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Discrete Orthogonal Harmonic Transforms
Speaker: Chun-Lin Liu,Advisor: Soo-Chang Pei Ph. D
Image Processing Laboratory, EEII 530,Graduate Institute of Communication Engineering,
National Taiwan University.
May 26, 2012
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 1 / 37
Outline
1 Introduction
2 Computation of 1D Discrete Orthogonal FunctionsProposed method to the 1D discrete orthogonal functionsSimulation results
3 Discrete implementation of 2D Fourier Transform EigenfunctionsThe general form of 2D FT eigenfunctions and the discreteimplementationSimulation Results
4 Conclusions
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 2 / 37
Outline
1 Introduction
2 Computation of 1D Discrete Orthogonal FunctionsProposed method to the 1D discrete orthogonal functionsSimulation results
3 Discrete implementation of 2D Fourier Transform EigenfunctionsThe general form of 2D FT eigenfunctions and the discreteimplementationSimulation Results
4 Conclusions
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 3 / 37
Introduction
Some well-known discrete orthogonal transforms areI Discrete Fourier transform (DFT),I Discrete Sine/Cosine/Hartley transforms,I Discrete wavelet transforms.
Properties of orthogonal transformsI More coefficients lead to smaller reconstruction error.I Perfect reconstruction.
ProblemI Explicit definitions of the transform kernels. For instance, the DFT is
defined as
Fk =
N−1∑n=0
fne−j 2πnk
N , n, k = 0, 1, 2, . . . , N − 1. (1)
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 4 / 37
A simple method to other discrete transforms
Choose a continuous orthogonal transform and take samples of thetransform kernel.
Pros:I Good approximation to the continuous transforms.
Cons:I Not orthogonal. For instance, assume that the continuous transform
kernel {ψn(x)}∞n=0 satisfies the orthogonal relation:∫ ∞−∞
ψ∗m(x)ψn(x) dx = δm,n. (2)
However, taking the samples x = l∆x does not ensure orthogonality:∑l
ψ∗m (l∆x)ψn (l∆x) ∆x 6= δm,n. (3)
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 5 / 37
Problem formulation
The 1D discrete orthogonal transform is characterized by thetransform matrix Ψ, as the following relationship
F =
F0
...FN−1
=[ψ0 . . . ψN−1
]H f0...
fN−1
= ΨHf , (4)
I f is the input signal, F is the transformed signal,I {ψn}
N−1n=0 are the discrete transform kernels,
I f , F, and ψn are all N -by-1 column vectors.
Problem statement
We want to the discrete orthogonal transform to be
General ( there are general ways to find entries of Ψ).
Good approximation (ψn is close to the samples of ψn(x)).
Orthogonal (Ψ is an unitary matrix).
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 6 / 37
Outline
1 Introduction
2 Computation of 1D Discrete Orthogonal FunctionsProposed method to the 1D discrete orthogonal functionsSimulation results
3 Discrete implementation of 2D Fourier Transform EigenfunctionsThe general form of 2D FT eigenfunctions and the discreteimplementationSimulation Results
4 Conclusions
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 7 / 37
Hermite Gaussian functions and Hermite transforms
The Hermite Gaussian functions (HGFs) are defined by
hn(x) =
(1
2nn!√π
)1/2
Hn(x)e−x2/2, (5)
where Hn(x) is the Hermite polynomial.
Differential equation (Dx = d / dx is the differential operator):(D2x − x2
)hn(x) = −(2n+ 1)hn(x). (6)
Properties:I hn(x) is the eigenfunction of the Fourier transform with eigenvalue
(−j)n.I Complete and orthonormal basis for L2(R).
Hermite transforms are defined by
an =
∫ ∞−∞
f(x)hn(x) dx. (7)
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 8 / 37
Outline
1 Introduction
2 Computation of 1D Discrete Orthogonal FunctionsProposed method to the 1D discrete orthogonal functionsSimulation results
3 Discrete implementation of 2D Fourier Transform EigenfunctionsThe general form of 2D FT eigenfunctions and the discreteimplementationSimulation Results
4 Conclusions
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 9 / 37
Implement discrete HGFs by differential equations
Implement the differential equation as the matrix eigen-problem.1 Convert L into L.2 Solve the eigenvectors and eigenvalues of L numerically.
