Contents. PLUS Factorization of Matrices and Its Applications. 1. Background. 1. Background 2. Main Achievements 3. Applications 4.

Transcription

1 PLUS Factorization of Matrices and Its Applications Pengwei Hao Center for Inforation Science, Peking University Contents 1 Background 2 Main Achieveents 3 Applications 4 Unsolved Probles Queen Mary, University of London Where the proble fro 1 Background Lossless? B/W iages Color iages Multiple Coponent iages Multiple Coponent iages Color iages B/W iages Color Space MC Inverse MC Inverse Color Spatial Inverse Spatial rans Coding Decoding Storage transiss ion Lossless? Lossless? Modeling of the General Proble What s integer reversible transfor? Visual Inforation Reotely Sensed Other types of Inforation Signal Acquisition digital signal integers Signal transfor doain integers Encoding Storage ransission Processing Applications he transfor aps integer x to integer y, and its inverse recovers integer x exactly the sae fro integer y If the inverse transfor is also required to be integer reversible, the transfor has to be one-to-one integer apping Matheatics : floating-point coputation integer reversible? 1

2 A Siple Exaple A siple 2D transfor: y1 4 x1 y = 2 1/4 x 2 What happens if apping integers directly? / Roundoff / It s not one-to-one integer apping, so not reversible! What if ore bits kept? What if take ore bits? In 2000, soe NASA researchers were trying to find out how any bits behind the point are needed for lossless transfor heir experients show that it doesn t help if ore bits kept And the cost is about 1 bit for 1 bit! Analysis of Direct Integer ation By direct rounding after transforation: y = Ax+ b yˆ = [ Ax+ b] he rounding error: e = y yˆ = y [ Ax + b] = Ax + b [ Ax + b] If the error is independent of integers: [ n+ c] = n+ [ c] hen, the transfor atrix ust satisfy: x = [ A ( yˆ b)] = [ A ( Ax+ b e b)] = [ x A e] = x + [ A e] ie we ust have [ 1 1 A e] = 0 and thus, A 1 If the inverse transfor is also integer reversible: A However, A A AA = 1 1 herefore, we have A = A = 1 his condition is too strict It s why NASA failed Research in Integer s Integer version of siple wavelet transfors S transfor (Blue & Fand, 1989) S transfor (Zandi et al, 1995) S+P transfor (Said & Pearlan, 1996) Ladder structure (Bruekers & van den Enden, 1992) he lifting schee (2D, Sweldens, 1996) Rough color space transfor (3D, Gorish et al, 1997) Factorization of wavelet transfors (2D, QY Shi, 1998, Daubechies et al, 1998) Previous Work S- reversible wavelets H Blue and A Fand, "Reversible and Irreversible Copression Using the S and Lepel-Ziv coding", Proc SPIE Medical Iaging III: Capture and Display, vol 1091, pp 2-18, 1989 Forward transfor: sn ( ) = ( x(2 n) + x(2n+ 1) ) 2 dn ( ) x(2 n) x(2n+ 1) Its Inverse: x(2 n) = s( n) + ( d( n) + 1) 2 x(2n+ 1) = s( n) d( n) 2 Carefully cobined!!! Previous Work S- reversible wavelets A Zandi, J D Allen, E L Schwartz, and M Boliek "CREW: Copression with reversible wavelets ", In Proc of IEEE Copression Conference, pp , Snowbird, U, 1995 Forward : s( n) = ( x(2 n) + x(2n+ 1) ) 2 d( n) x(2n 2) x(2n 3) ( s( n 2) s( n) 2) 4 = pn ( ) = dn ( 1) Its Inverse: ( sn ( + 1) sn ( 1) + 2) 4 x(2 n) = s( n) + ( p( n) + 1) 2 x(2n 1) s( n) p( n) 2 + = Carefully cobined! 2

