Discrete Orthogonal Harmonic Transforms

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 Functions Proposed method to the 1D discrete orthogonal functions Simulation results 3 Discrete implementation of 2D Fourier Transform Eigenfunctions The general form of 2D FT eigenfunctions and the discrete implementation Simulation Results 4 Conclusions CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 2 / 37

Introduction Some well-known discrete orthogonal transforms are Discrete Fourier transform (DFT), Discrete Sine/Cosine/Hartley transforms, Discrete wavelet transforms. Properties of orthogonal transforms More coefficients lead to smaller reconstruction error. Perfect reconstruction. Problem Explicit definitions of the transform kernels. For instance, the DFT is defined as F k = N 1 n=0 2πnk j f n e 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 the transform kernel. Pros: Good approximation to the continuous transforms. Cons: Not orthogonal. For instance, assume that the continuous transform kernel {ψ n (x)} n=0 satisfies the orthogonal relation: ψ m(x)ψ n (x) d x = δ m,n. (2) However, taking the samples x = l x does not ensure orthogonality: ψm (l x ) ψ n (l x ) x δ m,n. (3) l CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 5 / 37

Problem formulation The 1D discrete orthogonal transform is characterized by the transform matrix Ψ, as the following relationship F 0 F =.. = [ f 0 ] H ψ 0... ψ. N 1. = Ψ H f, (4) F N 1 f N 1 f is the input signal, F is the transformed signal, {ψ n } N 1 n=0 are the discrete transform kernels, f, F, and ψ n 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 (ψ n is close to the samples of ψ n (x)). Orthogonal (Ψ is an unitary matrix). CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 6 / 37

Hermite Gaussian functions and Hermite transforms The Hermite Gaussian functions (HGFs) are defined by ( ) 1 1/2 h n (x) = 2 n n! H n (x)e x2 /2, (5) π where H n (x) is the Hermite polynomial. Differential equation (D x = d / d x is the differential operator): ( D 2 x x 2) h n (x) = (2n + 1)h n (x). (6) Properties: hn (x) is the eigenfunction of the Fourier transform with eigenvalue ( j) n. Complete and orthonormal basis for L 2 (R). Hermite transforms are defined by a n = f(x)h n (x) d x. (7) CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 8 / 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 ({}}{ D 2 x x 2) h n (x) = (2n + 1) h n (x),... Discrete:........ = λ n.. Three basic operations }... {{ } L }{{} h n 1 Multiplied by a constant c. (ci) f(x) = cf(x). 2 Multiplied by x. X f(x) = xf(x). 3 Differentiate with respect to x, D x = d / d x. }{{} h n CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 10 / 37

Discrete version of ci and X 1 ci N is the discrete version of ci. I N is the identity matrix of size N. Continuous: ci f(x) = cf(x), c 0... 0 f 1 cf 1 0 c... 0 f 2 Discrete:.......... = cf 2... 0 0... c f N 1 cf N 1 2 Assume that the functions are sampled uniformly on x = n x, n = [ N 1 2, N 3 2,..., N 1 2 ]T. X = diag(x) represents the discrete version of X. X f(x) = xf(x), N 1 2 0... 0 f 1 N 1 0 N 3 2 xf 1 2... 0 f 2 x.......... = N 3 2 xf 2.. N 1 0 0... f N 1 2 N 1 2 xf N 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, backward difference, and central difference. However, they are simple but inaccurate. We derive the differential operator from Fourier transforms. D x f(x) = F 1 FD x f(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-x operator. CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 12 / 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 the continuous samples. Note that the results are identical to those of the n 2 -matrix 1. 1 S. C. Pei, J. J. Ding, W. L. Hsue and K. W. Chang, Generalized commuting matrices 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 0.4 0.3 Discrete Continuous 0.4 0.3 0.2 Discrete Continuous ψ 0 (x) 0.2 0.1 0 ψ 1 (x) 0.1 0 0.1 0.2 0.3 0.1 0.4 0.3 10 5 0 5 10 x Discrete Continuous 0.4 0.3 0.2 10 5 0 5 10 x Discrete Continuous 0.2 0.1 ψ 2 (x) 0.1 0 0.1 0.2 ψ 3 (x) 0 0.1 0.2 0.3 0.3 10 5 0 5 10 x 0.4 10 5 0 5 10 x CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 15 / 37