Continuous:
L︷ ︸︸ ︷(D2x − x2
)hn(x) = −(2n+ 1) hn(x),
Discrete:
· . . . ·.... . .
...· . . . ·
︸ ︷︷ ︸
L
·...·
︸︷︷︸hn
= λn
·...·
︸︷︷︸hn
.
Three basic operations1 Multiplied by a constant c. (cI) f(x) = cf(x).2 Multiplied by x. Xf(x) = xf(x).3 Differentiate with respect to x, Dx = d / dx.
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 10 / 37
Discrete version of cI and X
1 cIN is the discrete version of cI. IN is the identity matrix of size N .
Continuous: cI f(x) = cf(x),
Discrete:
c 0 . . . 00 c . . . 0...
.... . .
...0 0 . . . c
f1f2...
fN−1
=
cf1cf2...
cfN−1
.2 Assume that the functions are sampled uniformly on x = n∆x,
n = [−N−12 ,−N−3
2 , . . . , N−12 ]T . X = diag(x) represents the discreteversion of X .
X f(x) = xf(x),
∆x
−N−1
2 0 . . . 0
0 −N−32 . . . 0
......
. . ....
0 0 . . . N−12
f1f2...
fN−1
=
−N−1
2 ∆xf1−N−3
2 ∆xf2...
N−12 ∆xfN−1
.CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 11 / 37
Discrete version of differential operators
In numerical analysis, there are forward difference, backwarddifference, and central difference. However, they are simple butinaccurate.
We derive the differential operator from Fourier transforms.
Dxf(x) = F−1FDxf(x) = F−1 (jX )Ff(x), (8)
where F denotes the Fourier transform operator.
Proposed discrete differential operator
3 The discrete differential operator is obtained by
D = F−1 (jX) F, (9)
where F is the DFT matrix and X is the discrete multiplied-by-xoperator.
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 12 / 37
Outline
1 Introduction
2 Computation of 1D Discrete Orthogonal FunctionsProposed method to the 1D discrete orthogonal functionsSimulation results
3 Discrete implementation of 2D Fourier Transform EigenfunctionsThe general form of 2D FT eigenfunctions and the discreteimplementationSimulation Results
4 Conclusions
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 13 / 37
Simulation on the Discrete HGFs
The number of discrete points N = 101.
The sample interval ∆x =√
2π/N ≈ 0.2494.
The eigenvalues are all negative and sorted in the descent order.
We want to verify whether The eigenvectors are close to thecontinuous samples.
Note that the results are identical to those of the n2-matrix1.
1S. C. Pei, J. J. Ding, W. L. Hsue and K. W. Chang, “Generalized commutingmatrices and their eigenvectors for DFTs, offset DFTs, and other periodic operations,”IEEE Trans. on Signal Processing, vol.56, No.8, pp.3891-3904, Aug. 2008
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 14 / 37
First four Discrete HGFs
−10 −5 0 5 10−0.1
0
0.1
0.2
0.3
0.4
x
ψ0(x
)
Discrete
Continuous
−10 −5 0 5 10−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
x
ψ1(x
)
Discrete
Continuous
−10 −5 0 5 10−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
x
ψ2(x
)
Discrete
Continuous
−10 −5 0 5 10−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
x
ψ3(x
)
Discrete
Continuous
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 15 / 37
Outline
1 Introduction
2 Computation of 1D Discrete Orthogonal FunctionsProposed method to the 1D discrete orthogonal functionsSimulation results
3 Discrete implementation of 2D Fourier Transform EigenfunctionsThe general form of 2D FT eigenfunctions and the discreteimplementationSimulation Results
4 Conclusions
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 16 / 37
2D discrete orthogonal functions
Another question is raised: How do we implement high dimensionaldiscrete orthogonal functions while the orthogonality is kept?
