 3, 13 5 / 56
6 Transfrm Cding Linear Blck Transfrms Linear Blck Transfrm Each cmpnent f the N-dimensinal utput vectr represents a linear cmbinatin f the N cmpnents f the N-dimensinal input vectr Can be written as matrix multiplicatin Analysis transfrm Synthesis transfrm u = A s (1) s = B u () Vectr interpretatin: s is represented as a linear cmbinatin f clumn vectrs f B s = N 1 n= u n b n = u b + u 1 b u N 1 b N 1 (3) January 3, 13 6 / 56
7 Transfrm Cding Linear Blck Transfrms (cnt d) Perfect Recnstructin Prperty Cnsider case that n quantizatin is applied (u = u) Optimal synthesis transfrm: B = A 1 (4) Recnstructed samples are equal t surce samples s = B u = B A s = A 1 A s = s (5) Optimal Synthesis Transfrm (in presence f quantizatin) Optimality: Minimum MSE distrtin amng all synthesis transfrms B = A 1 is ptimal if A is invertible and prduces independent transfrm cefficients the cmpnent quantizers are centridal quantizers If abve cnditins are nt fulfilled, a synthesis transfrm B A 1 may reduce the distrtin January 3, 13 7 / 56
8 Transfrm Cding Orthgnal Blck Transfrms Orthnrmal Basis An analysis transfrm A frms an rthnrmal basis if basis vectrs (matrix rws) are rthgnal t each ther basis vectrs have t length 1 The crrespnding transfrm is called an rthgnal transfrm The transfrm matrices are called unitary matrices Unitary matrices with real entries are called rthgnal matrix Inverse f unitary matrices: Cnjugate transpse A 1 = A (fr rthgnal matrices: A 1 = A T ) (6) Why are rthgnal transfrms desirable? MSE distrtin can be minimized by independent scalar quantizatin f the transfrm cefficients Orthgnality f the basis vectrs sufficient: Vectr nrms can be taken int accunt in quantizer design January 3, 13 8 / 56
9 Transfrm Cding Prperties f Orthgnal Blck Transfrms Transfrm cding with rthgnal transfrm and perfect recnstructin B = A 1 = A preserves MSE distrtin d N (s, s ) = 1 N (s s ) (s s ) = 1 N ( A 1 u B u ) ( A 1 u B u ) = 1 N ( A u A u ) ( A u A u ) = 1 N (u u ) A A 1 (u u ) = 1 N (u u ) (u u ) = d N (u, u ) (7) Scalar quantizatin that minimizes MSE in transfrm dmain als minimizes MSE in riginal signal space Fr the special case f rthgnal matrices: ( ) = ( ) T January 3, 13 9 / 56
10 Transfrm Cding Prperties f Orthgnal Blck Transfrms (cnt d) Cvariance matrix f transfrm cefficients C UU = E { (U E {U})(U E {U}) T } = E {A } (S E {S})(S E {S}) T A T = A C SS A 1 (8) Since the trace f a matrix is similarity-invariant, tr(x) = tr(p X P 1 ), (9) and the trace f an autcvariance matrix is the sum f the variances f the vectr cmpnents, we have N 1 1 σi = σ N S. (1) i= The arithmetic mean f the variances f the transfrm cefficients is equal t the variances f the surce January 3, 13 1 / 56
11 Transfrm Cding Gemetrical Interpretatin f Orthgnal Transfrms Inverse -d transfrm matrix (= transpse f frward transfrm matrix) B = [ [ ] ] b b 1 = = A T 1 1 Vectr interpretatin fr -d example s = u b + u 1 b ] [ ] 1 [ = u s + u [ ] [ ] [ ] = [ s yielding transfrm cefficients u = 3.5 u 1 =.5 An rthgnal transfrm is a rtatin frm the signal crdinate system int the crdinate system f the basis functins 1 1 ] b s 1 u b u 1 b 1 b 1 s s January 3, / 56
12 Transfrm Cding Transfrm Example N = Adjacent samples f Gauss-Markv surce with different crrelatin factrs ρ ρ= 4 S1 S S1 S ρ = ρ = S1 S ρ =.95 S S 4 Transfrm cefficients fr rthnrmal D transfrm ρ= 4 4 U1 4 U1 4 4 ρ = U U1 U ρ =.95 U ρ =.9 U U January 3, / 56
13 Transfrm Cding Example fr Wavefrms (Gauss-Markv Surce with ρ =.95) Tp: signal s[k] Middle: transfrm cefficient u [k/] als called dc cefficient Bttm: transfrm cefficient u 1 [k/] als called ac cefficient Number f transfrm cefficients u is half the number f samples s Number f transfrm cefficients u 1 is half the number f samples s 4 s[k] k u [k/] k/ u 1 [k/] k/ January 3, / 56
14 Transfrm Cding Scalar Quantizatin in Transfrm Dmain Cnsider Transfrm Cding with Orthgnal Transfrms direct cding transfrm cding transfrm cding quantizatin cells quantizatin cells quantizatin cells in transfrm dmain in signal space Quantizatin cells are hyper-rectangles as in scalar quantizatin but rtated and aligned with the transfrm basis vectrs Number f quantizatin cells with appreciable prbabilities is reduced = indicates imprved cding efficiency fr crrelated surces January 3, / 56
15 Transfrm Cding Bit Allcatin fr Transfrm Cefficients Prblem: Distribute bit rate R amng the N transfrm cefficients such that the resulting distrtin D is minimized min D(R) = 1 N N D i (R i ) i=1 subject t 1 N N R i R (11) with D i (R i ) being the p. distrtin-rate functins f the scalar quantizers Apprach: Minimize Lagrangian cst functin: J = D + λr ( N ) N D i (R i ) + λ R i = D i(r i ) + λ =! (1) R i R i i=1 Slutin: Paret cnditin i=1 i=1 D i (R i ) R i = λ = cnst (13) Mve bits frm cefficients with small distrtin reductin per bit t cefficients with larger distrtin reductin per bit Similar t bit allcatin prblem in discrete sets: min D i + λr i January 3, / 56
16 Transfrm Cding Bit Allcatin fr Transfrm Cefficients (cnt d) Operatinal distrtin-rate functin f scalar quantizers can be written as D i (R i ) = σ i g i (R i ) (14) Justified t assume that g i(r i) is a cntinuus strictly cnvex functin g i(r i) has a cntinuus strictly increasing derivative g i(r i) with g i( ) = Paret cnditin becmes σ i g i(r i ) = λ (15) If λ σi g i (), the quantizer fr u i cannt be perated at the given slpe = Set the crrespnding cmpnent rate t R i = Bit allcatin rule { : σ ( ) i g i R i = () λ η i λ : σ σi i g i () > λ (16) where η i ( ) dentes the inverse f the derivative g i ( ) January 3, / 56
17 Transfrm Cding Apprximatin fr Gaussian Surces Transfrm cefficients have als a Gaussian distributin Experimentally fund apprximatin fr entrpy-cnstrained scalar quantizatin fr Gaussian surces (a.95) Use parameter Bit allcatin rule θ = λ g(r) = πe 6a ln(a R + 1) (17) 3 (a + 1) πe ln with θ σ max (18) R i (θ) = Resulting cmpnent distrtins { : θ σ ( ) i 1 lg σ i θ (a + 1) a : θ < σi. (19) { σ i ( ) : θ σi D i (θ) = ε ln a σi lg 1 θ a σi a+1 : θ < σi. () January 3, / 56
18 Transfrm Cding High-Rate Apprximatin Assumptin: High-rate apprximatin valid fr all cmpnent quantizers High-rate apprximatin fr distrtin-rate functin f cmpnent quantizers D i (R i ) = ε i σ i Ri (1) where ε i depends n transfrm cefficient distributin and quantizer Paret cnditin R i D i (R i ) = ln ε i σ i Ri = ln D i (R i ) = λ = cnst () states that all quantizers are perated at the same distrtin Bit allcatin rule R i (D) = 1 ( ε lg i σ ) i D Overall peratinal rate-distrin functin R(D) = 1 N N 1 i= R i (D) = 1 N 1 N i= lg ( σ i ε i D ) (3) (4) January 3, / 56
