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Phase in Signals and Images Phase in signals and images Brigitte Forster Fakultat¨ fur¨ Informatik und Mathematik Universitat¨ Passau Brigitte Forster (Passau) September 2013 1 / 47 Marie Curie Team MAMEBIA at the IBB Research Topic Mathematical Methods in Biological Image Analysis MAMEBIA ! Development of theoretical and concrete Researchm Topic:athematical methods Development of theoretical and concrete mathematical methods ! To model and analyse biological image data To model and analyse biomedical image data With! emphasisWith em onph complex-valuedasis on com methodsplex-v andalue phased methods informationand phase information Biological Mathematical Numerical Algorithms & " Image Theory Analysis Software Analysis Brigitte Forster (Passau) September 2013 2 / 47 1. Phase in signals and images Mostly real-valued methods used for (biological) image analysis, because “Images are real-valued.” Real-valued methods need less storage space. Exception: Fourier transform. Brigitte Forster (Passau) September 2013 3 / 47 1. Phase in signals and images But: There are good reasons to consider complex-valued methods. First reason: Well interpretable Fourier transform: f f where f (!) = f (t)e−i!t dt: 7! ZR Interpretation: f (!)bamplitudeb to the frequency !. j j bCoefficient = Amplitude Phase term · f (!) = f (!) ei'(!) j j · b b Brigitte Forster (Passau) September 2013 4 / 47 1. Phase in signals and images Similar amplitude-phase decomposition for signals in time domain: Analytic signal (Dennis Gabor, 1946) fa(t) = f (t) + iH(f )(t); where H denotes the Hilbert transform. Then there is the decomposition iφ(t) fa(t) = a(t)e with instantaneous amplitude a(t) and instantaneous frequency φ0(t), Brigitte Forster (Passau) September 2013 5 / 47 1. Phase in signals and images Second reason: Dogma “Images are real-valued” is just not true. There are complex-valued data and tasks: e.g. Magnetic resonance images as Fourier inverses of non-symmetric measurements Phase retrieval in digital holography, . (Left: Johnson & Becker, The whole brain atlas, Harvard / MIT. Right: Bob Mellish, Wikipedia.) Brigitte Forster (Passau) September 2013 6 / 47 1. Phase in signals and images Third reason: Real bases cannot analyze single-sided frequency bands: 1.5 0.8 1.0 0.6 0.5 0.4 0.2 -4 -2 2 4 -0.5 2 4 6 8 10 -0.2 -1.0 -0.4 -1.5 Left: Mexican hat wavelet. Right: Its Fourier transform and spectral envelope. This is another reason, why Gabor invented the analytical signal. Brigitte Forster (Passau) September 2013 7 / 47 1. Phase in signals and images Fourth reason: The phase of images contains edge and detail information. Reconstruction from Fourier amplitude and phase: (Cf. A. V. Oppenheim and J. S. Lim, The importance of phase in signals, Proc. IEEE, 1981.) (Left image: Laurent Condat’s Image Database http://www.greyc.ensicaen.fr/∼lcondat/imagebase.html.) Brigitte Forster (Passau) September 2013 8 / 47 1. Phase in signals and images Fourth reason: The phase of images contains edge and detail information. Reconstruction from Fourier amplitude and phase (denoised): (Cf. A. V. Oppenheim and J. S. Lim, The importance of phase in signals, Proc. IEEE, 1981.) (Left image: Laurent Condat’s Image Database http://www.greyc.ensicaen.fr/∼lcondat/imagebase.html.) Brigitte Forster (Passau) September 2013 9 / 47 1. Phase in signals and images Another example: Interchanging amplitude and phase Brigitte Forster (Passau) September 2013 10 / 47 1. Phase in signals and images Fifth reason: The dogma “Images are real-valued” neglects various classes of powerful mathematical transforms, e.g. complex-valued wavelet transforms, analytic and monogenic signals instantaneous frequency, ! conformal monogenic signals curvature term, and others. ! Brigitte Forster (Passau) September 2013 11 / 47 Outline 1 Introduction 2 Signal analysis: B-splines and their complex relatives — Interpolation 3 Image Analysis: Phase information in higher dimensions Brigitte Forster (Passau) September 2013 12 / 47 2. B-splines Cardinal B-splines Bn of order n, n N, with knots in Z are defined recursively: 2 B1 = χ[ 0;1 ], Bn = B − B for n 2. n 1 ∗ 1 ≥ Isaac J. Schoenberg (Source: The MacTutor History Piecewise polynomials of degree at most n and of Mathematics archive) − smoothness Cn 1(R). Brigitte Forster (Passau) September 2013 13 / 47 2. B-splines Why are splines useful? Approximation: Approximation of a function f C[a; b] with sums of translated 2 and dilated B-Splines Bn(( τ)/α) 2 2 . f • − gτ N,α D Sharp estimates of the best approximation with splines are known for numerous function