1 PLUS Factorization of Matrices and Its Applications Pengwei Hao Center for Information Science, Peking University Queen Mary, University of London Contents 1. Background 2. Main Achievements 3. Applications 4. Unsolved Problems 1. Background Where the problem from Storage transmiss ion Image Coding Spatial Transform B/W images Color images Multiple Component images Color Space Transform MCT Image Decoding Inverse Spatial Trans. Color images Multiple Component images Inverse Color Transform Inverse MCT B/W images Lossless? Lossless? Lossless? Modeling of the General Problem Visual Information Remotely Sensed Data Other types of Information Processing Storage Transmission Encoding Signal Transform Signal Acquisition digital signal Applications integers floating-point computation transform domain integers Mathematics : integer reversible? What’s integer reversible transform? The transform maps integer x to integer y, and its inverse recovers integer x exactly the same from integer y. If the inverse transform is also required to be “integer reversible”, the transform has to be one-to-one integer mapping.
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1
PLUS Factorization of Matrices and Its Applications
Pengwei Hao
Center for Information Science, Peking University
Queen Mary, University of London
Contents
1. Background
2. Main Achievements
3. Applications
4. Unsolved Problems
1. Background
Where the problem from
Storage
transmission
Image C
oding
Spatial Transform
B/W images
Color images
Multiple Component images
Color Space Transform
MCT
Image D
ecoding
Inverse Spatial Trans.
Color images
Multiple Component images
Inverse Color Transform
Inverse MCT
B/W images
Lossless?
Lossless? Lossless?
Modeling of the General Problem
Visual Information
Remotely Sensed Data
Other types of
Information
Processing
Storage
Transmission
En
codin
g
Sign
al Tran
sform
Sign
al Acq
uisition
digital signal
Applications
integers
floating-point computation
transform domain
integers
Mathematics : integer reversible?
What’s integer reversible transform?
The transform maps integer x to integer y, and its inverse recovers integer xexactly the same from integer y.
If the inverse transform is also required to be “integer reversible”, the transform has to be one-to-one integer mapping.
2
A Simple Example
A simple 2D transform:
What happens if mapping integers directly?
It’s not one-to-one integer mapping, so not reversible!
What if more bits kept?
1 1
2 2
4
1/ 4
y x
y x
= ⋅
4 1 4
1/ 4 0 0 4Roundoff
04 1 4
1/ 4 1 0.25
⋅ ⇒ ⇒ ⇒ ⋅ ⇒
What if take more bits?In 2000, some NASA researchers were trying to find out how many bits behind the point are needed for lossless transform.
Their experiments show that it doesn’t help if more bits kept. And the cost is about 1 bit for 1 bit!
By direct rounding after transformation:
The rounding error:
If the error is independent of integers:Then, the transform matrix must satisfy:
i.e. we must have and thus, If the inverse transform is also integer reversible: However, Therefore, we have
This condition is too strict.It’s why NASA failed.
11 ==∞
−∞
AA
Analysis of Direct Integer Transformation
][][ˆ bAxbAxbAxyyye +−+=+−=−=
ˆ [ ]= + ⇒ = +y Ax b y Ax b
1 1 1 1ˆ[ ( )] [ ( )] [ ] [ ]x− − − −= − = + − − = − = + −x A y b A Ax b e b x A e A e
[ ] [ ]n c n c+ = +
0=− − ][ 1eA 11 ≤∞
−A
1∞≤A
111 =≥∞
−
∞
−∞
AAAA
Research in Integer Transforms
• Integer version of simple wavelet transforms
S transform (Blume & Fand, 1989)
TS transform (Zandi et al, 1995)
S+P transform (Said & Pearlman, 1996)
• Ladder structure (Bruekers & van den Enden, 1992)
• The lifting scheme (2D, Sweldens, 1996)
• Rough color space transform (3D, Gormish et al, 1997)
• Factorization of wavelet transforms (2D, QY Shi, 1998, Daubechies et al, 1998)
Previous WorkS-Transform – reversible wavelets
H. Blume and A. Fand, "Reversible and Irreversible Image Data Compression Using the S Transform and Lempel-Ziv coding", Proc. SPIE Medical Imaging III: Image Capture and Display, vol. 1091, pp. 2-18, 1989.
Forward transform:
Its Inverse:
Carefully combined!!!