Possible approachI Take the samples of the continuous functions.
F Very simple but not orthogonal.
I Start from the differential equations.F Partial differential equations require a very huge matrix.F Large scale numerical eigen-decomposition is very difficult.F Different coordinate systems lead to multiple solutions.
Due to these difficulties, we narrow our discussion to theeigenfunctions of 2D Fourier transforms2.
2S. C. Pei and C. L. Liu, “A general form of 2D Fourier transformeigenfunctions,” 2012 IEEE International Conference on Acoustics, Speech andSignal Processing, Kyoto, Japan,, March 2012.
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 17 / 37
Known eigenfunctions of 2D FT
There are three known eigenfunctions for 2D FT.
Separable HGFs (SHGFs) in Cartesian coordinates
hm,n(x, y) = hm(x)hn(y). (10)
Rotated HGFs (RHGFs) in rotated Cartesian coordinates
hm,n(α;x, y) = hm,n(x cosα+ y sinα,−x sinα+ y cosα). (11)
Laguerre-Gaussian functions (LGFs) in polar coordinates
lm,n(r, θ) = Np,lr|l|L|l|p
(r2)e−r
2/2e−jlθ, (12)
where p = min {m,n}, l = m− n, Np,l is the normalization factor,and Llp (·) is the associated Laguerre polynomial.
All of them form a complete and orthonormal basis for L2(R2).
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 18 / 37
Observations on the 2D Fourier transform eigenfunctions
Their differential equations have identical operators, identicaleigenvalues, but different eigenfunctions.
(∂2
∂x2+
∂2
∂y2− x2 − y2
)hm,n(x, y) = −2 (m+ n+ 1)hm,n(x, y),(
∂2
∂x2+
∂2
∂y2− x2 − y2
)hm,n(α;x, y) = −2 (m+ n+ 1)hm,n(α;x, y),(
1
r
∂
∂rr∂
∂r+
1
r2∂2
∂θ2− r2
)lm,n(r, θ) = −2 (m+ n+ 1) lm,n(r, θ).
Eigenvalues of the 2D FT are the same, (−j)m+n, for each solution.
Our question
Is there a general form to represent these eigenfunctions?
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 19 / 37
Outline
1 Introduction
2 Computation of 1D Discrete Orthogonal FunctionsProposed method to the 1D discrete orthogonal functionsSimulation results
3 Discrete implementation of 2D Fourier Transform EigenfunctionsThe general form of 2D FT eigenfunctions and the discreteimplementationSimulation Results
4 Conclusions
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 20 / 37
The general form of the 2D FT eigenfunctions
The 2D Fourier transform eigenfunctions ψm,n(x, y) is
ψm,n(x, y) =m+n∑p=0
cm,np hp,m+n−p(x, y), (13)
ψ0,m+n(x, y)ψ1,m+n−1(x, y)
...ψm+n,0(x, y)
=
c0,m+n0 c0,m+n
1 . . . c0,m+nm+n
c1,m+n−10 c1,m+n−1
1 . . . c1,m+n−1m+n
......
. . ....
cm+n,00 cm+n,0
1 . . . cm+n,0m+n
︸ ︷︷ ︸
T
h0,m+n(x, y)h1,m+n−1(x, y)
...hm+n,0(x, y)
m,n = 0, 1, 2, . . . . cm,np are the combination coefficients.
ψm,n(x, y) corresponds eigenvalue (−j)m+n of the 2D FT.
Mathematical interpretation: linear combination of the eigenfunctionswith the same eigenvalue/eigenspace gives another eigenfunctions.
ψm,n(x, y) are orthogonal if and only if T is unitary.
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 21 / 37
Known combination coefficients
The SHGFs, RHGFs, and LGFs all satisfy the general for of 2DFourier transform eigenfunctions.
SHGFs hm,n(x, y) and cm,np = δ[p−m].