2D discrete orthogonal functions Another question is raised: How do we implement high dimensional discrete orthogonal functions while the orthogonality is kept? Possible approach Take the samples of the continuous functions. Very simple but not orthogonal. Start from the differential equations. Partial differential equations require a very huge matrix. Large scale numerical eigen-decomposition is very difficult. Different coordinate systems lead to multiple solutions. Due to these difficulties, we narrow our discussion to the eigenfunctions of 2D Fourier transforms 2. 2 S. C. Pei and C. L. Liu, A general form of 2D Fourier transform eigenfunctions, 2012 IEEE International Conference on Acoustics, Speech and Signal 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 h m,n (x, y) = h m (x)h n (y). (10) Rotated HGFs (RHGFs) in rotated Cartesian coordinates h m,n (α; x, y) = h m,n (x cos α + y sin α, x sin α + y cos α). (11) Laguerre-Gaussian functions (LGFs) in polar coordinates l m,n (r, θ) = N p,l r l L l ( p r 2 ) e r2 /2 e jlθ, (12) where p = min {m, n}, l = m n, N p,l is the normalization factor, and L l p ( ) is the associated Laguerre polynomial. All of them form a complete and orthonormal basis for L 2 (R 2 ). 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, identical eigenvalues, but different eigenfunctions. ( ) 2 x 2 + 2 y 2 x2 y 2 h m,n (x, y) = 2 (m + n + 1) h m,n (x, y), ( ) 2 x 2 + 2 y 2 x2 y 2 h m,n (α; x, y) = 2 (m + n + 1) h m,n (α; x, y), ( 1 r r r r + 1 2 ) r 2 θ 2 r2 l m,n (r, θ) = 2 (m + n + 1) l m,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

The general form of the 2D FT eigenfunctions The 2D Fourier transform eigenfunctions ψ m,n (x, y) is ψ m,n (x, y) = ψ 0,m+n(x, y) ψ 1,m+n 1(x, y) =. ψ m+n,0(x, y) m+n c m,n p h p,m+n p (x, y), (13) p=0 c 0,m+n 0 c 0,m+n 1... c 0,m+n m+n c 1,m+n 1 0 c 1,m+n 1 1... c 1,m+n 1 m+n........ } c m+n,0 0 c m+n,0 1 {{... c m+n,0 m+n } T h 0,m+n(x, y) h 1,m+n 1(x, y). h m+n,0(x, y) m, n = 0, 1, 2,.... c m,n p are the combination coefficients. ψ m,n (x, y) corresponds eigenvalue ( j) m+n of the 2D FT. Mathematical interpretation: linear combination of the eigenfunctions with the same eigenvalue/eigenspace gives another eigenfunctions. ψ m,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 2D Fourier transform eigenfunctions. SHGFs h m,n (x, y) and c m,n p = δ[p m]. RHGFs h m,n (α; x, y) 3, c m,n p = p!(m + n p)! m!n! where P n (α,β) ( ) is the Jacobi polynomial. LGFs l m,n (r, θ) 4, (sin α) m p (cos α) n p P (m p,n p) p (cos 2α), c m,n p (LGF) = j p c m,n p (RHGF) α=π/4. (14) 3 A. Wünsche, Hermite and Laguerre 2D polynomials, Journal of Computational and Applied Mathematics, vol. 133, pp. 665-678, 2001. 4 M. 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 HGFs in each dimension. 3 Compute combination coefficients c m,n p. 4 Obtain discrete 2D Fourier transform eigenfunctions according to ψ m,n (x, y) = m+n p=0 c m,n p h p,m+n p (x, y). Advantages: More general than the explicit expression. Different combination coefficients yield different eigenfunctions. Samples are always on the Cartesian grids. Orthogonal, as long as the matrix composed of c m,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 h 1,5 (α; x, y). The implementation steps indicates 1 We have only 6 1D discrete HGFs: h 0, h 1, h 2, h 3, h 4, h 5. 2 Construct separable HGFs. Then compute the combination coefficients. 3 We have a problem in the last step. Write out the general form c 2,4 0 h 6h T 0 + c 2,4 1 h 5h T 1 + c 2,4 2 h 4h T 2 + c 2,4 3 h 3h T 3 +c 2,4 4 h 2h T 4 + c 2,4 5 h 1h T 5 + c 2,4 6 h 0h T 6 The problem is the absence of h 6 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. h 1,5 (α; x, y) c 2,4 0 h 6h T 0 + c 2,4 1 h 5h T 1 + + c 2,4 5 h 1h T 5 + c 2,4 6 h 0h T 6 Not orthogonal High accuracy 2 Replace the combination coefficients with those of lower-order eigenfunctions. The concept is to mirror the coefficients. h 0,4 (α; x, y) d 2,4 0 h 4h T 0 + + d 2,4 4 h 0h T 4 h 1,5 (α; x, y) c 2,4 0 h 6h T 0 +c 2,4 1 h 5h T 1 + + c 2,4 5 h 1h T 5 + c 2,4 6 h 0h T 6 Orthogonal 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 L m,n = f(x, y)lm,n(r, θ) d 2 r. (15) R 2 According to the general form, (13), we have m+n ( ) L m,n = c m,n p f(x, y)h p,m+n p (x, y) d 2 r. (16) p=0 R } 2 {{} 2D separable Hermite transform The Laguerre Gaussian transform is composed of 1 2D separable Hermite transform 2 Linear transformation associated with c m,n p. In terms of discrete implementation: 1 Separable transforms and the linear transformation are faster than non-separable transforms. 2 High-order L m,n are adjusted either by truncating or by mirroring the transformation coefficients c m,n p. CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 26 / 37

