lecture unitarytransforms

7/27/2019 Lecture UnitaryTransforms

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Unitary Trans form s

Borrowed from UMD ENEE631Spring04


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Image Trans form : A Revis i t

With A Coding Perspect ive

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Why Do Transform s?

Fast computation

E.g., convolution vs. multiplication for filter with wide support

Conceptual insights for various image processing

E.g., spatial frequency info. (smooth, moderate change, fast change, etc.)

Obtain transformed data as measurement

E.g., blurred images, radiology images (medical and astrophysics) Often need inverse transform

May need to get assistance from other transforms

For efficient storage and transmission Pick a few representatives (basis)

Just store/send the contribution from each basis

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Basic Process o f Trans form Coding

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Figure is from slides at Gonzalez/ Woods DIP book website (Chapter 8)


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1-D DFT and Vecto r Form

{ z(n) }

{ Z(k) }n, k = 0, 1, , N-1

WN= exp{ - j2/ N }~ complex conjugate of primitive Nth root of unity

Vector form and interpretation for inverse transform

z = kZ(k) akak= [ 1 WN

-k WN-2k WN

-(N-1)k]T/N

Basis vectors

akH= ak

* T= [ 1 WNk WN

2k WN(N-1)k] /N

Use akHas row vectors to construct a matrix F

Z = F zz = F*TZ = F* Z

F is symmetric (FT=F) and unitary (F-1 = FHwhere FH= F*T)

1

0

1

0

)(1)(

)(1

)(

N

k

nk

N

N

n

nk

N

WkZN

nz

WnzN

kZ

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Basis Vecto rs and Basis Images

A basis for a vector space ~ a set of vectors

Linearly independent ~ ai vi = 0 if and only if all ai=0 Uniquely represent every vector in the space by their linear combination

~ bi vi ( spanning set {vi} )

Orthonormal basis

Orthogonality ~ inner product = y*Tx= 0

Normalized length ~ ||x ||2 = = x*Tx= 1

Inner product for 2-D arrays

= mn f(m,n) g*(m,n) = G1

*TF1 (rewrite matrix into vector)

!! Dont do FG ~ may not even be a valid operation for MxN matrices! 2D Basis Matrices (Basis Images)

Represent any images of the same size as a linear combination of basisimages

A vector space consists of a set of vectors, a

field of scalars, a vector addition operation,

and a scalar multiplication operation.

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1-D Unitary Trans form

Linear invertible transform

1-D sequence { x(0), x(1), , x(N-1) } as a vector

y = A x andA is invertible

Unitary matrix ~A-1 = A*T

DenoteA*T

asAH

~ Hermitianx = A-1 y = A*Ty = ai

*Ty(i)

Hermitian of row vectors of A form a set of orthonormal basis vectorsai

*T = [a*(i,0), , a*(i,N-1)] T

Orthogonal matrix ~A-1 = AT Real-valued unitary matrixis also an orthogonal matrix

Row vectors of real orthogonal matrix A form orthonormal basis vectors

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Propert ies o f 1-D Unitary Transfo rmy = A x

Energy Conservation

||y ||2 = ||x ||2

|| y ||2 = || Ax ||2= (Ax)*T(Ax)= x*TA*TA x = x*Tx = || x ||2

Rotation

The angles between vectors are preserved

A unitary transformation is a rotation of a vector in an

N-dimension space, i.e., a rotation of basis coordinatesUMCP

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Propert ies o f 1-D Unitary Transfo rm(contd)

Energy Compaction

Many common unitary transforms tend to pack a large fraction of signalenergy into just a few transform coefficients

Decorrelation

Highly correlated input elements quite uncorrelated output coefficients

Covariance matrix E[ ( yE(y) ) ( yE(y) )*T

] small correlation implies small off-diagonal terms

Example: recall the effect of DFT

Question: What unitary transform gives the best compaction and decorrelation?

=> Will revisit this issue in a few lectures

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Review : 1-D Discrete Cos ine Transfo rm(DCT)

Transform matrix C

c(k,n) = (0) for k=0

c(k,n) = (k) cos[(2n+1)/2N] for k>0

Cis real and orthogonal

rows ofCform orthonormal basis Cis not symmetric!

DCT is not the real part of unitary DFT! See Assignment#3

related to DFT of a symmetrically extended signal

Nk

N

N

knkkZnz

N

knknzkZ

N

k

N

n

2)(,

1)0(

2

)12(cos)()()(

2

)12(cos)()()(

1

0

1

0

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Period ic i ty Impl ied by DFT and DCT

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Figure is from slides at Gonzalez/ Woods DIP book website (Chapter 8)


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Examp le of 1-D DCT

100

50

0

-50

-100

0 1 2 3 4 5 6 7

n

z(n)

100

50

0

-50

-100

0 1 2 3 4 5 6 7

k

Z(k)

DCT

From Ken Lams DCT talk 2001 (HK Polytech)

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Examp le of 1-D DCT (contd)

k

Z(k)

Transform coeff.

