Compensating for Quasi-periodic Motion in Robotic Radiosurgery
Compensating for Quasi-periodic Motion in Robotic Radiosurgery outlines the techniques needed to accurately track and compensate for respiratory and pulsatory motion during robotic radiosurgery. The algorithms presented within the book aid in the treatmen
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Mathematical Addenda
A.1 The a` trous wavelet transform The basic idea of this transform, called the a` trous wavelet decomposition [6, 10–12, 17], is to iteratively convolve the input signal with an increasingly dilated wavelet function. By doing so, the original signal is transformed into a set of band-pass filtered components, so-called scales W j , and the residual or continuum cJ . We first have to choose a scaling function φ and an associated discrete low-pass filter h such that φ satisfies the dilation equation ∞ 1 x
φ = ∑ h(n)φ (x − n) . 2 2 n=−∞
If we now assume that there is a function f (x) representing the signal we want to decompose, we can define how the successively smoothed signals c j are computed using the scaling function φ . Additionally, let c0,k = f (k). + , x−k 1 , j>0 (A.1) c j,k = j f (x), φ 2 2j The above equation can be rewritten as c j+1,k =
∞
∑
h(n)c j,k+2 j n ,
j>0
(A.2)
n=−∞
since we work with discrete signals. Using these differently smoothed versions of the signal, we can compute the wavelet coefficients W j by means of W j+1,k = c j,k − c j+1,k ,
j ≥ 0.
(A.3)
It is also possible to directly compute the wavelet coefficients by using the wavelet function ψ derived from the scaling function φ :
F. Ernst, Compensating for Quasi-periodic Motion in Robotic Radiosurgery, DOI 10.1007/978-1-4614-1912-9, © Springer Science+Business Media, LLC 2012
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A Mathematical Addenda
ψ(x) = 2φ (2x) − φ (x),
1 W j,k = j 2
+
f (x), ψ
x−k 2j
,
By now deciding on a maximum level of decomposition J, say, we can deduce the reconstruction formula of the wavelet decomposition: J
yk = c0,k = cJ,k + ∑ W j,k
(A.4)
j=1
Performing the a` trous wavelet decomposition now boils down to selecting a suitable scaling function. The main limitation in our case is the inherent non-symmetricity of time: our time series is finite since we want to do real time processing of the measured input. This means that, if our current position in time is N, for computing the values c j,N and W j,N , we cannot use values f (k) with k > N. One thing that is commonly done to perform wavelet decomposition of finite signals is to add zeros to the end of the signal or to mirror the signal at its end. These methods, however, are not feasible for us since they add artefacts at the signal’s most important part. Consequently, the only wavelets we can actually use are those with one-sided support. The easiest such wavelet is the Haar wavelet. This is important since we want to process an incoming signal in real time. The corresponding functions φ , ψ and h for the Haar wavelet are given in equation A.5 and are plotted in figure A.1. φ (x) =
1 −1 < x ≤ 0 , 0 else
⎧ ⎨ 1 − 12 < x ≤ 0 ψ(x) = −1 −1 < x ≤ − 12 , ⎩ 0 else
h(x) =
1
2 −1 ≤ x ≤ 0 0 else
(A.5)
1.5
1.5
1
1
1
0.5
0.5
0.5
1.5
0
0
0
−0.5
−0.5
−0.5
−1
−1
−1
−1.5 −2
−1.5 −2
−1.5 −2
−1.5
−1
−0.5
0 φ(x)
0.5
1
1.5
(a) Haar scaling function φ (x)
2
−1.5
−1
−0.5
0 ψ(x)
0.5
1
(b) Haar wavelet ψ(x)
1.5
2
−1.5
−1
−0.5
0 h(x)
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