19 Transfrm Cding High-Rate Apprximatin (cnt d) Overall peratinal rate-distrin functin with gemetric means R(D) = 1 N 1 ( σ lg i ε i N D i= ) = 1 ( lg ε σ ) D (5) σ = ( N 1 i= σ i ) 1 N Overall distrtin-rate functin and ε = ( N 1 i= ε i ) 1 N (6) D(R) = ε σ R (7) Fr Gaussian surces (transfrm cefficients are als Gaussian) and entrpy-cnstrained scalar quantizers, we have ε i = ε = πe 6, yielding D G (R) = πe 6 σ R (8) January 3, / 56
20 Transfrm Cding Transfrm Cding Gain at High Rates Transfrm cding gain is the rati f the distrtin fr scalar quantizatin and the distrtin fr transfrm cding with G T = ε S σ S R ε σ R σ S : variance f the input signal = ε S σ S ε σ (9) ε S : factr f high-rate apprximatin fr direct scalar quantizatin High-rate transfrm cding gain fr Gaussian surces G T = σ 1 S σ = N N 1 i= σ i N N 1 i= σ i (3) Rati f arithmetic and gemetric mean f the transfrm cefficient variances The high-rate transfrm cding gain fr Gaussian surces is maximized if the gemetric mean is minimized (= Karhunen Lève Transfrm) January 3, 13 / 56
21 Transfrm Cding Example: Orthgnal Transfrmatin with N = Input vectr and transfrm matrix [ ] s s = and A = 1 [ ] Transfrmatin u = [ u s 1 u 1 ] = A s = 1 [ ] [ s s 1 ] (31) (3) Cefficients u = 1 (s + s 1 ), u = 1 (s s 1 ) (33) Inverse transfrmatin A 1 = A T = A = 1 [ ] (34) January 3, 13 1 / 56
22 Transfrm Cding Example: Orthgnal Transfrmatin with N = (cnt d) Variance f transfrm cefficients σ = E { { } U } 1 = E (S + S 1 ) = 1 ( { } { } E S + E S 1 + E {S S 1 } ) = 1 ( σ S + σs + σsρ ) = σs(1 + ρ) (35) σ1 = E { U1 } = σ s (1 ρ) (36) Crss-crrelatin f transfrm cefficients E{U U 1 } = 1 E{ (S + S 1 ) (S S 1 ) } = 1 E {( S S1)} = σ s σs = (37) Transfrm cding gain fr Gaussian (assuming ptimal bit allcatin) G T = σ S σ + σ 1 = 1 1 ρ (38) January 3, 13 / 56
23 Transfrm Cding Example: Analysis f Transfrm Cding fr N = R-d cst befre transfrm J () = (D + λr) (fr samples) R-r cst after transfrm J (1) = (D + D 1 ) + λ(r + R 1 ) (fr bth transfrm cefficients) Gain in r-d cst due t transfrm at same rate (R + R 1 = R) J = J () J (1) = D D D 1 (39) Fr Gaussian surces, input and utput f transfrm have Gaussian pdf With peratinal distrtin-rate functin fr an entrpy-cnstrained scalar quantizer at high rates (D = ε σ R with ε = πe/6), we have J = ε σ S( R+1 (1 + ρ) R (1 ρ) R1) (4) By eliminating R 1 using R 1 = R R, we get J = ε σ S( R+1 (1 + ρ) R (1 ρ) (R R)) (41) January 3, 13 3 / 56
24 Transfrm Cding Example: Analysis f Transfrm Cding fr N = (cnt d) Gain in rate-distrtin due t transfrm J = ε σ S( R+1 (1 + ρ) R (1 ρ) (R R)) (4) T maximize gain, we set R J = ln (1 + ρ) R ln (1 ρ) 4R+R! = (43) yielding the bit allcatin rule R = R + 1 lg 1 + ρ 1 ρ (44) Same expressin is btained by using the previusly derived high rate bit allcatin rule R i = 1 ( ε lg σ ) i (45) D Operatinal high-rate distrtin-rate functin (Gaussian, ECSQ, N = ) D(R) = πe 6 1 ρ σ S R (46) January 3, 13 4 / 56
25 Transfrm Cding Example: Experimental R-D curves fr N = TC1 Theretical.5 TC1 experimental (ECSQ) Distrtin >.4.3 Optimal bit allcatin using Paret cnditin. Clud f RD pints by cmbinatin f RD values f TC1 thrugh TC Bit rate > January 3, 13 5 / 56
26 Transfrm Cding Example: Experimental R-D curves fr N = TC1 theretical 1 Distrtin > Optimal bit allcatin using Paret cnditin TC1 experimental (ECSQ) Clud f RD pints by cmbinatin f RD values f TC1 thrugh TC Bit rate > January 3, 13 6 / 56
27 Transfrm Cding General Bit Allcatin fr Transfrm Cefficients Fr Gaussian surces, the fllwing pints need t be cnsidered: High-rate apprximatins are nt valid fr lw bit rates; better apprximatins shuld be used fr lw rates Fr lw rates, Paret cnditins cannt be fulfilled fr all transfrm cefficients, since the cmpnent rates R i must nt be less then Slutin: Use generalized apprximatin f D i(r i) fr cmpnents quantizers Set cmpnents rates R i t zer fr all transfrm cefficients, fr which the Paret cnditin R i D(R i) = λ cannt be fullfilled fr R i Distribute rate amng remaining cefficients Fr nn-gaussian surces, the fllwing needs t be cnsidered in additin The transfrm cefficients have different (nn-gaussian) distributins (except fr large transfrm sizes) Using the same quantizer design fr all transfrm cefficients with D i (R i ) = σ i g(r i) is subptimal January 3, 13 7 / 56
28 Transfrm Cding Linear Transfrm Examples b WHT DFT DCT KLT WHT: Walsh Hadamard Transfrm b 1 b DFT: Discrete Furier Transfrm b 3 b 4 b 5 DCT: Discrete Csine Transfrm b 6 b 7 KLT: Karhunen Lève Transfrm January 3, 13 8 / 56
29 Transfrm Cding Karhunen Lève Transfrm (KLT) Karhunen Lève Transfrm Orthgnal transfrm that decrrelates the input vectrs Transfrm matrix depends n the surce Autcrrelatin matrix f input vectrs s { } R SS = E SS T (47) Autcrrelatin matrix f transfrm cefficient vectrs u R UU = E{UU T } = E{(AS)(AS) T } = AE{SS T }A T = A R SS A T (48) By multiplying with A 1 = A T frm the frnt, we get R SS A T = A T R UU (49) T get uncrrelated transfrm cefficients, we need t btain a diagnal autcrrelatin matrix R UU fr the transfrm cefficients January 3, 13 9 / 56
30 Transfrm Cding Karhunen Lève Transfrm (cnt d) Expressin fr autcrrelatin matrices R SS A T = A T R UU (5) R UU is a diagnal matrix if the eigenvectr equatin R SS b i = ξ i b i (51) is fulfilled fr all basis vectrs b i (clumn vectrs f A T, rw vectrs f A) The transfrm matrix A decrrelates the input vectrs if its rws are equal t the unit-nrm eigenvectrs v i f R SS [ ] T A KLT = v v 1 v N 1 (5) The resulting autcrrelatin matrix R UU is a diagnal matrix with the eigenvalues f R SS n its main diagnal ξ ξ 1 R UU =..... (53). ξ N 1 January 3, 13 3 / 56
31 Transfrm Cding Optimality f KLT fr Gaussian Surces Transfrm cding with rthgnal N N transfrm matrix A and B = A T Scalar quantizatin using scaled quantizers D(R, A k ) = N 1 i= σ i (A k ) g(r i ) (54) with σ i (A k) being variance f ith transfrm cefficient and A k being the transfrm matrix Cnsider an arbitrary rthgnal transfrm matrix A and an arbitrary bit allcatin given by the vectr r = [R, R N 1 ] T with N 1 i= R i = R Starting with arbitrary rthgnal matrix A, apply iterative algrithm that generates a series f rthnrmal transfrm matrices {A k }, k = 1,,... Iteratin A k+1 = J k A k cnsists f Jacbi rtatin and re-rdering = Transfrm matrix appraches a KLT matrix Can shw that fr all A k : D(R, A k+1 ) D(R, AA k ) = KLT is ptimal transfrm fr Gaussian surces (minimizes MSE) January 3, / 56