1.0

0.0

-1.0

1.0

0.0

-1.0

1.0

0.0

-1.0

1.0

0.0

-1.0

1.0

0.0

-1.0

1.0

0.0

-1.0

1.0

0.0

-1.0

1.0

0.0

-1.0

Basis vectors

100

0

-100

100

0

-100

100

0

-100

100

0

-100

100

0

-100

100

0

-100

100

0

-100

100

0

-100u=0

u=0 to 1

u=0 to 4

u=0 to 5

u=0 to 2

u=0 to 3

u=0 to 6

u=0 to 7

Reconstructions

n

z(n)

Original signal

From Ken Lams DCT talk 2001 (HK Polytech)


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2-D DCT

Separable orthogonal transform

Apply 1-D DCT to each row, then to each columnY = C X CTX = CTY C = mn y(m,n) Bm,n

DCT basis images:

Equivalent to representan NxN image with a setof orthonormal NxNbasis images

Each DCT coefficient

indicates the contributionfrom (or similarity to) thecorresponding basis image

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2-D Transfo rm : General Case

A general 2-D linear transform {ak,l(m,n)}

y(k,l) is a transform coefficient for Image {x(m,n)}

{y(k,l)} is Transformed Image

Equiv to rewriting all from 2-D to 1-D and applying 1-D transform

Computational complexity N2 values to compute

N2 terms in summation per output coefficient

O(N4) for transforming an NxN image!

),(),(),(

),(),(),(

1

0

1

0

,

1

0

1

0,

nmhlkynmx

nmanmxlky

N

k

N

l

lk

N

m

N

nlk

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2-D Separab le Unitary Trans fo rms

Restrict to separable transform

ak,l(m,n) = ak(m) bl(n) , denote this as a(k,m) b(l,n)

Use 1-D unitary transform as building block

{ak(m)}kand {bl(n)}lare 1-D complete orthonormal sets of basis vectors

use as row vectors to obtain unitary matrices A={a(k,m)} & B={b(l,n)}

Apply to columns and rows Y = A X BT

often choose same unitary matrix asA andB (e.g., 2-D DFT)

For square NxN imageA: Y = A X ATX = AHY A*

For rectangular MxN imageA: Y = AMX ANTX = AM

HY AN*

Complexity ~ O(N3)

May further reduce complexity if unitary transf. has fast implementation

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Basis Images

X = AHY A* => x(m,n) = kl a*(k,m)a*(l,n) y(k,l)

RepresentXwith NxN basis images weighted by coeff. Y

Obtain basis image by setting Y={(k-k0, l-l0)} & getting X

{ a*(k0,m)a*(l0,n) }m,n in matrix form A*

k,l= a*

ka

l

*T ~ a*k

is kth column vector of

AH

trasnf. coeff. y(k,l) is the inner product ofA*k,lwiththe image

(Jains e.g.5.1, pp137)

2

1

2

12

1

2

1

A

43

21X

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Common Unitary Transform s and BasisImages

DFT DCT Haar transform K-L transform

See also: Jains Fig.5.2 pp136


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2-D DFT

2-D DFT is Separable

Y = F X FX = F* Y F*

Basis images Bk,l= (ak)(al

)T

where ak= [ 1 WN-k WN

-2k WN-(N-1)k]T/N

1

0

1

0

1

0

1

0

),(1

),(

),(1

),(

N

k

N

l

mk

N

nl

N

N

m

N

n

mk

N

nl

N

WWlkYN

nmX

WWnmXN

lkY


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Summary and Review (1)

1-D transform of a vector

Represent an N-sample sequence as a vector in N-dimension vector space

Transform

Different representation of this vector in the space via different basis

e.g., 1-D DFT from time domain to frequency domain

Forward transform In the form of inner product

Project a vector onto a new set of basis to obtain N coefficients

Inverse transform

Use linear combination of basis vectors weighted by transform coeff.

to represent the original signal

2-D transform of a matrix

Rewrite the matrix into a vector and apply 1-D transform

Separable transform allows applying transform to rows then columns

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Summary and Review (1) contd

Vector/matrix representation of 1-D & 2-D sampled signal

Representing an image as a matrix or sometimes as a long vector

Basis functions/vectors and orthonomal basis

Used for representing the space via their linear combinations

Many possible sets of basis and orthonomal basis

Unitary transform on inputx ~ A-1 = A*T

y = A xx = A-1 y = A*Ty = ai*Ty(i) ~ represented by basis vectors{ai

*T}

Rows (and columns) of a unitary matrix form an orthonormal basis

General 2-D transform and separable unitary 2-D transform 2-D transform involves O(N4) computation

Separable: Y = A X AT= (A X) AT ~ O(N3) computation

Apply 1-D transform to all columns, then apply 1-D transform to rows

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Optimal Trans form

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Optimal Transfo rm

Recall: Why use transform in coding/compression?