Simulation results 1 Compare the continuous-sampled Laguerre Gaussian functions with the discrete Laguerre Gaussian functions. 2 Compare the high-order continuous-sampled LGFs with the discrete LGFs with either truncated coefficients or mirrored coefficients. 3 Image expansion and reconstruction using discrete Laguerre Gaussian transforms. CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 28 / 37

Simulation 1: Accuracy of the discrete eigenfunctions l 8,3 (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 Number of points in each dimension N = 101. Sampling interval x = y = 2π/N. Disc. 10 5 0 5 10 5 0 5 Error comes from very small values ( 10 14 ) on the boundaries. 10 10 10 5 0 5 10 10 5 0 5 10 CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 29 / 37

Simulation 2: High-order discrete LGFs Continuous samples l 63,1 (r, θ) Truncate Mirror Number of points in each dimension N = 64. Sampling interval x = y = 2π/N. Compare the magnitudes of l m,n (r, θ). The number of rings should be min {m, n} + 1. For the mirrored discrete function, coefficients are borrowed from those of l 62,0 (r, θ). Error comes from Inaccurate combination terms. 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 n L m L m,n K Image expansion: Convert input image I into its LGT L m,n. Image reconstruction: partial L m,n and the inverse LGT yields the reconstructed image Î. N = 90, x = y = 2π/N. The input image I is the 2D squared image. We have two type of reconstruction schemes, shown on the left: Fix L, increase K gradually. 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 L m,n n m K = 0, 2, 4, 6 K 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 L m n L m,n L = 0, 2, 4, 6 L = 8, 10, 12, 14 CL-Liu (GICE, NTU) Discrete Orthogonal Harmonic Transforms May 26, 2012 33 / 37

Conclusion on 1D discrete orthogonal functions In the first part, we proposed an alternative method to the discrete orthogonal functions. As long as the differential equations are known, we can replace the continuous operators L with discrete operator L. This approach is more general and the explicit form is not required. We can apply this procedure to solve any linear differential equations. Classical orthogonal polynomials Schrödinger equations in quantum mechanics Scale transforms Fractional Fourier transforms 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 transform eigenfunctions along with the transform derived from these eigenfunctions. The general form of 2D Fourier transform eigenfunctions. Deal with high-order discrete eigenfunctions. We can truncate or mirror the combination coefficients. 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