32 Transfrm Cding Asympttic Rate Distrtin Efficiency fr KLT f Gaussian Surces at High Rates Transfrm cefficient variances σi are equal t the eigenvalues ξ i f R SS High-rate apprximatin fr Gaussian surce and ptimal ECSQ D(R) = πe 6 σ R = πe 6 ξ R = πe 6 1 N 1 N i= lg ξi R (55) Fr N, we can apply Szegös therem fr infinite Teplitz matrices: If all eigenvalues ξ i f an infinite autcrrelatin matrix (N ) are finite and G(ξ i ) is any cntinuus functin ver all eigenvalues, lim N N 1 1 N i= G(ξ i ) = 1 π G(Φ(ω))dω (56) π π Resulting distrtin-rate functin fr KLT f infinite size fr high rates DKLT (R) = πe 6 1 π π π lg Φss(ω) dω R (57) January 3, 13 3 / 56
33 Transfrm Cding Asympttic Rate Distrtin Efficiency fr KLT f Gaussian Surces at High Rates (cnt d) Asympttic distrtin-rate functin fr KLT f infinite size fr high rates DKLT (R) = πe 6 1 π π π lg Φss(ω) dω R (58) Infrmatin distrtin-rate functin (fundamental bund) is by a factr ε = πe/6 smaller D(R) = 1 π π π lg Φss(ω) dω R (59) Asympttic transfrm gain (N ) at high rates G T = ε σs R 1 π DKLT (R) = π Φ SS(ω)dω π 1 π π π lg Φ SS(ω)dω (6) Asympttic transfrm gain (N ) at high rates is identical t the asympttic predictin gain at high rates January 3, / 56
34 Transfrm Cding KLT Transfrm Size Gain fr Gauss-Markv at High Rates Operatinal distrtin-rate functin fr KLT f size N, ECSQ, and ptimum bit allcatin fr Gauss-Markv surces with crrelatin factr ρ D N (R) = πe 6 σ s (1 ρ ) 1 1/N R (61) 1 lg 1 D N (R) D 1(R) [db] G T = 7.1 db ρ =.9 transfrm size N January 3, / 56
35 Transfrm Cding Distrtin-Rate Functins fr Gauss-Markv Distrtin-rate curves fr cding a first-rder Gauss-Markv surce with crrelatin factr ρ =.9 and different transfrm sizes N SNR [db] ECSQ+KLT, N N = 16 space-filling gain: 1.53 db N = 8 N = 4 distrtin-rate N = functin D(R) G T = 7.1 db 5 EC-Llyd (n transfrm) bit rate [bit/sample] January 3, / 56
36 Transfrm Cding KLTs fr Gauss-Markv Surces ρ =.1 ρ =.5 ρ =.9 ρ =.95 b b b b b b b b January 3, / 56
37 Transfrm Cding Walsh-Hadamard Transfrm Very simple rthgnal transfrm (nly additins & final scaling) Fr transfrm sizes N that are psitive integer pwer f A N = 1 [ ] AN/ A N/ with A A N/ A 1 = [1]. (6) N/ Transfrm matrix fr N = A 8 = Piecewise-cnstant basis vectrs Image & vide cding: Prduces subjectively disturbing artifacts when cmbined with strng quantizatin (63) January 3, / 56
38 Transfrm Cding Discrete Furier Transfrm (DFT) Discrete versin f the Furier transfrm Frward Transfrm Inverse Transfrm u[k] = 1 N 1 πkn j s[n]e N (64) N n= s[n] = 1 N 1 u[k]e j πkn N (65) N DFT is an rthnrmal transfrm (specified by a unitary transfrm matrix) k= Prduces cmplex transfrm cefficients Fr real inputs, it beys the symmetry u[k] = u [N k], s that N real samples are mapped nt N real values FFT is a fast algrithm fr DFT cmputatin, uses sparse matrix factrizatin Implies peridic signal extensin: Differences between left and right signal bundary reduces rate f cnvergence f Furier series Strng quantizatin = significant high-frequent artifacts January 3, / 56
39 Transfrm Cding Discrete Furier Transfrm vs. Discrete Csine Transfrm (a) Input time-dmain signal (b) Time-dmain replica in case f DFT (c) Time-dmain replica in case f DCT-II January 3, / 56
40 Transfrm Cding Derivatin f DCT High-frequent DFT quantizatin errr cmpnents can be reduced by intrducing implicit symmetry at the bundaries f the input signal and applying a DFT f apprximately duble length Signal with mirrr symmetry { s s[n 1/] : n < N [n] = s[n n 3/] : N n < N Transfrm cefficients (rthnrmal: divide u [] by ) u [k] = = = = N 1 1 s j πkn [i]e N N i= N 1 1 s[n 1/] (e j N π kn + e j π k(n n 1)) N N n= N 1 1 s[n] (e j N π k(n+ 1 ) + e j N π k(n+ 1 ) ) N N n= N 1 n= ( ( π s[n] cs N k n + 1 )) (66) (67) January 3, 13 4 / 56
41 Transfrm Cding Discrete Csine Transfrm (DCT) Implicit peridicity f DFT leads t lss f cding efficiency This can be reduced by intrducing mirrr symmetry at the bundaries and applying a DFT f apprximately duble size Due t mirrr symmetry, imaginary sine terms get eliminated and nly csine terms remain Mst cmmn DCT is the s-called DCT-II (mirrr symmetry with sample repetitins at bth sides: n = 1 ) DCT and IDCT Type-II are given by N 1 u[k] = α k s[n] = n= N 1 k= [ ( s[n] cs k n + 1 ) π ] N α k u[k] cs [ ( k n + 1 ) π ] N (68) (69) fr n 1 where α = N and α n = N January 3, / 56
42 Transfrm Cding DCT vs. KLT Crrelatin matrix f a first-rder Markv prcesses can be written as 1 ρ ρ ρ N 1 R SS = σs ρ 1 ρ ρ N..... ρ N 1 ρ N ρ N 3 1 (7) DCT is a gd apprximatin f the eigenvectrs f R SS DCT basis vectrs apprach the basis functins f the KLT fr first-rder Markv prcesses fr ρ 1 DCT des nt depend n input signal Fast algrithms fr cmputing frward and inverse transfrm Justificatin fr wide usage f DCT (r integer apprximatins theref) in image and vide cding: JPEG, H.61, H.6/MPEG-4, H.63, MPEG-4, H.64/AVC, H.65/HEVC January 3, 13 4 / 56
43 Transfrm Cding KLT Cnvergence Twards DCT fr ρ 1 b KLT, ρ = DCT-II Difference between the transfrm matrices f KLT and DCT-II b b.5.5 δ(ρ) = A KLT (ρ) A DCT b δ(ρ).5.5 b b b b ρ January 3, / 56
44 Transfrm Cding Bit Allcatin fr Audi Signals Bit Allcatin based n human respnse t sund signals - Psychacustic Mdels (PM) After transfrm, set f frequencies are gruped tgether t frm bands Fr each band, PM gives maximum allwed quantizatin nise s that the distrtin cannt be heard, i.e. nise is masked The gals f the encder are Enfrcing the bit rate specified by the user Implementing the PM threshld Adding nise in less ffensive places when there are nt enugh bits As can be seen, transfrm prvides a means fr eliminating nt nly redundancy but als irrelevancy January 3, / 56
45 Transfrm Cding Transfrm Type Cding efficiency fr a speech signal [Zelinski and Nll, 1977] January 3, / 56
46 Transfrm Cding Tw-dimensinal Transfrms -D linear transfrm: input image is represented as a linear cmbinatin f basis images An rthnrmal transfrm is separable and symmetric, if the transfrm f a signal blck s f size N N can be expressed as, u = A s A T (71) where A is the transfrmatin matrix and u is the matrix f transfrm cefficients, bth f size N N. The inverse transfrm is s = A T s A (7) Great practical imprtance: transfrm requires matrix multiplicatins f size N N instead ne multiplicatin f a vectr f size 1 N with a matrix f size N N Reductin f the cmplexity frm O(N 4 ) t O(N 3 ) January 3, / 56