spaces and topologies. Well interpretable coefficients. Multiresolution analyses Interpolation: Spline interpolation does not show oscillations — in contrast to interpolation with polynomials. Brigitte Forster (Passau) September 2013 14 / 47 2. B-splines Classical B-splines B = χ ; Bn = B − B for n 2; 1 [ 0;1 ] n 1 ∗ 1 ≥ n 1 e−i! B (!) = − n i! have a discrete indexing.b have a discrete order of approximation. are real-valued. have a symmetric spectrum. Brigitte Forster (Passau) September 2013 15 / 47 2. Complex B-splines Idea: Complex-valued B-splines defined in Fourier domain: z 1 e−i! B (!) = B (!) = − ; z α+iγ i! 1 where z C, withb parametersb Re z > , Im z R. 2 2 2 (BF, Blu, Unser. ACHA, 2006) Brigitte Forster (Passau) September 2013 16 / 47 2. Complex B-splines Why is this definition reasonable and useful? z 1−e−i! z Bz (!) = Bα+iγ(!) = i! = Ω(!) is well defined, since Ω(R) R− = . b b \ ; z Re z −Im z arg(Ω(!)) Bz (!) = Ω(!) = Ω(!) e . j j j j j j 6 b 5 Im 4 0.5 3 2 Re 0.5 1 -0.5 -15 -10 -5 5 10 15 Ω: R C B3+iγ for γ = 0; 1; 2. ! j j Approximately single-sided frequencyb analysis! Brigitte Forster (Passau) September 2013 17 / 47 2. Complex B-splines What do they look like? Representation in time-domain 1 k z z−1 Bz (t) = ( 1) (t k) Γ(z) − k − + kX≥0 pointwise for all t R and in the L2(R) norm. 2 —— Compare: The cardinal B-spline Bn, n N, has the representation 2 n 1 k n n−1 Bn(t) = ( 1) (t k) (n 1)! − k − + − k=0 1 X 1 n = ( 1)k (t k)n−1: Γ(n) − k − + kX=0 Brigitte Forster (Passau) September 2013 18 / 47 2. Complex B-splines Complex B-splines Bz are piecewise polynomials of complex degree. Smoothness and decay tunable via the parameter Re z. Recursion: Bz Bz = Bz +z 1 ∗ 2 1 2 They are scaling functions and generate multiresolution analyses and wavelets. Simple implementation in Fourier domain Fast algorithm. ! Nearly optimal time frequency localization. Relate fractional or complex differential operators with difference operators. (Here Re z; Re z1; Re z2 > 1.) Brigitte Forster (Passau) September 2013 19 / 47 2. Interpolation with complex B-splines How about interpolation with complex B-splines? Some interpolating complex splines for Re z = 4: (Real and imaginary part) 1.0 0.15 0.8 0.10 0.6 0.05 0.4 -6 -4 -2 2 4 6 - 0.2 0.05 -0.10 -6 -4 -2 2 4 6 -0.15 Joint work with Peter Massopust, Ramunas¯ Garunkstis,ˇ and Jorn¨ Steuding. (Complex B-splines and Hurwitz zeta functions. LMS Journal of Computation and Mathematics, 16, pp. 61–77, 2013.) Brigitte Forster (Passau) September 2013 20 / 47 2. Interpolation with complex B-splines Aim: Interpolating splines Lz of fractional or complex order. Interpolation problem: (z) Lz (m) = c Bz (m k) = δm;0; m Z; k − 2 Xk2Z Fourier transform: B (!) 1=!z L (!) = z = ; Re z 1: z 1 Bz (! + 2πk) ≥ z k2 b (! + 2πk) b XZ k2Z b X Question: When is the denominator well-defined? Brigitte Forster (Passau) September 2013 21 / 47 2. Interpolation with complex B-splines Rescale denominator. 1=!z Lz (!) = ; Re z 1 1 ≥ (! + 2πk)z b 2 Xk Z Let ! = 0 and a := !=2π. 6 1 1 1 1 1 = + e±iπz (k + a)z (k + a)z (k + 1 a)z Xk2Z kX=0 kX=0 − = ζ(z; a) + e±iπz ζ(z; 1 a): − This is a sum of Hurwitz zeta functions ζ(z; a), 0 < a < 1. Brigitte Forster (Passau) September 2013 22 / 47 2. Interpolation with complex B-splines Theorem: (Spira, 1976) If Re z 1 + a, then ζ(z; a) = 0. ≥ 6 Thus: For 0 < a < 1 and for σ R with σ 2 and σ = 2N + 1: 2 ≥ 2 ζ(σ; a) + e±iπσ ζ(σ; 1 a) = 0 : − 6 Brigitte Forster (Passau) September 2013 23 / 47 2. Interpolation with complex B-splines 1 0.5 2 4 6 8 -0.5 Interpolating spline Lσ in time domain for σ = 2:0; 2:1;:::; 2:9. σ = 2: Classical case of linear interpolation. Brigitte Forster (Passau) September 2013 24 / 47 2. Interpolation with complex B-splines Question: Are there complex exponents z C with Re z > 3 and Re z = 2N + 1, such that the denominator of the2 Fourier representation of the2 fundamental spline satisfies ζ(z; a) + e±iπz ζ(z; 1 a) = 0 − 6 for all 0 < a < 1. Consequence: The interpolation construction also holds for certain B-splines Lz of complex order z. Brigitte Forster (Passau) September 2013 25 / 47 2. Interpolation with complex B-splines Why was the problem on zero-free regions so resisting? ±iπz f±(z; a) := ζ(z; a) + e ζ(z; 1 a) − Involves the Hurwitz zeta function aside the “interesting domain” 1 around Re z = 2 . Rouche’s´ theorem seems to be suitable to find zeros, but not to exclude them. The problem is that the estimates have to be uniform in a. Numerical examples fail, because the Hurwitz zeta function is implemented via meromorphic approximations.
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