3 Previous Work S+P reversible wavelets A Said, W A Pearlan, "An iage ultiresolution representation for lossless and lossy copression", IEEE ransactions on Processing, vol 5, pp , Sept 1996 S- + Prediction Carefully cobined! Previous Work Color Space M J Gorish, E L Schwartz, A F Keith, M P Boliek, and A Zandi, "Lossless and nearly lossless copression for high quality iages", Proceedings of SPIE, vol 3025, pp 62-70, 1997 Yr = ( R + 2G + B) 2 A Rough approxiation: Ur = R G Vr = B G G = Yr ( Ur+ Vr) 4 Its Inverse: R = Ur + G B = Vr + G Carefully cobined!!! Previous Work Ladder Structure F A M L Bruekers, A W M van den Enden, "New networks for perfect inversion and perfect reconstruction", IEEE J on Selected Areas in Counications, vol 10, pp , 1992 Proposed a new network for perfect inversion Previous Work he Lifting Schee W Sweldens, "he Lifting Schee: A new philosophy in biorthogonal wavelet constructions", in A F Laine and M Unser, editors, Wavelet Applications in Signal and Processing III, Proc SPIE, vol 2569, pp 68-79, 1995 W Sweldens, "he lifting schee: a custo-design construction of biorthogonal wavelets", J of Applied and Coputational Haronic Analysis, vol 3, No 2, pp , 1996 W Sweldens, "he lifting schee: A construction of second generation wavelets", SIAM J Math Anal, vol 29, No 2, pp , 1997 In order to increase the oents of wavelets Previous Work Integer apping with lifting schee A R Calderbank, I Daubechies, W Sweldens, B-L Yeo, "Wavelet transfors that ap integers to integers", J of Applied and Coputational Haronic Analysis, 5(3): , 1998 Previous Work Factoring wavelet transfor into lifting steps I Daubechies, W Sweldens, "Factoring wavelet transfors into lifting steps", J of Fourier Analysis and Application, vol 4, No 3, pp , 1998 Qingyun Shi, "Biorthogonal wavelet theory and techniques for iage coding", Proceedings of SPIE, vol 3545(ISMIP'98, Oct 1998): 24-32, 1998 Split the polyphase representation into even and odd coefficients: First tie relate lifting schee to integer apping e( ) o h z h ( z) 1 i s ( z) 1 0 K 0 P( z) = g e ( z) g o ( z) = i= t i ( z) K A general theory of 2 channel polyphase atrix factorizations 3

4 hen for N-D? Can anyone cobine an integer reversible ipleentation for higher diensional transfors? Even for experts? oo difficult!!! We ust find soe new way out Proble in Coputer Graphics Shears can be ipleented fast by eoryoving with hardware x shear y shear A viewing transfor can be represented by a atrix if hoogeneous coordinates are eployed Proble: Can we use soe shears to ipleent the transfor so as to speed-up it? Research in Shear Factorizations 2-shear-scaling factorization for 2D rotation (Catull & Sith, SIGGRAPH, 1980) 3-shear factorization for 2D rotation (Paeth,1986) 3-shear-scaling factorization for 3D affine transfors (Hanrahan, SIGGRAPH, 1990) 3-shear factorization for 3D rotation (Wittenbrink & Soani, SIGGRAPH, 1993) Shear-warp factorization for 3D projection (Lacroute & Levoy, SIGGRAPH, 1994) 4-shear factorization for 3D rotation (Chen & Kaufan, 2000) Relations between Shear Factorizations and Integer Reversible s A shear atrix is a unit triangular atrix Shear factorizations are to find those factors of unit triangular atrices Shear factorizations and the factorization for reversible integer transforation are equivalent in atheatics: Custoizable Factorization Generic Linear s If we can copute in soe special way We find soe Eleentary Reversible Matrices can do it! Our Solution: PLUS Factorization Pengwei Hao and Qingyun Shi, "Invertible linear transfors ipleented by integer apping", Science in China, Series E, (in Chinese), vol 30(2), pp , 2000 Pengwei Hao and Qingyun Shi, "Proposal of reversible integer ipleentation for ultiple coponent transfors", ISO/IEC JC1/SC29/WG1N 1720, Arles, France, July 3-7, 2000 Pengwei Hao and Qingyun Shi, "Matrix factorizations for reversible integer apping", IEEE ransactions on Signal Processing, vol 49, No 10, pp , Oct 2001 Yiyuan She and Pengwei Hao, "Block ERM Factorization of Unifor Block Matrices", Science in China (Series F), Vol 47, No 4, pp , 2004 Pengwei Hao, "Custoizable riangular Factorizations of Matrices", Linear Algebra and Its Applications, Vol 382, pp , May 2004 A general theory of finite diensional atrix factorizations 4