47 Transfrm Cding DCT Example Image blck DCT clumn-wise 1-d DCT is applied clumn-wise n image blck t btain DCT clumn-wise result Ntice the energy cncentratin in the first rw (DC cefficients) January 3, / 56
48 Transfrm Cding DCT-Example (cntd.) DCT clumn-wise DCT rw-wise Fr cnvenience, DCT clumn-wise f previus slide is repeated n left side 1-d DCT is applied rw-wise n DCT clumn-wise result t btain final result Ntice the energy cncentratin in the first cefficient January 3, / 56
49 Transfrm Cding Entrpy cding f transfrm cefficients AC cefficients are very likely equal t zer (fr mderate quantizatin) Fr -d, rdering f the transfrm cefficients by zig-zag (r similar) scan Example fr zig-zag scanning in case f a -d transfrm Huffman cde fr events {number f leading zers, cefficient value} r events {end-f-blck, number f leading zers, cefficient value} Arithmetic cding: Fr example, use prbabilities that particular cefficient is unequal t zer when quantizing with a particular step size January 3, / 56
50 Transfrm Cding Analysis f DCT Cefficient Distributins fr Images Let s x,y dente pixel intensity in a blck. The s x,y s are assumed t be identically distributed, but nt necessarily Gaussian DCT is a weighted summatin f s x,y s By the central limit therem, the weighted summatin f identically distributed randm variables can be well apprximated as having a Gaussian distributin Therefre, DCT cefficients f this blck shuld be apprximately distributed as Gaussian f(u σ ) 1 e u σ (73) πσ In typical images, variance f the blcks has apprximately an expnential distributin f(σ ) λe λσ (74) Can shw that pdf f each transfrm cefficient then has apprximately a Laplacian distributin λ f(u) λ u e (75) January 3, 13 5 / 56
51 Transfrm Cding Example Histgrams f DCT Cefficients Picture Lena Histgram f transfrm ceffcients fr picture Lena January 3, / 56
52 Transfrm Cding Distributin f Variances f(σ ) λe λσ σ January 3, 13 5 / 56
53 Transfrm Cding Calculatin f Apprximate Distributin f DCT Cefficients Distributin f DCT cefficients can be written as f(u) = With the cnditinal pdf assumptin f(u σ ) f(σ ) dσ (76) f(u σ ) = 1 πσ e u σ (77) we get a Laplacian distributin fr the transfrm cefficients f(u) = = = π λ π λ1 1 πσ e u σ λe λσ dσ e u σ λσ dσ π λ e λu = Nte (frm integral table): e ax bx dx = 1 π a e ab λ λ u e (78) January 3, / 56
54 Transfrm Cding Mdified Discrete Csine Transfrm (MDCT) MDCT intrduced by [Princen, Jhnsn, and Bradley 1987] t avid blck bundaries artifacts MDCT is a lapped transfrm that maps N dimensinal data t N dimensinal data F : R N R N Frward transfrm u k = Inverse transfrm N 1 n= [ ( π s n cs n + 1 N + N ) ( k + 1 )] (79) s n = 1 N N 1 k= [ ( π u k cs n + 1 N + N ) ( k + 1 )] (8) Perfect recnstructin is achieved by adding the IMDCTs f subsequent verlapping blcks Used extensively in audi cding - MP3, AAC, Dlby AC3, etc. January 3, / 56
55 Transfrm Cding MDCT Blck Diagram Frame k Frame k+1 Frame k+ Frame k+3 N N N N N MDCT N N MDCT N N MDCT N N IMDCT N N IMDCT N N IMDCT N N N Frame k+1 Frame k+ January 3, / 56
56 Transfrm Cding Summary n Transfrm Cding Orthnrmal transfrm: rtatin f crdinate system in signal space Purpse f transfrm: decrrelatin, energy cncentratin = Align quantizatin cells with primary axis f jint pdf KLT achieves ptimum decrrelatin, but signal dependent and, hence, withut a fast algrithm DCT shws reduced blcking artifacts cmpared t DFT Fr Gauss-Markv and ρ : DCT appraches KLT Fr Gaussian surces: Bit allcatin prprtinal t lgarithm f variance similar t bit allcatin in discrete sets (D + λr) Fr high rates: Optimum bit allcatin yields equal cmpnent distrtin Larger transfrm size increases gain fr Gauss-Markv surce Fr picture cding: decrrelating transfrm + entrpy-cnstrained quantizatin + zig-zag scan + entrpy cding is widely used tday (e.g. JPEG, MPEG-1//4, ITU-T H.61//3/4) Fr pictures: pdf f transfrm cefficients is Laplacian because f expnential distributin f blck variances Fr audi cding: MDCT is widely used January 3, / 56
Predictive Coding. U n " S n
Intrductin Predictive Cding The better the future f a randm prcess is predicted frm the past and the mre redundancy the signal cntains, the less new infrmatin is cntributed by each successive bservatin
More informationSource Coding and Compression
Surce Cding and Cmpressin Heik Schwarz Cntact: Dr.-Ing. Heik Schwarz heik.schwarz@hhi.fraunhfer.de Heik Schwarz Surce Cding and Cmpressin September 22, 2013 1 / 60 PartI: Surce Cding Fundamentals Heik
More informationQuantization. Quantization is the realization of the lossy part of source coding Typically allows for a trade-off between signal fidelity and bit rate
Quantizatin Quantizatin is the realizatin f the lssy part f surce cding Typically allws fr a trade-ff between signal fidelity and bit rate s! s! Quantizer Quantizatin is a functinal mapping f a (cntinuus
More informationProbability, Random Variables, and Processes. Probability
Prbability, Randm Variables, and Prcesses Prbability Prbability Prbability thery: branch f mathematics fr descriptin and mdelling f randm events Mdern prbability thery - the aximatic definitin f prbability
More informationSource Coding Fundamentals
Surce Cding Fundamentals Surce Cding Fundamentals Thmas Wiegand Digital Image Cmmunicatin 1 / 54 Surce Cding Fundamentals Outline Intrductin Lssless Cding Huffman Cding Elias and Arithmetic Cding Rate-Distrtin
More informationVideo Encoder Control
Vide Encder Cntrl Thmas Wiegand Digital Image Cmmunicatin 1 / 41 Outline Intrductin Encder Cntrl using Lagrange multipliers Lagrangian ptimizatin Lagrangian bit allcatin Lagrangian Optimizatin in Hybrid
More informationinitially lcated away frm the data set never win the cmpetitin, resulting in a nnptimal nal cdebk, [2] [3] [4] and [5]. Khnen's Self Organizing Featur
Cdewrd Distributin fr Frequency Sensitive Cmpetitive Learning with One Dimensinal Input Data Aristides S. Galanpuls and Stanley C. Ahalt Department f Electrical Engineering The Ohi State University Abstract
More informationMATHEMATICS SYLLABUS SECONDARY 5th YEAR
Eurpean Schls Office f the Secretary-General Pedaggical Develpment Unit Ref. : 011-01-D-8-en- Orig. : EN MATHEMATICS SYLLABUS SECONDARY 5th YEAR 6 perid/week curse APPROVED BY THE JOINT TEACHING COMMITTEE
More informationBuilding to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve real-world and mathematical problems.