RHGFs hm,n(α;x, y)3,
cm,np =
√p!(m+ n− p)!
m!n!(sinα)m−p (cosα)n−p P (m−p,n−p)
p (cos 2α) ,
where P(α,β)n (·) is the Jacobi polynomial.
LGFs lm,n(r, θ) 4,
cm,np (LGF) = jp cm,np (RHGF)∣∣α=π/4
. (14)
3A. Wunsche, “Hermite and Laguerre 2D polynomials,” Journal of Computationaland Applied Mathematics, vol. 133, pp. 665-678, 2001.
4M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman,“Astigmatic laser mode converters and transform of orbital angular momentum,” Opt.Comm., vol. 96, pp. 123-132, 1993.
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 22 / 37
Discrete 2D Fourier transform eigenfunctions
Implementation of discrete 2D Fourier transform eigenfunctions
1 Solve the 1D discrete Hermite Gaussian functions.
2 Construct discrete separable HGFs by multiplying 1D discrete HGFsin each dimension.
3 Compute combination coefficients cm,np .
4 Obtain discrete 2D Fourier transform eigenfunctions according to
ψm,n(x, y) =
m+n∑p=0
cm,np hp,m+n−p(x, y).
Advantages:I More general than the explicit expression.I Different combination coefficients yield different eigenfunctions.I Samples are always on the Cartesian grids.I Orthogonal, as long as the matrix composed of cm,n
p is unitary.
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 23 / 37
Problem on very high order discrete eigenfunctions
An illustrative example
The number of discrete points in each dimension N = 6.
We want to get the discrete function of h1,5(α;x, y).
The implementation steps indicates1 We have only 6 1D discrete HGFs: h0,h1,h2,h3,h4,h5.2 Construct separable HGFs. Then compute the combination coefficients.3 We have a problem in the last step. Write out the general form
c2,40 h6hT0 + c2,41 h5h
T1 + c2,42 h4h
T2 + c2,43 h3h
T3
+c2,44 h2hT4 + c2,45 h1h
T5 + c2,46 h0h
T6
The problem is the absence of h6 in the 1D discrete HGFs.
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 24 / 37
Two possible solutions to high-order discrete eigenfunctions
1 Truncate the combination coefficients.
h1,5(α;x, y)⇒ c2,40 h6hT0 + c2,41 h5h
T1 + · · ·+ c2,45 h1h
T5 + c2,46 h0h
T6
I Not orthogonalI High accuracy
2 Replace the combination coefficients with those of lower-ordereigenfunctions. The concept is to mirror the coefficients.
h0,4(α;x, y)⇒ d2,40 h4hT0 + · · ·+ d2,44 h0h
T4
↓ ↓h1,5(α;x, y)⇒ c2,40 h6h
T0 +c2,41 h5h
T1 + · · ·+ c2,45 h1h
T5 + c2,46 h0h
T6
I OrthogonalI Low accuracy
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 25 / 37
The discrete Laguerre Gaussian transform
The Laguerre Gaussian transform is
Lm,n =
∫R2
f(x, y)l∗m,n(r, θ) d2 r. (15)
According to the general form, (13), we have
Lm,n =
m+n∑p=0
(cm,np
)∗ ∫R2
f(x, y)hp,m+n−p(x, y) d2 r︸ ︷︷ ︸2D separable Hermite transform
. (16)
The Laguerre Gaussian transform is composed of1 2D separable Hermite transform2 Linear transformation associated with cm,n
p .In terms of discrete implementation:
1 Separable transforms and the linear transformation are faster thannon-separable transforms.
2 High-order Lm,n are adjusted either by truncating or by mirroring thetransformation coefficients cm,n
p .