5 Reversible Condition for just One Nuber 2 Main Achieveents If two integers x and y satisfy both equation: y=ax+b x=(y b)/a hen, a, b, 1/a and b/a ust be all integers If integer b is replaced by rounding a fractional nuber: [b] a and 1/a ust be soe special integers: integer factors (integer units) Integer Factors Integer units Integer factor: j Real nubers: 1, -1 Coplex nubers: 1, -1, i, -i Integer reversibility: y=jx+[b] Property 1: integer factors don t change the agnitude of an integer Property 2: the inverse of an integer factor is still an integer factor For Integer Nubers, How Reversible Mapping Works? Eleentary structure of integer nuber apping j is an integer factor, 1, -1, i, -i b b b [ ] [ ] + - x j y 1 x + + /j y=jx+[b] x=(1/j)(y+[b]) Rounding arithetic: chopping, rounding, or bits before decial point If j=1, it is the sae as the ladder structure by Bruekers & van den Enden and the lifting schee by Sweldens Ipleentation of Linear s Eleentary coputing units are the sae as the transforation of nubers: y=jx+[b] atrices: he diagonal eleents are integer factors Forward transfor: b ust be independent of x Inverse transfor: b ust be independent of y Eleentary Reversible Matrix (ERM) Eleentary Reversible Matrices Diagonal eleents are all integer factors riangular ERM (ERM) Upper ERM Lower ERM Single-row ERM (SERM) S = J + e s Other atrices that can be converted into ERMs by row perutations only 5

6 with ERMs Upper ERMs Forward Reverse Lower ERMs Forward Reverse with SERMs Forward transfor calculation : y = jx + snxn n = j x + Its reverse : [] b 1 x = y b j ( []) Properties of ERMs he product of two upper ERMs is also an upper ERM, and the product of two lower ERMs akes a lower ERM he deterinant of a ERM is an integer factor he inverse of a ERM is a ERM A 2D ERM is also a SERM Properties of SERMs Unit SERMs : where s = e { } s n 1 S I es S =, det = 1 S S = I + e s + s For any SERM : S = I + es S 0 = I + en s0 the -th standard basis vector fored as the -th colun of the identity atrix a vector that the -th eleent s is 0 ( ),1,2,1,2 det S is an integer factor Probles We Must Solve Can a atrix be factorized into soe ERMs? If not always, what s the sufficient/necessary conditions? Under the conditions it can be factorized, how? Is the factorization unique? If not unique, what s the optial factorization? What s the error between the original theoretical transfor and its integer reversible ipleentation? Factorization of wo Special Matrices 2D rotation: cosθ sinθ sinθ 1 0 = sinθ cosθ (1 cosθ ) sinθ (1 cosθ ) sinθ 1 1 (cosθ 1) sinθ (cosθ 1) sinθ = 0 1 sinθ Scaling: α α 0 1α = 1α α α α 1 = 1α