( )( ) (2 ) (2 1) 2
( ) (2 ) (2 1)
s n x n x n
d n x n x n
= + + +
( )(2 ) ( ) ( ) 1 2
(2 1) ( ) ( ) 2
x n s n d n
x n s n d n
= + +
+ =
Previous WorkTS-Transform – reversible wavelets
A. Zandi, J. D. Allen, E. L. Schwartz, and M. Boliek. "CREW: Compression with reversible wavelets ", In Proc. of IEEE Data Compression Conference, pp. 212-221, Snowbird, UT, 1995.
Forward :
Its Inverse:
Carefully combined!
( )( )
( ) (2 ) (2 1) 2
( ) (2 2) (2 3) ( 2) ( ) 2 4
s n x n x n
d n x n x n s n s n
= + +
= + + + + +
( )( )
( ) ( 1) ( 1) ( 1) 2 4
(2 ) ( ) ( ) 1 2
(2 1) ( ) ( ) 2
p n d n s n s n
x n s n p n
x n s n p n
= − − + − − + = + + + = −
3
Previous WorkS+P Transform – reversible wavelets
A. Said, W. A. Pearlman, "An image multiresolution representation for lossless and lossy compression" , IEEE Transactions on Image Processing, vol. 5, pp. 1303-1310, Sept. 1996.
S-Transform + Prediction
Carefully combined!
Previous WorkColor Space Transform
M. J. Gormish, E. L. Schwartz, A. F. Keith, M. P. Boliek, and A.Zandi, "Lossless and nearly lossless compression for high quality images", Proceedings of SPIE, vol. 3025, pp. 62-70, 1997.
A Rough approximation:
Its Inverse:
Carefully combined!!!
( )2 2Yr R G B
Ur R G
Vr B G
= + + = − = −
( ) 4G Yr Ur Vr
R Ur G
B Vr G
= − + = + = +
Previous WorkLadder 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 Communications, vol. 10, pp. 130-137, 1992.
Proposed a new network for perfect inversion
Previous WorkThe Lifting Scheme
W. Sweldens, "The Lifting Scheme: A new philosophy inbiorthogonal wavelet constructions", in A. F. Laine and M. Unser, editors, Wavelet Applications in Signal and Image Processing III, Proc. SPIE, vol. 2569, pp. 68-79, 1995.
W. Sweldens, "The lifting scheme: a custom-design construction of biorthogonal wavelets", J. of Applied and Computational Harmonic Analysis, vol. 3, No. 2, pp. 186-200, 1996.
W. Sweldens, "The lifting scheme: A construction of second generation wavelets", SIAM J. Math. Anal, vol. 29, No. 2, pp. 511-546, 1997.
In order to increase the moments of wavelets
Previous WorkInteger mapping with lifting scheme
A. R. Calderbank, I. Daubechies, W. Sweldens, B-L. Yeo, "Wavelet transforms that map integers to integers", J. of Applied and Computational Harmonic Analysis, 5(3): 332-369 , 1998.
First time relate lifting scheme to integer mapping
Previous WorkFactoring wavelet transform into lifting steps
I. Daubechies, W. Sweldens, "Factoring wavelet transforms into lifting steps", J. of Fourier Analysis and Application, vol. 4, No. 3, pp. 247-269, 1998.
Qingyun Shi, "Biorthogonal wavelet theory and techniques for image coding", Proceedings of SPIE, vol. 3545(ISMIP'98, Oct. 1998): 24-32, 1998.
Split the polyphase representation into even and odd coefficients:
A general theory of 2 channel polyphase matrix factorizations
1
( ) ( ) 1 01 ( ) 0( )
( ) ( ) ( ) 10 1 0 1
me o i
ie o i
h z h z s z Kz
g z g z t z K=
= = ⋅
∏P
4
Then…… for N-D?
Can anyone combine an integer reversible implementation for higher dimensional transforms?
Even for experts?
Too difficult!!!
We must find some new way out.
• Shears can be implemented fast by memory-moving with hardware.
x yshear shear
• A viewing transform can be represented by a matrix if homogeneous coordinates are employed.
• Problem: Can we use some shears to implement the transform so as to speed-up it?