Building t Transfrmatins n Crdinate Axis Grade 5: Gemetry Graph pints n the crdinate plane t slve real-wrld and mathematical prblems. 5.G.1. Use a pair f perpendicular number lines, called axes, t define
More informationT Algorithmic methods for data mining. Slide set 6: dimensionality reduction
T-61.5060 Algrithmic methds fr data mining Slide set 6: dimensinality reductin reading assignment LRU bk: 11.1 11.3 PCA tutrial in mycurses (ptinal) ptinal: An Elementary Prf f a Therem f Jhnsn and Lindenstrauss,
More informationPattern Recognition 2014 Support Vector Machines
Pattern Recgnitin 2014 Supprt Vectr Machines Ad Feelders Universiteit Utrecht Ad Feelders ( Universiteit Utrecht ) Pattern Recgnitin 1 / 55 Overview 1 Separable Case 2 Kernel Functins 3 Allwing Errrs (Sft
More informationPhysical Layer: Outline
18-: Intrductin t Telecmmunicatin Netwrks Lectures : Physical Layer Peter Steenkiste Spring 01 www.cs.cmu.edu/~prs/nets-ece Physical Layer: Outline Digital Representatin f Infrmatin Characterizatin f Cmmunicatin
More informationChapter 9 Vector Differential Calculus, Grad, Div, Curl
Chapter 9 Vectr Differential Calculus, Grad, Div, Curl 9.1 Vectrs in 2-Space and 3-Space 9.2 Inner Prduct (Dt Prduct) 9.3 Vectr Prduct (Crss Prduct, Outer Prduct) 9.4 Vectr and Scalar Functins and Fields
More informationDetermining the Accuracy of Modal Parameter Estimation Methods
Determining the Accuracy f Mdal Parameter Estimatin Methds by Michael Lee Ph.D., P.E. & Mar Richardsn Ph.D. Structural Measurement Systems Milpitas, CA Abstract The mst cmmn type f mdal testing system
More informationChapter 3: Cluster Analysis
Chapter 3: Cluster Analysis } 3.1 Basic Cncepts f Clustering 3.1.1 Cluster Analysis 3.1. Clustering Categries } 3. Partitining Methds 3..1 The principle 3.. K-Means Methd 3..3 K-Medids Methd 3..4 CLARA
More information, which yields. where z1. and z2
The Gaussian r Nrmal PDF, Page 1 The Gaussian r Nrmal Prbability Density Functin Authr: Jhn M Cimbala, Penn State University Latest revisin: 11 September 13 The Gaussian r Nrmal Prbability Density Functin
More informationSections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic.
Tpic : AC Fundamentals, Sinusidal Wavefrm, and Phasrs Sectins 5. t 5., 6. and 6. f the textbk (Rbbins-Miller) cver the materials required fr this tpic.. Wavefrms in electrical systems are current r vltage
More information3.4 Shrinkage Methods Prostate Cancer Data Example (Continued) Ridge Regression
3.3.4 Prstate Cancer Data Example (Cntinued) 3.4 Shrinkage Methds 61 Table 3.3 shws the cefficients frm a number f different selectin and shrinkage methds. They are best-subset selectin using an all-subsets
More informationDistributions, spatial statistics and a Bayesian perspective
Distributins, spatial statistics and a Bayesian perspective Dug Nychka Natinal Center fr Atmspheric Research Distributins and densities Cnditinal distributins and Bayes Thm Bivariate nrmal Spatial statistics
More informationCOMP 551 Applied Machine Learning Lecture 11: Support Vector Machines
COMP 551 Applied Machine Learning Lecture 11: Supprt Vectr Machines Instructr: (jpineau@cs.mcgill.ca) Class web page: www.cs.mcgill.ca/~jpineau/cmp551 Unless therwise nted, all material psted fr this curse
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 4: Mdel checing fr ODE mdels In Petre Department f IT, Åb Aademi http://www.users.ab.fi/ipetre/cmpmd/ Cntent Stichimetric matrix Calculating the mass cnservatin relatins
More informationReview Problems 3. Four FIR Filter Types
Review Prblems 3 Fur FIR Filter Types Fur types f FIR linear phase digital filters have cefficients h(n fr 0 n M. They are defined as fllws: Type I: h(n = h(m-n and M even. Type II: h(n = h(m-n and M dd.