7 Factorization of Nonsingular Matrices Any nonsingular atrix : A=PLDU Proble: perutation P, unit triangular L and U are all ERMs, how to factorize D? λ = dd 1 2 d D R = diag( 1,,1, λn ) D = diag( d1, d2,, d N ) = D O D E D R = D E D O D R diag( λ1,1 λ1, λ3,1 λ3,, λn 1,1 λn 1) if N is even DO = diag( λ1,1 λ1, λ3,1 λ3,, λn 2,1 λn 2,1) if N is odd diag(1, λ2,1 λ2, λ4,1 λ4,, λn 2,1 λn 2,1) if N is even DE = diag(1, λ2,1 λ2, λ4,1 λ4,, λn 1,1 λn 1 ) if N is odd With 7 ERMs: A = PLVO 1V O2VO3V O4VE1V E2V E3V E4DRU = PVVVVVVV D R he Least Nuber of Factor Matrices? We prove: the necessary and sufficient condition to factorize a atrix into up to 3 ERMs A = PLDRUS 0 is det P A det D 0 = R ( ) DR = Diag 1,1,,1, det( P A) where S I + e s = I + e s, s,, s 0 = N 0 N [ 1 2 N 1,0] heore of SERM Factorizations Matrix A has a unit SERM factorization of A=PS N S N-1 S 1 S 0 if and only if det A = j is an integer factor, where S ( = 0,1,2,, N) are SERMs SERM Factorization Owing to the equation: A11 A12 I 0 A11 A12 = 1 1 A21 A22 A21A11 A22 A21A11 A12 0 I A11 A12 1 det = det( A22 A21A11 A12) det( A11 ) A21 A22 SERM factorization can be derived fro ERM factorization: A=PLUS 0 1 LU = LUS S ( 1 ) ( ) = LUS S S S = = LUS S S S S S ( 1 ) 1 1 N = S S S N N A=PSNSN-1 S1S0 Necessary and Sufficient Condition for ERM factorizations he absolute value of the deterinant of the atrix is 1 det(a) =1 is uch looser than the reversible condition for direct rounding 1 A = A = 1 If the Deterinant is Not 1 For nonsingular atrices, we can apply soe scaling to ake the deterinant 1: o ultiply a diagonal atrix or a nonzero scalar nuber Scaling can also: Reduce the nuber of factor atrices Control the transfor dynaic range Increase the transforation efficiency 7

8 Factorization Algorith 1 Factorize into 3 ERMs : A=PLUS 0 siilar to LU factorization 2 Further factorize LU into N SERMs : LU=S N S N-1 S 1 extract row-by-row Exaple: = 1 0 1/ / / /2 = /2 1 0 = ERM Factorization for Integer Coputational flow chart x S0 A=PLUS 0 U L P y SERM Factorization for Integer Coputational flow chart Coputational Coplexity A=PS N S N-1 S 1 S 0 Ipleentation Addition Multiplication Rounding Perutation x S0 S1 S2 S3 SN-1 SN y Original N N N 2 N No ERM factorization N 2 1 N 2 1 2N 1 Needed SERM factorization N 2 1 N 2 1 N + 1 Needed P A ERM or a SERM-factorized transfor at ost has a coputational coplexity very close to the naive transforation Error Estiation A = PLUS 0 ( u + L ( u + Uu ) = P( u + Lu ) u = P LUu3 ( u + Lu + LUu ) P( u + Lu ) u = P LUu3 u u 1 ( + L LUe ) + N Properties of ERMs S 0 is not unique, so neither are ERM factorizations A real atrix has a real ERM factorization Error bounds depend on factorization itself, are independent of the data to be transfored 8

9 Advantages of ERM Factorization Best approxiation of given linear transfor Integer reversible ( lossless ) Flexible rounding arithetic ( [b] ) In-place calculation ( add to: += ) Siple inverse ( subtract fro: -= ) Acceptable coputation coplexity Generalization - Custoizable A is just nonsingular for alost every transfors P is unit triangular pseudo-perutation U has custoizable diagonal freely designated by users S has any possible fors freely chosen by users PLUS factorization: A=PLUS (custoizable triangular factorization) Fors of Special Matrix S Single-row Single-colun Bidiagonal Other? Factorization of Block Matrices Factorization of block atrices is a generalization for eleent atrices For parallel and efficient coputation Proble: soe blocks ay be singular Questions: How to block? What perutation works? How to ake it ore efficient? Solution: define a deterinant atrix BLUS factorization (Yiyuan She) Solved Probles 3 Applications 1 linear invertible deta =1 A = A = 1 9