Problem in Computer Graphics
• 2-shear-scaling factorization for 2D rotation (Catmull& Smith, SIGGRAPH, 1980)
• 3-shear factorization for 2D rotation (Paeth ,1986)• 3-shear-scaling factorization for 3D affine transforms
(Hanrahan, SIGGRAPH, 1990)• 3-shear factorization for 3D rotation (Wittenbrink &
Somani, SIGGRAPH, 1993)• Shear-warp factorization for 3D projection (Lacroute
& Levoy, SIGGRAPH, 1994)• 4-shear factorization for 3D rotation (Chen & Kaufman,
2000)
Research in Shear Factorizations
• A shear matrix is a unit triangular matrix
• Shear factorizations are to find those factors of unit triangular matrices
• Shear factorizations and the factorization for reversible integer transformation are equivalent in mathematics: Customizable Factorization
Relations between Shear Factorizations and Integer Reversible Transforms
Generic Linear Transforms
If we can compute in some special way…
We find some
Elementary Reversible Matrices
can do it!
Our Solution: PLUS FactorizationPengwei Hao and Qingyun Shi, "Invertible linear transforms implemented by integer mapping", Science in China, Series E, (in Chinese), vol. 30(2), pp. 132-141, 2000.
Pengwei Hao and Qingyun Shi, "Proposal of reversible integer implementation for multiple component transforms", ISO/IEC JTC1/SC29/WG1N 1720, Arles, France, July 3-7, 2000.
Pengwei Hao and Qingyun Shi, "Matrix factorizations for reversible integer mapping", IEEE Transactions on Signal Processing, vol. 49, No. 10, pp. 2314-2324, Oct. 2001.
Yiyuan She and Pengwei Hao, "Block TERM Factorization of Uniform Block Matrices", Science in China (Series F), Vol. 47, No. 4, pp. 421-436, 2004.
Pengwei Hao, "Customizable Triangular Factorizations of Matrices", Linear Algebra and Its Applications, Vol. 382, pp. 135-154, May 2004.
A general theory of finite dimensional matrix factorizations
5
2. Main Achievements
Reversible Condition for just One Number
If two integers x and y satisfy both equation:
y=ax+b
x=(y b)/a
Then, a, b, 1/a and b/a must be all integers
If integer b is replaced by rounding a fractional number: [b]
a and 1/a must be some special integers: integer factors (integer units)
Integer Factors
Integer units
Integer factor: j
• Real numbers: 1, -1
• Complex numbers: 1, -1, i , -i
Integer reversibility: y=jx+[b]
Property 1: integer factors don’t change the magnitude of an integer
Property 2: the inverse of an integer factor is still an integer factor
b
[ ]
x j y+
+
b
x1/j
b
[ ]
-
+
Elementary structure of integer number mappingj is an integer factor, 1, -1, i, -i
For Integer Numbers,How Reversible Mapping Works?
y=jx+[b] x=(1/j).(y+[b])
If j=1, it is the same as the ladder structure byBruekers & van den Enden and the lifting scheme by Sweldens.
Rounding arithmetic: chopping, rounding, or bits before decimal point
Implementation of Linear Transforms
Elementary computing units are the same as the transformation of numbers: y=jx+[b]
Transform matrices: The diagonal elements are integer factors
Forward transform: b must be independent of x
Inverse transform: b must be independent of y
Elementary Reversible Matrix (ERM)
Elementary Reversible Matrices
• Diagonal elements are all integer factors
• Triangular ERM (TERM)– Upper TERM
– Lower TERM
• Single-row ERM (SERM)–
• Other matrices that can be converted into ERMs by row permutations only
Tm m m= +S J e s
6
Transform with TERMsUpper TERMs
Forward Reverse
Lower TERMs
Forward Reverse
[ ]
[ ]( )1
m m m mn nn m
m m
m mm
y j x s x
j x b
x y bj
≠
= + = +
= −
∑
Transform with SERMs
Forward transform calculation :
Its reverse :
Properties of TERMs
• The product of two upper TERMs is also an upper TERM, and the product of two lower TERMs makes a lower TERM.
• The determinant of a TERM is an integer factor.
• The inverse of a TERM is a TERM.
• A 2D TERM is also a SERM.For any SERM : is an integer factor.
Properties of SERMs
Unit SERMs :
where
Tmmm seIS +=
( )
1
,1 ,2 ,1 ,2
, det 1Tm m m m
T
m m m m m
− = − =
⋅ = + +
S I e s S
S S I e s s
TN 00 seIS +=
the m-th standard basis vector formed as the m-th column of the identity matrix
a vector that the m-th element smm is 0
me
{ }mnm s=s
det mS
Problems We Must Solve
• Can a matrix be factorized into some 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 optimal factorization?
