The Cascading Haar Wavelet algorithm for computing the Walsh-Hadamard Transform

Transcription

1 The Cascading Haar Wavelet algorithm for computing the Walsh-Hadamard Transform Andrew Thompson (University of Oxford) Global Conference on Signal and Information Processing, Montreal, Canada 15 November 2017

2 The Walsh-Hadamard Transform (WHT) The WHT is a staple of the signal processing world. It is widely used......in coding in wireless communications...in image processing...as a proxy for the Fast Fourier Transform (FFT). Cascading Haar Wavelet algorithm Andrew Thompson 2 / 15

3 The Walsh-Hadamard Transform (WHT) The WHT is a staple of the signal processing world. It is widely used......in coding in wireless communications...in image processing...as a proxy for the Fast Fourier Transform (FFT). Givenx R n wheren = 2 m, its WHT (with coefficients in dyadic/paley order) is equal toh m x, whereh m is a Walsh-Hadamard matrix defined by the recursion H 0 := 1; H r+1 = 1 2 [ Hr ( 1 1 ) H r ( 1 1 ) ] for r 0. Cascading Haar Wavelet algorithm Andrew Thompson 2 / 15

4 The Walsh-Hadamard Transform (WHT) The WHT is a staple of the signal processing world. It is widely used......in coding in wireless communications...in image processing...as a proxy for the Fast Fourier Transform (FFT). Givenx R n wheren = 2 m, its WHT (with coefficients in dyadic/paley order) is equal toh m x, whereh m is a Walsh-Hadamard matrix defined by the recursion H 0 := 1; H r+1 = 1 2 [ Hr ( 1 1 ) H r ( 1 1 ) ] for r 0. Divide-and-conquer algorithms (e.g. Cooley-Tukey) exist which require exactlynlog 2 n operations. Cascading Haar Wavelet algorithm Andrew Thompson 2 / 15

5 The Walsh-Hadamard Transform (WHT) H 6 : blue= +, red= Cascading Haar Wavelet algorithm Andrew Thompson 3 / 15

6 The Haar wavelet transform The Haar wavelet transform is the oldest and simplest discrete wavelet transform (Haar, 1910). Cascading Haar Wavelet algorithm Andrew Thompson 4 / 15

7 The Haar wavelet transform The Haar wavelet transform is the oldest and simplest discrete wavelet transform (Haar, 1910). Givenx R m wheren = 2 m, its Haar wavelet transform is equal toψ m x whereψ m is defined by the recursion Ψ 0 := 1; Ψ r+1 = 1 [ Ψr ( 1 1 ) ] 2 I r ( 1 1 ) for r 0. Cascading Haar Wavelet algorithm Andrew Thompson 4 / 15

8 The Haar wavelet transform The Haar wavelet transform is the oldest and simplest discrete wavelet transform (Haar, 1910). Givenx R m wheren = 2 m, its Haar wavelet transform is equal toψ m x whereψ m is defined by the recursion Ψ 0 := 1; Ψ r+1 = 1 [ Ψr ( 1 1 ) ] 2 I r ( 1 1 ) for r 0. It requires 2n 2 operations. Cascading Haar Wavelet algorithm Andrew Thompson 4 / 15

9 The Haar wavelet transform H 6 : blue= +, red= Cascading Haar Wavelet algorithm Andrew Thompson 5 / 15

10 Known connections between WHT and Haar Computing the WHT ofxis equivalent to computing mini-whts of the coefficients in each scale of the Haar wavelet transform ofx(fino 1972, Falkowski/Rahardja 1996). 1 H 0 H m Ψ T m = H 1... Hm 1 Cascading Haar Wavelet algorithm Andrew Thompson 6 / 15

11 Known connections between WHT and Haar Computing the WHT ofxis equivalent to computing mini-whts of the coefficients in each scale of the Haar wavelet transform ofx(fino 1972, Falkowski/Rahardja 1996). 1 H 0 H m Ψ T m = H 1... Hm 1 This gives an alternative algorithm for computing the WHT: via a detour into the Haar wavelet domain, also requiringnlog 2 n operations. Cascading Haar Wavelet algorithm Andrew Thompson 6 / 15

12 Known connections between WHT and Haar Computing the WHT ofxis equivalent to computing mini-whts of the coefficients in each scale of the Haar wavelet transform ofx(fino 1972, Falkowski/Rahardja 1996). 1 H 0 H m Ψ T m = H 1... Hm 1 This gives an alternative algorithm for computing the WHT: via a detour into the Haar wavelet domain, also requiringnlog 2 n operations. The result also describes the asymptotic mutual coherence of the Walsh-Hadamard and Haar bases. Cascading Haar Wavelet algorithm Andrew Thompson 6 / 15

13 A new decomposition formula Theorem (Thompson 2017): H m = { m 1 r=1 I r 1 [ Im r 0 0 Ψ m r ] } Ψ m, for m 1. Cascading Haar Wavelet algorithm Andrew Thompson 7 / 15