10 Applications in Coding Signals acquired and stored are alost integers Lossless data/signal copression Unified lossy/lossless copression syste Region-of-interest (ROI) decoding Lossy copression with lossless reproduction of ROI Progressive lossy-lossless transission Integer-reversible ipleentation of transfors akes lossy coding ethods lossless and reusable App1 Color Coding Color s R G B Y 0299 Cr = 0500 Cb 0169 Color rans C1 C2 C3 Coding G R 0500 B he integer reversibility of the color transfor is the key proble of lossless color iage coding App1 Color (JPEG 2000) JPEG2000 irreversible transfor Y R Cr = G Cb B JPEG2000 reversible transfor Y = [( R + 2 G + B) 4] G = Y [( Cr + Cb) 4] Cr = R G R = Cr + G Cb = B G B = Cb + G Roughly approxiated Different fro the transfor for lossy coding App1 Integer Reversible Color Our atrix factorization ethod: c1 c2 1 P L APR = 1 b1 1 b2 1 1 a a d d k k det P AP = ±1 = L R It can be applied to all 3D linear transfors, thus all possible transfors can be copared for lossless coding App2 Multiple Coponent s XB - Hyperspectral iage - 80 bands (highly correlated) - 346x512 - Road, water, rice paddy App2 Multiple Coponent Decorrelating transfors of ultiple coponent - No transfor (No) - Discrete wavelet transfor (DW) - Direct prediction transfor (DP) - asselled cap transfor for M (C) - Karhunen-Loeve transfor (KL) - DC, DH, 多成分图像数据 Input Cube JPEG 2000 copression on transfored coponents 10

11 App2 MC Coding App2 Lossless Copression (XB, Bit Rate, BPPPB) Previous solutions: Different ethods for lossy and lossless coding Lossy transfor + Lossless residue coding MC atrix ay be of large size, 100x100, 512x512 Our solution (JPEG 2000 proposal No 1720): reversible integer ipleentation by using atrix factorization for MC (including KL) RKL gives the best perforance App2 Lossy Copression (XB, 05BPPPB, PSNR, db) RKL gives the perforance close to the best Preview(1/256) & ROI 46%, Lossless 2%, 338dB Coding Based on ROI M5-3-2,Beijing(Huairou) Matrix Factorization for Reversible Integer Mapping Applications in Inforation Security App4 Hiding transfor without encryption with cipher still keeps soe good properties, such as orthogonal Covert counication: to hide soe secret data in a carrier signal Digital waterarking: for copyright protection, source detection, Original Secret with Encryption transfer Extraction Inverse hiding extraction Decrypt with Hidden 11