• What’s the error between the original theoretical transform and its integer reversible implementation?
Factorization of Two Special Matrices
−
−
−
=
−1sin)cos1(
01
10
sin1
1sin)cos1(
01
cossin
sincos
θθθ
θθθθθθ
−
−=
10
sin)1(cos1
1sin
01
10
sin)1(cos1 θθθ
θθ
0 1 0 1 1 1 0 1 1
0 1 1 1 1 0 1 1 1 0 1
1 0 1 1 1 0 1 1 1
1 1 0 1 1 1 0 1
α αα α α
α αα
− = − − − −
= −
2D rotation:
Scaling:
7
Factorization of Nonsingular Matrices
Any nonsingular matrix : A=PLDU
Problem: permutation P, unit triangular Land U are all ERMs, how to factorize D?
With 7 TERMs:
1 2( , , , )N O E R E O Rdiag d d d= ⋅⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅D D D D D D D
⋅⋅⋅⋅⋅⋅
=−−
−−
odd is if )1,1,,,1,,1,(
even is if )1,,,1,,1,(
223311
113311
Ndiag
Ndiag
NN
NNO λλλλλλ
λλλλλλD
⋅⋅⋅⋅⋅⋅
=−−
−−
odd is if )1,,,1,,1,,1(
even is if )1,1,,,1,,1,,1(
114422
224422
Ndiag
Ndiag
NN
NNE λλλλλλ
λλλλλλD
1 2m md d dλ = ⋅⋅⋅
1 2 3 4 1 2 3 4
1 2 3 4 5 6 7
O O O O E E E E R
R
==
A PLV V V V V V V V D U
PVV V V V V V D
),1,,1( NR diag λ=D …
We prove: the necessary and sufficient condition to factorize a matrix into up to 3 TERMs
is
where
The Least Number of Factor Matrices?
0USPLDA R=
0detdet ≠= RT DAP
]0,,,,[ 12100 −⋅+=+= NNT
N ssseIseIS( ))det(,1,,1,1 APD T
R Diag= ……
…
Theorem of SERM Factorizations
Matrix A has a unit SERM factorization of
A=PSNSN-1 … S1S0
if and only if is an integer factor,
where are SERMs.
j=Adet
),,2,1,0( Nmm ⋅⋅⋅=S
SERM Factorization
11 12 11 12
1 121 22 21 11 22 21 11 12
− −
= −
A A I 0 A A
A A A A A A A A 0 I
11 12 122 21 11 12 11
21 22
det det( ) det( )− = − ⋅
A AA A A A A
A A
Owing to the equation:
SERM factorization can be derived from TERM factorization: A=PLUS0
A=PSNSN-1 … S1S0
( )( )
( )
11 1
1 11 2 2 1
1 1 11 2 1 1 2 1
2 1
N N
N
−
− −
− − −− −
=
=
=
=
=
LU LUS S
LUS S S S
LUS S S S S S
S S S
…
… …
…
Necessary and Sufficient Condition for ERM factorizations
The absolute value of the determinant of the matrix is 1.
|det(A)|=1 is much looser than the reversible condition for direct rounding
11
==−
∞ AA
If the Determinant is Not 1
For nonsingular matrices, we can apply some scaling to make the determinant 1:
To multiply a diagonal matrix or a nonzero scalar number.
Scaling can also:• Reduce the number of factor matrices• Control the transform dynamic range• Increase the transformation efficiency• ……
8
Factorization Algorithm1. Factorize into 3 TERMs :
A=PLUS0 similar to LU factorization2. Further factorize LU into N SERMs :
LU=SNSN-1 … S1 extract row-by-rowExample:
2 3 0 1 1 0 1 4 1 0
3 4 1 0 1/ 2 1 0 1 1/ 2 1
= ⋅ ⋅ ⋅
2 0 0 1 1 0 1 1/ 2 1 0
2 1/ 2 1 0 2 1 0 1 2 1
1 1 1 0 1 1/ 2 1 0
0 1 1 1 0 1 2 1
= ⋅ ⋅ ⋅ −
− = ⋅ ⋅ ⋅
.