More informationECE 2100 Circuit Analysis
ECE 2100 Circuit Analysis Lessn 25 Chapter 9 & App B: Passive circuit elements in the phasr representatin Daniel M. Litynski, Ph.D. http://hmepages.wmich.edu/~dlitynsk/ ECE 2100 Circuit Analysis Lessn
More informationA Matrix Representation of Panel Data
web Extensin 6 Appendix 6.A A Matrix Representatin f Panel Data Panel data mdels cme in tw brad varieties, distinct intercept DGPs and errr cmpnent DGPs. his appendix presents matrix algebra representatins
More informationSource Coding and Compression
Surce Cding and Cmpressin Heik Schwarz Cntact: Dr.-Ing. Heik Schwarz heik.schwarz@hhi.fraunhfer.de Heik Schwarz Surce Cding and Cmpressin December 7, 2013 1 / 539 PartII: Applicatin in Image and Vide Cding
More informationPrincipal Components
Principal Cmpnents Suppse we have N measurements n each f p variables X j, j = 1,..., p. There are several equivalent appraches t principal cmpnents: Given X = (X 1,... X p ), prduce a derived (and small)
More informationLecture 10, Principal Component Analysis
Principal Cmpnent Analysis Lecture 10, Principal Cmpnent Analysis Ha Helen Zhang Fall 2017 Ha Helen Zhang Lecture 10, Principal Cmpnent Analysis 1 / 16 Principal Cmpnent Analysis Lecture 10, Principal
More informationLeast Squares Optimal Filtering with Multirate Observations
Prc. 36th Asilmar Cnf. n Signals, Systems, and Cmputers, Pacific Grve, CA, Nvember 2002 Least Squares Optimal Filtering with Multirate Observatins Charles W. herrien and Anthny H. Hawes Department f Electrical
More informationk-nearest Neighbor How to choose k Average of k points more reliable when: Large k: noise in attributes +o o noise in class labels
Mtivating Example Memry-Based Learning Instance-Based Learning K-earest eighbr Inductive Assumptin Similar inputs map t similar utputs If nt true => learning is impssible If true => learning reduces t
More informationMath 302 Learning Objectives
Multivariable Calculus (Part I) 13.1 Vectrs in Three-Dimensinal Space Math 302 Learning Objectives Plt pints in three-dimensinal space. Find the distance between tw pints in three-dimensinal space. Write
More informationChapter 3 Kinematics in Two Dimensions; Vectors
Chapter 3 Kinematics in Tw Dimensins; Vectrs Vectrs and Scalars Additin f Vectrs Graphical Methds (One and Tw- Dimensin) Multiplicatin f a Vectr b a Scalar Subtractin f Vectrs Graphical Methds Adding Vectrs
More informationSAMPLING DYNAMICAL SYSTEMS
SAMPLING DYNAMICAL SYSTEMS Melvin J. Hinich Applied Research Labratries The University f Texas at Austin Austin, TX 78713-8029, USA (512) 835-3278 (Vice) 835-3259 (Fax) hinich@mail.la.utexas.edu ABSTRACT
More informationCOMP 551 Applied Machine Learning Lecture 4: Linear classification
COMP 551 Applied Machine Learning Lecture 4: Linear classificatin Instructr: Jelle Pineau (jpineau@cs.mcgill.ca) Class web page: www.cs.mcgill.ca/~jpineau/cmp551 Unless therwise nted, all material psted
More informationENSC Discrete Time Systems. Project Outline. Semester
ENSC 49 - iscrete Time Systems Prject Outline Semester 006-1. Objectives The gal f the prject is t design a channel fading simulatr. Upn successful cmpletin f the prject, yu will reinfrce yur understanding
More informationIN a recent article, Geary [1972] discussed the merit of taking first differences
The Efficiency f Taking First Differences in Regressin Analysis: A Nte J. A. TILLMAN IN a recent article, Geary [1972] discussed the merit f taking first differences t deal with the prblems that trends
More informationLHS Mathematics Department Honors Pre-Calculus Final Exam 2002 Answers
LHS Mathematics Department Hnrs Pre-alculus Final Eam nswers Part Shrt Prblems The table at the right gives the ppulatin f Massachusetts ver the past several decades Using an epnential mdel, predict the
More informationRevision: August 19, E Main Suite D Pullman, WA (509) Voice and Fax
.7.4: Direct frequency dmain circuit analysis Revisin: August 9, 00 5 E Main Suite D Pullman, WA 9963 (509) 334 6306 ice and Fax Overview n chapter.7., we determined the steadystate respnse f electrical
More informationPure adaptive search for finite global optimization*
Mathematical Prgramming 69 (1995) 443-448 Pure adaptive search fr finite glbal ptimizatin* Z.B. Zabinskya.*, G.R. Wd b, M.A. Steel c, W.P. Baritmpa c a Industrial Engineering Prgram, FU-20. University
More informationInter-Picture Coding. Inter-Picture Coding. o Thomas Wiegand Digital Image Communication 1 / 62
Inter-Picture Cding Thmas Wiegand Digital Image Cmmunicatin 1 / 62 Outline Intrductin Accuracy f Mtin-Cmpensated Predictin Theretical Cnsideratins Chice f Interplatin Filters Mtin Vectr Accuracy Mtin Mdels
More informationSupport-Vector Machines
Supprt-Vectr Machines Intrductin Supprt vectr machine is a linear machine with sme very nice prperties. Haykin chapter 6. See Alpaydin chapter 13 fr similar cntent. Nte: Part f this lecture drew material
More informationLyapunov Stability Stability of Equilibrium Points
Lyapunv Stability Stability f Equilibrium Pints 1. Stability f Equilibrium Pints - Definitins In this sectin we cnsider n-th rder nnlinear time varying cntinuus time (C) systems f the frm x = f ( t, x),
More informationLecture 2: Supervised vs. unsupervised learning, bias-variance tradeoff
Lecture 2: Supervised vs. unsupervised learning, bias-variance tradeff Reading: Chapter 2 STATS 202: Data mining and analysis September 27, 2017 1 / 20 Supervised vs. unsupervised learning In unsupervised
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCurseWare http://cw.mit.edu 2.161 Signal Prcessing: Cntinuus and Discrete Fall 2008 Fr infrmatin abut citing these materials r ur Terms f Use, visit: http://cw.mit.edu/terms. Massachusetts Institute
More information7 TH GRADE MATH STANDARDS
ALGEBRA STANDARDS Gal 1: Students will use the language f algebra t explre, describe, represent, and analyze number expressins and relatins 7 TH GRADE MATH STANDARDS 7.M.1.1: (Cmprehensin) Select, use,
More information[COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t o m a k e s u r e y o u a r e r e a d y )
(Abut the final) [COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t m a k e s u r e y u a r e r e a d y ) The department writes the final exam s I dn't really knw what's n it and I can't very well
More informationLecture 2: Supervised vs. unsupervised learning, bias-variance tradeoff
Lecture 2: Supervised vs. unsupervised learning, bias-variance tradeff Reading: Chapter 2 STATS 202: Data mining and analysis September 27, 2017 1 / 20 Supervised vs. unsupervised learning In unsupervised
More informationCS 477/677 Analysis of Algorithms Fall 2007 Dr. George Bebis Course Project Due Date: 11/29/2007
CS 477/677 Analysis f Algrithms Fall 2007 Dr. Gerge Bebis Curse Prject Due Date: 11/29/2007 Part1: Cmparisn f Srting Algrithms (70% f the prject grade) The bjective f the first part f the assignment is
More informationEmphases in Common Core Standards for Mathematical Content Kindergarten High School
Emphases in Cmmn Cre Standards fr Mathematical Cntent Kindergarten High Schl Cntent Emphases by Cluster March 12, 2012 Describes cntent emphases in the standards at the cluster level fr each grade. These
More informationEDA Engineering Design & Analysis Ltd