12 App4 Hiding transfor with encryption Cipher App5 Hiding Experient with huge data: 70K bytes Original Secret Encryption Inverse hiding with transfer with Cipher Decryption Extraction extraction Hidden Original iage (512x512) PSNR = 3637dB App5 Hiding he hidden data (7K bytes) 北京大学视觉与听觉信息处理室验室视觉与听觉信息处理实验室于 1986 年 3 月通过专家委员会论证,1986 年 6 月经国家计委批准,1988 年 12 月通过国家验收, 是国家在北京大学建立的第一个重点实验室实验室学术委员会主任为中科院院士唐孝威教授, 实验室学术委员会顾问为中科院院士程民德教授实验室主任为唐世谓教授实验室的主要研究方向是紧密结合智能计算机, 智能机器人, 智能武器及指挥系统, 管理自动化等社会主义四个现代化的重大课题和发展需要, 开展机器视觉与听觉信息处理方面的基础研究与应用基础研究特别是关于机器视觉机器听觉语音文字的识别处理与自然语言理解, 图像识别和图像数据库, 智能系统与知识工程, 视觉与听觉的神经计算模型和人工神经网络等方面的基本理论与基本方法研究, 以及具有显著经济和社会效益的应用研究中的基础性工作以期为实现实用的机器视觉系统与听觉系统提供可用的方法与技术, 为实现自然的人机通信提供基础技术手段并在某些应用领域提供具有国际先进水平的设计思想与技术, 指导实用产品的开发为此实验室装备了先进的计算机及其他设备根据学术委员会确定的研究方向和目标, 实验室积极开展了相关课题的研究包括国家科技攻关 863 高技术攀登计划科学基金博士点基金以及部委合作项目等, 近 60 项在进行这些课题的研究中实验室发挥北京大学重视基础研究和应用基础研究的传统和学科相互交叉渗透的良好学风, 互相配合, 共同奋斗在所承担的一些项目上取得了创新成果 七 五 国家重点攻关项目 模式识别图像数据库 被鉴定委员会认为是取得一系列具有国际水平国际先进水平和国际领先水平的成果特别对实验室完成的 指纹自动识别系统 给予了极高的评价, 认为 系统采用领先的指纹软件技术成为第一个在工作站上实现的技术最先进的指纹自动识别系统 是一项居于国际领先地位的科技成果 这一成果获得了国家教委科技进步一等奖和国家科技进步? 等奖并在国际竞争中战胜了实力强大的对手进入国际市场 八五 以来该项目在上海, 珠海, 杭州, 广州等十个公安部门进一步推广应用, 取得了良好的社会效益此外在计算机视觉, 说话人识别, 图象数据库, 智能系统与知识工程以及视觉与听觉的生理与心理模型方面也取得了一些显著成果几年来实验室在国内外学术会议和刊物上发表了论文近 300 篇, 开展了广泛的学术交流活动 一些年青的硕士, 博士毕业生也很快的成长起来, 在研究上成为独挡一面的骨干目前在实验室课题指南涉及的各个方面有年青研究人员从事独立研究, 且正在成长为学术接班人实验室自 1988 年底通过国家验收以来, 还实施了八轮开放课题研究计划, 支持了 147 项课题 邮政编码 通讯地址北京大学信息科学中心电话 (010) 传真 (010) E-ail chi@cispkueducn URL HE NAIONAL LABORAORY ON MACHINE PERCEPION he proposal of setting up a National Laboratory on Machine Perception was discussed and accepted by the Expert Coittee in March 1986 he laboratory construction begun in June 1986 after being approved by the State Planning Coittee, and finished in Deceber 1988, which is the first National Key Laboratory settled in Peking University Our leaders: Chairan of the laboratory's Acadeic coittee: Acad Prof ANG XiaoWei Director of the laboratory: Prof ANG ShiWei Consultant of the Acadeic Coittee: Acad Prof CHENG Minde he laboratory's ain research objects are fundaental and applications oriented basic studies in the field of achine visual perception and auditory inforation processing hese researches are closely related to artificial intelligence, robotics, intelligent control systes, anageent autoation and other iportant subjects deanded by socialist odernization and developent Special concern is given to basic theory and basic ethodology for achine visual and auditory perception, voice and text recognition and natural language understanding, iage recognition and iage database, intelligent syste and knowledge engineering, neural coputational odeling and artificial neural networks, and other fundaental works with evident social and econoical ipact he purpose of these researches is to provide ethods and techniques for realization of the practically applicable achine visual and auditory perception syste, to provide basic technical solution for natural huan-achine counication systes, to provide, in certain application fields, principles for designing techniques at advanced international standards, to lead the developent of practical products According to those purposes, the laboratory is equipped with advanced coputers, workstations and other special facilities Based on the decision of the Acadeic Coittee for the research orientation and purpose, the laboratory has been actively carrying out researches under related projects hese include the National Key projects, including fundaental and high-tech projects the Foundation for, the Foundation for PhD projects, the National Natural Science Foundation projects, and other projects with collaboration to other Ministries and Coittees, totals alost 60 projects Researchers in the laboratory enhanced Peking University's tradition to proote the fundaental and application oriented basic researches, and in the good style of interdisciplinary study, collaborated closely with each other, obtained original results in certain projects On the7th National Five-year Plan's "Pattern Recognition and base" was considered by the Evaluation Coittee as having a series of results with internationally advanced achieveent Especially the Autoated Fingerprint Identification Syste, which was considered as the ost advanced syste, realized for the first tie on workstations adopting special software technique received a high praise It has received the first rank award fro the State Education Coittee for progress in science and technology and the second rank National Award for progress in science and technology Since the 8th National Five-Year Plan, this syste was applied in Shanghai, Zhuhai, Hangzhou, Guangzhou and other Police Departents, and obtained a good social effect It also entered the international arket Good results were also obtained in Coputer Vision, Speaker Identification, base, Intelligent Systes and Knowledge Engineering and Physiological and Psychological Modeling for Vision or Auditory Systes In last years, the laboratory published near 300 papers, collaboration and exchange activities been rapidly developed Now, ore and ore young research fellows and PhD candidates are actively participating in the laboratory's projects Since the end of 1988, the laboratory already supported 147 projects for guest scientists and visiting scholars Center for Inforation Sciences Peking University, Beijing, , P R of China el : Fax : E-ail : chi@cispkueducn URL : App6 Digital Waterarking with waterark (PSNR=3392dB) Waterark 北京大学信息科学中心电话 :(010) 传真 :(010) URL: cispku pkuedu educn cn/ (R) All rights reserved, 1999 年 3 月版权所有 Digital waterarks should survive coonly used iage processing, copression, deforation and cutting Applications in Coputer Graphics App7 s of Voluetric o speedup the geoetric transfors (affine or linear) by factorizing the into a series of shear transfors (3-4 shears in 2D, 4-5 shears in 3D) and a possible resize Original Scaled and Rotated or volue data display/rendering or volue data deforation or volue data registration or volue data erging Naive transforation: All the pixels after geoetric transfor have to be re-located Soe pixels have to be obtained by interpolation Auxiliary eory is often required ed data cannot be recovered losslessly 12