.
.
P
.
.
.
S0
.
.
.
.
.
.
x yLU
.
.
.
TERM Factorization for Integer TransformComputational flow chart
A=PLUS0
.
.
.
P
.
.
.
S0
.
.
.
.
.
.
x ySNS1
.
.
.
S2
.
.
.
S3
.
.
.
SN-1
.
.
.
. . .
. . .
. . .
. . .
. . .
. . .
SERM Factorization for Integer TransformComputational flow chart
THE NATIONAL LABORATORY ON MACHINE PERCEPTIONThe proposal of setting up a National Laboratory on Machine Perception was discussed and accepted
by the Expert Committee in March 1986. The laboratory construction begun in June 1986 after being approved by the State Planning Committee, and finished in December 1988, which is the first National Key Laboratory settled in Peking University.
Our leaders: Chairman of the laboratory's Academic committee: Acad. Prof. TANG XiaoWei Director of the laboratory: Prof. TANG ShiWei Consultant of the Academic Committee: Acad. Prof. CHENG Minde The laboratory's main research objects are fundamental and applications oriented basic studies in the
field of machine visual perception and auditory information processing. These researches are closely related to artificial intelligence, robotics, intelligent control systems, management automation and other important subjects demanded by socialist modernization and development.
Special concern is given to basic theory and basic methodology for machine visual and auditory perception, voice and text recognition and natural language understanding, image recognition and image database, intelligent system and knowledge engineering, neural computational modeling and artificial neural networks, and other fundamental works with evident social and economical impact. The purpose of these researches is to provide methods and techniques for realization of the practically applicable machine visual and auditory perception system, to provide basic technical solution for natural human-machine communication systems, to provide, in certain application fields, principles for designing techniques at advanced international standards, to lead the development of practical products. According to those purposes, the laboratory is equipped with advanced computers, workstations and other special facilities.
Based on the decision of the Academic Committee for the research orientation and purpose, the laboratory has been actively carrying out researches under related projects. These include the National Key projects, including fundamental and high-tech projects the Foundation for, the Foundation forPh.D. projects, the National Natural Science Foundation projects, and other projects with collaboration to other Ministries and Committees, totals almost 60 projects.
Researchers in the laboratory enhanced Peking University's tradition to promote the fundamental 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 Image Database" was considered by the Evaluation Committee as having a series of results with internationally advanced achievement. Especially the Automated Fingerprint Identification System, which was considered as the most advanced system, realized for the first time on workstations adopting special software technique received a high praise. It has received the first rank award from the State Education Committee 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 system was applied in Shanghai, Zhuhai,Hangzhou, Guangzhou and other Police Departments, and obtained a good social effect. It also entered the international market.
Good results were also obtained in Computer Vision, Speaker Identification, Image Database, Intelligent Systems and Knowledge Engineering and Physiological and Psychological Modeling for Vision or Auditory Systems.
In last years, the laboratory published near 300 papers, collaboration and exchange activities been rapidly developed.
Now, more and more young research fellows and Ph.D. 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 Information SciencesPeking University, Beijing, 100871, P. R. of ChinaTel. : 86-010-62751935Fax : 86-010-62755654E-mail : [email protected] : http://www.cis.pku.edu.cn/
App6. Digital Watermarking
Image with watermark(PSNR=33.92dB)
Watermark北京大学信息科学中心北京大学信息科学中心北京大学信息科学中心北京大学信息科学中心电话电话电话电话:(010)62752461:(010)62752461:(010)62752461:(010)62752461传真传真传真传真:(010)62755654:(010)62755654:(010)62755654:(010)62755654URL: http://www.URL: http://www.URL: http://www.URL: http://www.ciscisciscis....pkupkupkupku....eduedueduedu....cncncncn////(R) All rights reserved, 1999(R) All rights reserved, 1999(R) All rights reserved, 1999(R) All rights reserved, 1999年年年年3333月月月月
版权所有版权所有版权所有版权所有
Digital watermarks should survive commonly
used image processing, compression,
deformation and cutting
Watermark北京大学信息科学中心北京大学信息科学中心北京大学信息科学中心北京大学信息科学中心电话电话电话电话:(010)62752461:(010)62752461:(010)62752461:(010)62752461传真传真传真传真:(010)62755654:(010)62755654:(010)62755654:(010)62755654URL: http://www.URL: http://www.URL: http://www.URL: http://www.ciscisciscis....pkupkupkupku....eduedueduedu....cncncncn////(R) All rights reserved, 1999(R) All rights reserved, 1999(R) All rights reserved, 1999(R) All rights reserved, 1999年年年年3333月月月月
版权所有版权所有版权所有版权所有
Applications in Computer Graphics
To speedup the geometric transforms (affine or linear) by factorizing them into a series of shear transforms (3-4 shears in 2D, 4-5 shears in 3D) and a possible resize.