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 26 / 37
Outline
1 Introduction
2 Computation of 1D Discrete Orthogonal FunctionsProposed method to the 1D discrete orthogonal functionsSimulation results
3 Discrete implementation of 2D Fourier Transform EigenfunctionsThe general form of 2D FT eigenfunctions and the discreteimplementationSimulation Results
4 Conclusions
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 27 / 37
Simulation results
1 Compare the continuous-sampled Laguerre Gaussian functions withthe discrete Laguerre Gaussian functions.
2 Compare the high-order continuous-sampled LGFs with the discreteLGFs with either truncated coefficients or mirrored coefficients.
3 Image expansion and reconstruction using discrete Laguerre Gaussiantransforms.
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 28 / 37
Simulation 1: Accuracy of the discrete eigenfunctions
l8,3(r, θ) Magnitude Phase
Cont.
−10 −5 0 5 10
−10
−5
0
5
10
−10 −5 0 5 10
−10
−5
0
5
10
Disc.
−10 −5 0 5 10
−10
−5
0
5
10
−10 −5 0 5 10
−10
−5
0
5
10
Number of pointsin each dimensionN = 101.
Sampling interval∆x = ∆y =√
2π/N .
Error comes fromvery small values(≈ 10−14) on theboundaries.
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 29 / 37
Simulation 2: High-order discrete LGFs
Continuous samples
l63,1(r, θ)
Truncate Mirror
Number of points in eachdimension N = 64.
Sampling interval∆x = ∆y =
√2π/N .
Compare the magnitudes oflm,n(r, θ). The number of ringsshould be min {m,n}+ 1.
For the mirrored discretefunction, coefficients areborrowed from those of l62,0(r, θ).
Error comes fromI Inaccurate combination terms.I Inaccurate high-order Hermite
Gaussian functions.
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 30 / 37
Simulation 3: Image expansion and reconstruction
The input image I
Reconstruction schemes
L
Km
n
Lm,n
Image expansion: Convert inputimage I into its LGT Lm,n.
Image reconstruction: partial Lm,nand the inverse LGT yields thereconstructed image I.
N = 90, ∆x = ∆y =√
2π/N .
The input image I is the 2Dsquared image.
We have two type ofreconstruction schemes, shown onthe left:
I Fix L, increase K gradually.I Fix K, increase L gradually.
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 31 / 37
Image reconstruction along K (details in θ-direction)
The input image I
discrete LGT=⇒
Reconstruction scheme
m
nLm,n
K
K = 0, 2, 4, 6
K = 8, 10, 12, 14
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 32 / 37
Image reconstruction along L (details in r-direction)
The input image I
discrete LGT=⇒
Reconstruction scheme
m
n Lm,n
L
L = 0, 2, 4, 6
L = 8, 10, 12, 14
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 33 / 37
Outline
1 Introduction
2 Computation of 1D Discrete Orthogonal FunctionsProposed method to the 1D discrete orthogonal functionsSimulation results
3 Discrete implementation of 2D Fourier Transform EigenfunctionsThe general form of 2D FT eigenfunctions and the discreteimplementationSimulation Results
4 Conclusions
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 34 / 37
Conclusion on 1D discrete orthogonal functions
In the first part, we proposed an alternative method to the discreteorthogonal functions.
As long as the differential equations are known, we can replace thecontinuous operators L with discrete operator L. This approach ismore general and the explicit form is not required.
We can apply this procedure to solve any linear differential equations.I Classical orthogonal polynomialsI Schrodinger equations in quantum mechanicsI Scale transformsI Fractional Fourier transformsI Linear canonical transforms and their eigenfunctions
We utilize the 1D results to implement the 2D case.
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 35 / 37
Conclusion on 2D Fourier transform eigenfunctions
In the section part, we implement the 2D Fourier transformeigenfunctions along with the transform derived from theseeigenfunctions.
I The general form of 2D Fourier transform eigenfunctions.I Deal with high-order discrete eigenfunctions. We can truncate or
mirror the combination coefficients.I Convert the non-separable transform (LGT) into 2D separable Hermite
transforms and then a linear transformation (fast).
The concept can be generalized to three-dimensional eigenfunctions.
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 36 / 37
Thank you for your attention!
Q & A Time
CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 37 / 37