13 App7 Average ie for 3D s (s) volue size transfors x y translations z D xy /z slice xz/y shears yz/x x/y z bea y/xz shears z/xy x y resizes z D xy transposes yz naive rotation linear transfors App7 Speedup Geoetric s Shear-resize factorization: =D x LUS x =D x S x S y S x x Original shear (Ying Chen) Equivalent Scaled & Rotated 1 a 1 0 Sx = = 0 1 Sy b 1 y shear x resize x shear App8 Accelerate Voluetric Registration ake one as the reference and the other as the floating iage, and iteratively to find the optial geoetric transfor such that the two iages are registered App8 Multi-Modal Voluetric Registration - slices Side View Reference iage Floating iage op View Front View Reference Floating Registered App8 Multi-Modal Voluetric Registration - rendered App8 Multi-Modal Voluetric Registration - rendered and copared Side View op View Front View Reference Floating Registered 13

14 App8 ie for 3D Registration (s/iteration) Advantages of Shear-Resize Factorization using size ranslation Resize shears Shear-Resize Factorization Meory Reallocation otal Naive ation he transforation with our shear-resize factorization is about 10 ties faster than the naive transforation coputation is just once for each layer Interpolation can be accelerated No auxiliary eory is needed he data can be recovered losslessly if the nearest neighbor interpolation is eployed and the area or volue is not reduced 4 Unsolved Probles (I) Fors of Generic S for PLUS Factorization We have found the necessary and sufficient conditions for several fors of special atrix S, and two necessary conditions and one sufficient condition for generic S What necessary and sufficient condition should S satisfy so that the PLUS factorization exists? (II) PLUS Factorization with the Least Rounding Error Now that PLUS factorization is not unique and the rounding error bounds only depend on the factorization u = P u1 + Lu2 + LUu3 P u1 + Lu2 + LUu ( ) ( ) hen, what factorization is with the least rounding error? How to find it? What pivoting rules should be followed to find it? 3 (III) Stability of PLUS factorization When the atrix size is large, the PLUS factorization ay be unstable, the eleents in the factor atrices ay be extreely large Why does PLUS factorization turn unstable? What s the boundary condition? How to ake factorization stable? 14

15 (IV) Perturbation Analysis for PLUS Factorization A=PLUS A+dA =P(L+dL)(U+dU)(S+dS) What s the influence of perturbation? What atrices suffer ore fro perturbation? How to reduce the influence before factorization? How to reduce the influence during factorization? (V) Other Applications of PLUS Factorization With developent and disseination of PLUS factorization, we believe that ore applications will be found and PLUS factorization will ake ore applications possible and attractive hank You! phao@cispkueducn phao@dcsqulacuk 15