• Image or volume data display/rendering
• Image or volume data deformation
• Image or volume data registration
• Image or volume data merging
App7. Transforms of Volumetric Data
Original
Image
Scaled
and
Rotated
Naive transformation:
All the pixels after geometric transform have to be re-located
Some pixels have to be obtained by interpolation
Auxiliary memory is often required
Transformed data cannot be recovered losslessly
Transform
13
App7. Average Time for 3D Transforms (ms)
volume size
transforms 643 1283 2563 5123
x 0.26 5.73 46.97 361.55
y 0.10 3.94 36.52 293.40
z 0.09 4.13 50.88 445.38 translations
3D 0.22 5.61 58.19 462.84
xy/z 0.10 4.04 35.17 287.66
xz/y 0.19 7.20 51.82 366.14 slice
shears yz/x 1.98 183.28 1866.25 19261.0
x/yz 0.48 5.75 37.10 297.76
y/xz 1.37 14.46 115.89 2117.59 beam
shears z/xy 4.83 203.84 5422.13 34434.7
x 10.85 87.73 683.58 5435.71
y 0.39 6.78 47.67 380.84
z 0.12 6.39 44.96 363.26 resizes
3D 11.37 91.33 709.62 5608.07
xy 0.31 5.95 41.71 490.67 transposes
yz 0.38 5.66 48.21 1097.42
rotation 50.27 403.18 3570.23 29338.0 naive
transforms linear 56.08 471.58 4252.21 34347.3
App7. Speedup Geometric Transforms
Shear-resize factorization: T=DxLUSx=DxSxSySx
xshear
x shear
xresize
Equivalent
Scaled &
Rotated
1 1 0
0 1 1
a
b
= =
x yS S
(Ying Chen)
Originaly
shear
App8. Accelerate Volumetric Data Registration
Take one as the reference and the other as the floating image, and iteratively to find the optimal geometric transform such that the two images are registered.
Reference
imageFloating
image
App8. Multi-Modal Volumetric Data Registration - slices
Side
View
Top
View
Front
View
Reference Floating Registered
App8. Multi-Modal Volumetric Data Registration - rendered
Side
View
Top
View
Front
View
Reference Floating Registered
App8. Multi-Modal Volumetric Data Registration -rendered and compared
14
App8. Time for 3D Registration (ms/iteration)
Image size
Transform 643 1283 2563
Translation 0.30 6.39 54.07
Resize 1.79 28.15 208.80
3 shears 1.52 27.12 160.20
Memory
Reallocation 1.46 19.13 135.42
Transform
using
Shear-Resize
Factorization
Total 5.06 80.79 558.50
Naive Transformation 63.33 508.31 4163.69
The transformation with our shear-resize factorization is about 10 times faster than the naive transformation.
Advantages of Shear-Resize Factorization
Transform computation is just once for each layer
Interpolation can be accelerated
No auxiliary memory is needed
The data can be recovered losslessly if the nearest neighbor interpolation is employed and the area or volume is not reduced.
4. Unsolved Problems
(I) Forms of Generic S for PLUS Factorization
We have found the necessary and sufficient conditions for several forms of special matrix 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
Then, what factorization is with the least rounding error? How to find it? What pivoting rules should be followed to find it?
( ) ( )321321 LUuLuuPLUuLuuPu ++≤++=
(III) Stability of PLUS factorization
When the matrix size is large, the PLUS factorization may be unstable, the elements in the factor matrices may be extremely large.
Why does PLUS factorization turn unstable? What’s the boundary condition? How to make factorization stable?
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 matrices suffer more from perturbation?
How to reduce the influence before factorization?How to reduce the influence during factorization?
(V) Other Applications of PLUS Factorization
With development and dissemination of PLUS factorization, we believe that more applications will be found and PLUS factorization will make more applications possible and attractive.