EDA Engineering Design & Analysis Ltd THE FINITE ELEMENT METHOD A shrt tutrial giving an verview f the histry, thery and applicatin f the finite element methd. Intrductin Value f FEM Applicatins Elements
More informationCOMP 551 Applied Machine Learning Lecture 5: Generative models for linear classification
COMP 551 Applied Machine Learning Lecture 5: Generative mdels fr linear classificatin Instructr: Herke van Hf (herke.vanhf@mail.mcgill.ca) Slides mstly by: Jelle Pineau Class web page: www.cs.mcgill.ca/~hvanh2/cmp551
More informationCOMP 551 Applied Machine Learning Lecture 9: Support Vector Machines (cont d)
COMP 551 Applied Machine Learning Lecture 9: Supprt Vectr Machines (cnt d) Instructr: Herke van Hf (herke.vanhf@mail.mcgill.ca) Slides mstly by: Class web page: www.cs.mcgill.ca/~hvanh2/cmp551 Unless therwise
More informationFunction notation & composite functions Factoring Dividing polynomials Remainder theorem & factor property
Functin ntatin & cmpsite functins Factring Dividing plynmials Remainder therem & factr prperty Can d s by gruping r by: Always lk fr a cmmn factr first 2 numbers that ADD t give yu middle term and MULTIPLY
More informationDead-beat controller design
J. Hetthéssy, A. Barta, R. Bars: Dead beat cntrller design Nvember, 4 Dead-beat cntrller design In sampled data cntrl systems the cntrller is realised by an intelligent device, typically by a PLC (Prgrammable
More informationFebruary 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA
February 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA Mental Experiment regarding 1D randm walk Cnsider a cntainer f gas in thermal
More informationMultiple Source Multiple. using Network Coding
Multiple Surce Multiple Destinatin Tplgy Inference using Netwrk Cding Pegah Sattari EECS, UC Irvine Jint wrk with Athina Markpulu, at UCI, Christina Fraguli, at EPFL, Lausanne Outline Netwrk Tmgraphy Gal,
More informationPerformance Bounds for Detect and Avoid Signal Sensing
Perfrmance unds fr Detect and Avid Signal Sensing Sam Reisenfeld Real-ime Infrmatin etwrks, University f echnlgy, Sydney, radway, SW 007, Australia samr@uts.edu.au Abstract Detect and Avid (DAA) is a Cgnitive
More informationThe Kullback-Leibler Kernel as a Framework for Discriminant and Localized Representations for Visual Recognition
The Kullback-Leibler Kernel as a Framewrk fr Discriminant and Lcalized Representatins fr Visual Recgnitin Nun Vascncels Purdy H Pedr Mren ECE Department University f Califrnia, San Dieg HP Labs Cambridge
More informationand the Doppler frequency rate f R , can be related to the coefficients of this polynomial. The relationships are:
Algrithm fr Estimating R and R - (David Sandwell, SIO, August 4, 2006) Azimith cmpressin invlves the alignment f successive eches t be fcused n a pint target Let s be the slw time alng the satellite track
More informationResampling Methods. Chapter 5. Chapter 5 1 / 52
Resampling Methds Chapter 5 Chapter 5 1 / 52 1 51 Validatin set apprach 2 52 Crss validatin 3 53 Btstrap Chapter 5 2 / 52 Abut Resampling An imprtant statistical tl Pretending the data as ppulatin and
More information1 The limitations of Hartree Fock approximation
Chapter: Pst-Hartree Fck Methds - I The limitatins f Hartree Fck apprximatin The n electrn single determinant Hartree Fck wave functin is the variatinal best amng all pssible n electrn single determinants
More informationTree Structured Classifier
Tree Structured Classifier Reference: Classificatin and Regressin Trees by L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stne, Chapman & Hall, 98. A Medical Eample (CART): Predict high risk patients
More informationECEN620: Network Theory Broadband Circuit Design Fall 2012
ECEN60: Netwrk Thery Bradband Circuit Design Fall 01 Lecture 16: VCO Phase Nise Sam Palerm Analg & Mixed-Signal Center Texas A&M University Agenda Phase Nise Definitin and Impact Ideal Oscillatr Phase
More informationNUMBERS, MATHEMATICS AND EQUATIONS
AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t
More informationChapter 3 Digital Transmission Fundamentals
Chapter 3 Digital Transmissin Fundamentals Errr Detectin and Crrectin Errr Cntrl Digital transmissin systems intrduce errrs, BER ranges frm 10-3 fr wireless t 10-9 fr ptical fiber Applicatins require certain
More informationECE 2100 Circuit Analysis
ECE 00 Circuit Analysis Lessn 6 Chapter 4 Sec 4., 4.5, 4.7 Series LC Circuit C Lw Pass Filter Daniel M. Litynski, Ph.D. http://hmepages.wmich.edu/~dlitynsk/ ECE 00 Circuit Analysis Lessn 5 Chapter 9 &
More informationBootstrap Method > # Purpose: understand how bootstrap method works > obs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(obs) >
Btstrap Methd > # Purpse: understand hw btstrap methd wrks > bs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(bs) > mean(bs) [1] 21.64625 > # estimate f lambda > lambda = 1/mean(bs);
More information4th Indian Institute of Astrophysics - PennState Astrostatistics School July, 2013 Vainu Bappu Observatory, Kavalur. Correlation and Regression
4th Indian Institute f Astrphysics - PennState Astrstatistics Schl July, 2013 Vainu Bappu Observatry, Kavalur Crrelatin and Regressin Rahul Ry Indian Statistical Institute, Delhi. Crrelatin Cnsider a tw
More informationHomology groups of disks with holes
Hmlgy grups f disks with hles THEOREM. Let p 1,, p k } be a sequence f distinct pints in the interir unit disk D n where n 2, and suppse that fr all j the sets E j Int D n are clsed, pairwise disjint subdisks.
More informationKinematic transformation of mechanical behavior Neville Hogan
inematic transfrmatin f mechanical behavir Neville Hgan Generalized crdinates are fundamental If we assume that a linkage may accurately be described as a cllectin f linked rigid bdies, their generalized
More informationECEN620: Network Theory Broadband Circuit Design Fall 2014
ECEN60: Netwrk Thery Bradband Circuit Design Fall 014 Lecture 11: VCO Phase Nise Sam Palerm Analg & Mixed-Signal Center Texas A&M University Annuncements & Agenda HW3 is due tday at 5PM Phase Nise Definitin
More informationSlide04 (supplemental) Haykin Chapter 4 (both 2nd and 3rd ed): Multi-Layer Perceptrons
Slide04 supplemental) Haykin Chapter 4 bth 2nd and 3rd ed): Multi-Layer Perceptrns CPSC 636-600 Instructr: Ynsuck Che Heuristic fr Making Backprp Perfrm Better 1. Sequential vs. batch update: fr large
More information5 th grade Common Core Standards
5 th grade Cmmn Cre Standards In Grade 5, instructinal time shuld fcus n three critical areas: (1) develping fluency with additin and subtractin f fractins, and develping understanding f the multiplicatin
More informationComparing Several Means: ANOVA. Group Means and Grand Mean
STAT 511 ANOVA and Regressin 1 Cmparing Several Means: ANOVA Slide 1 Blue Lake snap beans were grwn in 12 pen-tp chambers which are subject t 4 treatments 3 each with O 3 and SO 2 present/absent. The ttal
More informationExperiment #3. Graphing with Excel
Experiment #3. Graphing with Excel Study the "Graphing with Excel" instructins that have been prvided. Additinal help with learning t use Excel can be fund n several web sites, including http://www.ncsu.edu/labwrite/res/gt/gt-
More information1996 Engineering Systems Design and Analysis Conference, Montpellier, France, July 1-4, 1996, Vol. 7, pp