14 A new decomposition formula Theorem (Thompson 2017): H m = { m 1 r=1 I r 1 [ Im r 0 0 Ψ m r ] } Ψ m, for m 1. The WHT can be computed by first computing the Haar wavelet transform, and then employing a divide-and-conquer approach also consisting of Haar wavelet transforms. Cascading Haar Wavelet algorithm Andrew Thompson 7 / 15

15 A new decomposition formula Expanding the product: H m = I 1 Ψ 1 I m 2 Ψ m 2 I 1 Ψ 1 I m 2 Ψ m 2... I1 [ Im 1 Ψ 1 Ψ m 1 ] Ψ m, Cascading Haar Wavelet algorithm Andrew Thompson 8 / 15

16 The Cascading Haar Wavelet (CHW) algorithm Cascading Haar Wavelet algorithm Andrew Thompson 9 / 15

17 Complexity Theorem (Thompson 2017): The CHW algorithm can be implemented in nlog 2 n operations. Cascading Haar Wavelet algorithm Andrew Thompson 10 / 15

18 Complexity Theorem (Thompson 2017): The CHW algorithm can be implemented in nlog 2 n operations. Proof: The CHW algorithm requires a single Haar wavelet transform of size2 m, and2 m 1 r Haar wavelet transforms of size2 r, forr = 1,2,...,m 1. Cascading Haar Wavelet algorithm Andrew Thompson 10 / 15

19 Complexity Theorem (Thompson 2017): The CHW algorithm can be implemented in nlog 2 n operations. Proof: The CHW algorithm requires a single Haar wavelet transform of size2 m, and2 m 1 r Haar wavelet transforms of size2 r, forr = 1,2,...,m 1. The total number of operations is therefore 2(2 m 1)+ = 2 m+1 2+ m 1 r=1 m 1 r=1 { 2 m 1 r 2(2 r 1) } 2 m m 1 r=1 2 m r = 2 m m (m 1) 2(2 m 1 1), which simplifies tom 2 m = nlog 2 n. Cascading Haar Wavelet algorithm Andrew Thompson 10 / 15

20 Proposal for a parallel implementation Collapse by removing the identity transformations a cascade of Haar wavelet transforms... Cascading Haar Wavelet algorithm Andrew Thompson 11 / 15

21 Proposal for a parallel implementation There is a natural parallelization in which each ofm 1nodes is devoted to the task of performing Haar wavelet transforms of a certain size. Here illustrated form = 4. Cascading Haar Wavelet algorithm Andrew Thompson 12 / 15

22 Features of the proposed parallelization Fixed tasks: Each node only needs to be programmed once to do a single fixed task. Cascading Haar Wavelet algorithm Andrew Thompson 13 / 15

23 Features of the proposed parallelization Fixed tasks: Each node only needs to be programmed once to do a single fixed task. Distributed memory: Nodes pass on their output to other nodes by predetermined rules, with no need for shared memory. Cascading Haar Wavelet algorithm Andrew Thompson 13 / 15

24 Features of the proposed parallelization Fixed tasks: Each node only needs to be programmed once to do a single fixed task. Distributed memory: Nodes pass on their output to other nodes by predetermined rules, with no need for shared memory. Asynchronous: Synchronization occurs automatically, when each node has received all of its inputs. It could therefore be implemented by a circuit which is not governed by a global clock. Cascading Haar Wavelet algorithm Andrew Thompson 13 / 15

25 Summary A novel algorithm is proposed for computing the WHT which involves a cascade of Haar wavelet transforms. Cascading Haar Wavelet algorithm Andrew Thompson 14 / 15

26 Summary A novel algorithm is proposed for computing the WHT which involves a cascade of Haar wavelet transforms. Its serial complexity is identical to the classical Cooley-Tukey algorithm. Cascading Haar Wavelet algorithm Andrew Thompson 14 / 15

27 Summary A novel algorithm is proposed for computing the WHT which involves a cascade of Haar wavelet transforms. Its serial complexity is identical to the classical Cooley-Tukey algorithm. There is a natural way to parallelize the algorithm which has a number of potentially beneficial features. Cascading Haar Wavelet algorithm Andrew Thompson 14 / 15

28 References Falkowski, B. and Rahardja, S. Walsh-like functions and their relations. IEEE Proceedings on Vision, Image and Signal Processing (1996). Fino, B. Relations Between Haar and Walsh/Hadamard Transforms. Proceedings of the IEEE (1972). Haar, A. Zur Theorie der orthogonalen Funktionensysteme. Mathematische Annalen (1910). Thompson, A. The Cascading Haar Wavelet algorithm for computing the Walsh-Hadamard Transform. Signal Processing Letters (2017). Cascading Haar Wavelet algorithm Andrew Thompson 15 / 15