Decorrelate the correlated data

Pack energy into a small number of coefficients

Interested in unitary/orthogonal or approximate orthogonal transforms

Energy preservation s.t. quantization effects can be better understood

and controlled

Unitary transforms weve dealt so far are data independent

Transform basis/filters are not depending on the signals we are processing

What unitary transform gives the best energy compaction and decorrelation?

Optimal in a statistical sense to allow the codec works well withmany images

Signal statistics would play an important role

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Review : Correlation A fter a Lin earTransform

Consider an Nx1 zero-mean random vectorx

Covariance (autocorrelation) matrixRx = E[ x xH]

give ideas of correlation between elements

Rxis a diagonal matrix for if all N r.v.s are uncorrelated

Apply a linear transform tox: y = Ax What is the correlation matrix fory ?

Ry = E[ y yH] = E[ (Ax) (Ax)H] = E[ A x xHAH]

= A E[ x xH] AH= A Rx AH

Decorrelation: try to search forA that can produce a decorrelatedy (equiv. a

diagonal correlation matrixRy)

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K-L Trans form (Princ ipal ComponentAnalys is)

Eigen decomposition of Rx: Rx uk= kuk

Recall the properties of Rx Hermitian (conjugate symmetric RH= R);

Nonnegative definite (real non-negative eigen values)

Karhunen-Loeve Transform (KLT)

y = UHxx = U y with U = [ u1, uN]

KLT is a unitary transform with basis vectors in U being the

orthonormalized eigenvectors of Rx

UH Rx

U = diag{1

, 2

, , N

} i.e. KLT performs decorrelation

Often order{ui} so that 12 N

Also known as the Hotelling transform or

the Principle Component Analysis (PCA)


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Propert ies o f K-L Trans form

Decorrelation

E[ y yH]= E[ (UHx) (UHx)H]= UHE[ x xH] U= diag{1, 2, , N}

Note: Other matrices (unitary or nonunitary) may also decorrelate

the transformed sequence [Jains e.g.5.7 pp166]

Minimizing MSE under basis restriction

If only allow to keep m coefficients for any 1m N, whats the

best way to minimize reconstruction error?

Keep the coefficients w.r.t. the eigenvectors of the first m largesteigenvalues

Theorem 5.1 and Proof in Jains Book (pp166)


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KLT Basis Restr ic t ion

Basis restriction

Keep only a subset of m transform coefficients and then performinverse transform (1 m N)

Basis restriction error: MSE between original & new sequences

Goal: to find the forward and backward transform matrices to minimize therestriction error for each and every m

The minimum is achieved by KLT arranged according to the

decreasing order of the eigenvalues of R

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K-L Trans form for Images

Work with 2-D autocorrelation function

R(m,n; m,n)= E[ x(m, n) x(m, n) ] for all 0m, m, n, nN-1

K-L Basis images is the orthonormalized eigenfunctions ofR

Rewrite images into vector form (N2x1)

Need solve the eigen problem forN2xN2 matrix! ~ O(N6)

Reduced computation for separableR

R(m,n; m,n)= r1(m,m) r2(n,n)

Only need solve the eigen problem for twoNxNmatrices ~ O(N3

) KLT can now be performed separably on rows and columns

Reducing the transform complexity from O(N4) to O(N3)

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Pros and Cons of K -L Trans form

Optimality

Decorrelation and MMSE for the same# of partial coeff.

Data dependent

Have to estimate the 2nd-order statistics to determine the transform

Can we get data-independent transform with similar performance?

DCT

Applications

(non-universal) compression

pattern recognition: e.g., eigen faces

analyze the principal (dominating) components

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Energy Compact ion o f DCT vs. KLT

DCT has excellent energy compaction for highlycorrelated data

DCT is a good replacement for K-L

Close to optimal for highly correlated data

Not depend on specific data like K-L does

Fast algorithm available

[ref and statistics: Jains pp153, 168-175]

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Energy Compaction of DCT vs. KLT (contd)

Preliminaries

The matrices R, R-1, and R-1 share the same eigen vectors

DCT basis vectors are eigenvectors of a symmetric tri-diagonal matrixQc

Covariance matrixR of 1st-order stationary Markov sequence with has an

inverse in the form of symmetric tri-diagonal matrix DCT is close to KLT on 1st-order stationary Markov

For highly correlated sequence, a scaled version ofR-1 approx. Qc

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Summary and Review on Uni tary Trans form

Representation with orthonormal basis Unitary transform

Preserve energy

Common unitary transforms

DFT, DCT, Haar, KLT

Which transform to choose?

Depend on need in particular task/application

DFT ~ reflect physical meaning of frequency or spatial

frequency

KLT ~ optimal in energy compaction

DCT ~ real-to-real, and close to KLTs energy compaction

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