THE POWER AND LIMIT OF NEURAL NETWORKS T. Y. Lin Department f Mathematics and Cmputer Science San Jse State University San Jse, Califrnia 959-003 tylin@cs.ssu.edu and Bereley Initiative in Sft Cmputing*
More informationA Correlation of. to the. South Carolina Academic Standards for Mathematics Precalculus
A Crrelatin f Suth Carlina Academic Standards fr Mathematics Precalculus INTRODUCTION This dcument demnstrates hw Precalculus (Blitzer), 4 th Editin 010, meets the indicatrs f the. Crrelatin page references
More informationa(k) received through m channels of length N and coefficients v(k) is an additive independent white Gaussian noise with
urst Mde Nn-Causal Decisin-Feedback Equalizer based n Sft Decisins Elisabeth de Carvalh and Dirk T.M. Slck Institut EURECOM, 2229 rute des Crêtes,.P. 93, 694 Sphia ntiplis Cedex, FRNCE Tel: +33 493263
More informationGreen Noise Digital Halftoning with Multiscale Error Diffusion
Green Nise Digital Halftning with Multiscale Errr Diffusin Yik-Hing Fung and Yuk-Hee Chan Center f Multimedia Signal Prcessing Department f Electrnic and Infrmatin Engineering The Hng Kng Plytechnic University,
More informationLecture 02 CSE 40547/60547 Computing at the Nanoscale
PN Junctin Ntes: Lecture 02 CSE 40547/60547 Cmputing at the Nanscale Letʼs start with a (very) shrt review f semi-cnducting materials: - N-type material: Obtained by adding impurity with 5 valence elements
More informationSequential Allocation with Minimal Switching
In Cmputing Science and Statistics 28 (1996), pp. 567 572 Sequential Allcatin with Minimal Switching Quentin F. Stut 1 Janis Hardwick 1 EECS Dept., University f Michigan Statistics Dept., Purdue University
More informationELT COMMUNICATION THEORY
ELT 41307 COMMUNICATION THEORY Matlab Exercise #2 Randm variables and randm prcesses 1 RANDOM VARIABLES 1.1 ROLLING A FAIR 6 FACED DICE (DISCRETE VALIABLE) Generate randm samples fr rlling a fair 6 faced
More informationCalculus Placement Review. x x. =. Find each of the following. 9 = 4 ( )
Calculus Placement Review I. Finding dmain, intercepts, and asympttes f ratinal functins 9 Eample Cnsider the functin f ( ). Find each f the fllwing. (a) What is the dmain f f ( )? Write yur answer in
More informationLEARNING : At the end of the lesson, students should be able to: OUTCOMES a) state trigonometric ratios of sin,cos, tan, cosec, sec and cot
Mathematics DM 05 Tpic : Trignmetric Functins LECTURE OF 5 TOPIC :.0 TRIGONOMETRIC FUNCTIONS SUBTOPIC :. Trignmetric Ratis and Identities LEARNING : At the end f the lessn, students shuld be able t: OUTCOMES
More informationSupplementary Course Notes Adding and Subtracting AC Voltages and Currents
Supplementary Curse Ntes Adding and Subtracting AC Vltages and Currents As mentined previusly, when cmbining DC vltages r currents, we nly need t knw the plarity (vltage) and directin (current). In the
More informationInterference is when two (or more) sets of waves meet and combine to produce a new pattern.
Interference Interference is when tw (r mre) sets f waves meet and cmbine t prduce a new pattern. This pattern can vary depending n the riginal wave directin, wavelength, amplitude, etc. The tw mst extreme
More informationLinear Classification
Linear Classificatin CS 54: Machine Learning Slides adapted frm Lee Cper, Jydeep Ghsh, and Sham Kakade Review: Linear Regressin CS 54 [Spring 07] - H Regressin Given an input vectr x T = (x, x,, xp), we
More informationPhysics 2B Chapter 23 Notes - Faraday s Law & Inductors Spring 2018
Michael Faraday lived in the Lndn area frm 1791 t 1867. He was 29 years ld when Hand Oersted, in 1820, accidentally discvered that electric current creates magnetic field. Thrugh empirical bservatin and
More informationPreparation work for A2 Mathematics [2018]
Preparatin wrk fr A Mathematics [018] The wrk studied in Y1 will frm the fundatins n which will build upn in Year 13. It will nly be reviewed during Year 13, it will nt be retaught. This is t allw time
More information22.54 Neutron Interactions and Applications (Spring 2004) Chapter 11 (3/11/04) Neutron Diffusion
.54 Neutrn Interactins and Applicatins (Spring 004) Chapter (3//04) Neutrn Diffusin References -- J. R. Lamarsh, Intrductin t Nuclear Reactr Thery (Addisn-Wesley, Reading, 966) T study neutrn diffusin
More informationLecture 7: Damped and Driven Oscillations
Lecture 7: Damped and Driven Oscillatins Last time, we fund fr underdamped scillatrs: βt x t = e A1 + A csω1t + i A1 A sinω1t A 1 and A are cmplex numbers, but ur answer must be real Implies that A 1 and
More informationMODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b
. REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but
More informationParticle Size Distributions from SANS Data Using the Maximum Entropy Method. By J. A. POTTON, G. J. DANIELL AND B. D. RAINFORD
3 J. Appl. Cryst. (1988). 21,3-8 Particle Size Distributins frm SANS Data Using the Maximum Entrpy Methd By J. A. PTTN, G. J. DANIELL AND B. D. RAINFRD Physics Department, The University, Suthamptn S9
More informationCHM112 Lab Graphing with Excel Grading Rubric
Name CHM112 Lab Graphing with Excel Grading Rubric Criteria Pints pssible Pints earned Graphs crrectly pltted and adhere t all guidelines (including descriptive title, prperly frmatted axes, trendline
More informationSection I5: Feedback in Operational Amplifiers
Sectin I5: eedback in Operatinal mplifiers s discussed earlier, practical p-amps hae a high gain under dc (zer frequency) cnditins and the gain decreases as frequency increases. This frequency dependence
More informationThe blessing of dimensionality for kernel methods
fr kernel methds Building classifiers in high dimensinal space Pierre Dupnt Pierre.Dupnt@ucluvain.be Classifiers define decisin surfaces in sme feature space where the data is either initially represented
More informationMath Foundations 10 Work Plan
Math Fundatins 10 Wrk Plan Units / Tpics 10.1 Demnstrate understanding f factrs f whle numbers by: Prime factrs Greatest Cmmn Factrs (GCF) Least Cmmn Multiple (LCM) Principal square rt Cube rt Time Frame
More informationCAPACITY OF MULTI-ANTENNA ARRAY SYSTEMS IN INDOOR WIRELESS ENVIRONMENT
CAPACITY OF MULTI-ANTENNA ARRAY SYSTEMS IN INDOOR WIRELESS ENVIRONMENT Chen-Nee Chuah, Jseph M. Kahn and David Tse -7 Cry Hall, University f Califrnia Berkeley, CA 947 e-mail: {chuah, jmk, dtse}@eecs.berkeley.edu
More informationDifferentiation Applications 1: Related Rates
Differentiatin Applicatins 1: Related Rates 151 Differentiatin Applicatins 1: Related Rates Mdel 1: Sliding Ladder 10 ladder y 10 ladder 10 ladder A 10 ft ladder is leaning against a wall when the bttm
More informationMethods for Determination of Mean Speckle Size in Simulated Speckle Pattern
0.478/msr-04-004 MEASUREMENT SCENCE REVEW, Vlume 4, N. 3, 04 Methds fr Determinatin f Mean Speckle Size in Simulated Speckle Pattern. Hamarvá, P. Šmíd, P. Hrváth, M. Hrabvský nstitute f Physics f the Academy
More informationModelling of NOLM Demultiplexers Employing Optical Soliton Control Pulse
Micwave and Optical Technlgy Letters, Vl. 1, N. 3, 1999. pp. 05-08 Mdelling f NOLM Demultiplexers Emplying Optical Slitn Cntrl Pulse Z. Ghassemly, C. Y. Cheung & A. K. Ray Electrnics Research Grup, Schl
More information