The inverse problem of a Gaussian convolution and

Home

Search

Collections

Journals

About

Contact us

My IOPscience

The inverse problem of a Gaussian convolution and its application to the finite size of the measurement chambers/detectors in photon and proton dosimetry

This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2003 Phys. Med. Biol. 48 707 (http://iopscience.iop.org/0031-9155/48/6/302) View the table of contents for this issue, or go to the journal homepage for more

Download details: IP Address: 132.187.191.1 The article was downloaded on 10/10/2011 at 15:04

Please note that terms and conditions apply.

INSTITUTE OF PHYSICS PUBLISHING

PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 48 (2003) 707–727

PII: S0031-9155(03)54301-9

The inverse problem of a Gaussian convolution and its application to the finite size of the measurement chambers/detectors in photon and proton dosimetry W Ulmer and W Kaissl VARIAN Medical Systems, Neuenhoferstr. 107 CH-5400 Baden, Switzerland

Received 3 October 2002 Published 5 March 2003 Online at stacks.iop.org/PMB/48/707 Abstract A Gaussian convolution kernel K is deduced as a Green’s function of a Lie operator series. The deconvolution of a Gaussian kernel is developed by the inverse Green’s function K −1 . A practical application is the deconvolution of measured profiles Dm (x) of photons and protons with finite detector size to determine the profiles Dp (x) of point-detectors or Monte Carlo Bragg curves of protons. The presented algorithms work if Dm (x) is either an analytical function or only given in a numerical form. Some approximation methods of the deconvolution are compared (differential operator expansion to analytical adaptations of 2 × 2 cm2 and 4 ×4 cm2 profiles, Hermite expansions to measured 6 × 6 cm2 and 20 × 20 cm2 profiles and Bragg curves of 80/180 MeV protons, FFT to an analytical 4 × 4 cm2 profile). The inverse problem may imply ill-posed problems, and, in particular, the use of FFT may be susceptible to them.

1. Introduction Based on the normalized Gaussian convolution kernel K(s, u − x) = K(s, x − u) = (sπ 1/2 )−1 exp(−(u − x)2 /s 2 ) the integral transform

(1)




Dm (x) =

K(s, u − x)Dp (u) du

(1a)

is considered in various problems of applied physics and dosimetry, e.g. calculation of transverse profiles of photon, proton and electron beams, Monto Carlo calculations to smooth statistical fluctuations, inclusion of the energy straggling of proton beams, connection between dose profiles and the size of the measurement chamber/detector. Equation (1a) defines a correspondence between the source (input) function Dp (x) and image (output) function Dm (x), which is broadened by kernel (1) according to the characteristic parameter s. 0031-9155/03/060707+21$30.00

© 2003 IOP Publishing Ltd Printed in the UK

707

708

W Ulmer and W Kaissl

By taking lim s ⇒ 0 the kernel K(s, u − x)  assumes δ(u − x), and equation (1a) takes the shape of a δ-distribution yielding Dm (x) = δ(u−x)Dp (u) du = Dp (x). A modified version of kernel (1) resulting from the substitution s 2 ⇒ 2s 2 and yielding K(s, u − x) = s −1 (2π)−1/2 exp(−(u − x)2 /2s 2 )

(1b)

is also usual in the literature. The inverse problem appears if one wants to perform a reverse transform from Dm (x) to the initial profile Dp (x), which is not broadened by a Gaussian convolution. The inverse kernel K −1 is derived in this paper and subjected to some practical problems of dosimetry. In order to include the influence of the finite extension of detectors and ionization chambers, the convolution problem of profile functions has been considered in many investigations in the past decade (Chang et al 1996, Charland et al 1998, Garcia-Vicente et al 1998, 2000, Higgins et al 1995, Metcalfe et al 1993, Sibata et al 1991). With respect to the appropriate convolution kernel one should, in particular, expect a finite boundary kernel, which is restricted to the detector volume and vanishes outside. However, experimental investigations (Garcia-Vicente et al 1998) showed that the assumption of a Gaussian convolution kernel for the inclusion of the influence of the finite detector size is much more adequate than the restriction to a fixed volume with respect to the convolution. At first glance, this fact might appear to be surprising, because a Gaussian has a tail outside the detector wall. But one should take into account that ionization chambers or detectors cannot only be considered as passive systems, which receive signals from the immediate environment. They interact with this environment, because the photons, protons, electrons, etc of the radiation beam also are scattered by the chambers/detectors (in particular, at the wall) without any production of dose signals, which can be recorded by the measurement system. As a result there is a correspondence between the parameter s of kernel (1) and the detector/chamber size. Thus, the use of a Gaussian convolution for the description of a measured dose profile Dm (x) seems to be justified. On the other side, the task is that the influence of the convolution may be removed by an appropriate deconvolution in order to get an approach to the dose distribution Dp (x) provided by a perfect passive point-detector. 2. Methods 2.1. Determination of dose profiles Measured transverse dose profiles of 6 MV and 18 MV photon beams recorded by the Farmer ionization chambers with sensitive volumes of 0.6 cm3 have been made available by VARIAN Medical Systems, Palo Alto (USA) and of 0.7 cm3 by the Strahlentherapiezentrum Raschplatz/Klinikum Siloah, Hannover (Germany). Calculations of monoenergetic proton beams have been performed with the Monte Carlo codes GEANT-Fluka 1997 and GEANTFluka 2000 within the framework of a proton project (Ulmer et al 2000). 2.2. Mathematical aspects of a Gaussian kernel K(s, u − x) and its reverse kernel K −1 (s, u − x) According to the arguments of the previous section it is assumed (see e.g. Garcia-Vicente et al 1998, 2000) that between point detection of a profile Dp (x) and the measured profile Dm (x) equations (1), (1a) hold. The parameter s stands in a close relationship with the chamber radius rch . However, this specification of Dp (x) and Dm (x) is only considered with regard to the applications of this paper. In general, Dm (x) and Dp (x) may refer to the input/output

The inverse problem of a Gaussian convolution and its application

709

functions with rather different practical applications. Equations (1) and (1a) result from the differential equation ADm (x) = exp(−0.25s 2 d2 /dx 2 )Dm (x) = Dp (x).

(2)

K(s, u − x) is referred to as a Green’s function of the operator A in equation (2). This operator A is defined as a Lie series A = 1 − s 2 d2 /dx 2 /4 · 1! + s 4 d4 /dx 4 /4 · 4 · 2! − · · · + · · · − · · · , which formally represents a Taylor expansion of the operators. The Lie series of operator functions is a generalization of the Heaviside calculus of operators. The inverse operator A−1 satisfying A−1 A = AA−1 = 1 is defined by A−1 Dp (x) = exp(0.25s 2 d2 /dx 2 )Dp (x) = Dm (x). By combining equation (3) with equation (1a) we obtain
Dm (x) = A−1 Dp (x) = exp(0.25s 2 d2 /dx 2 )Dp (x) = K(s, u − x)Dp (u) du.

(3)

(3a)

Equation (3a) provides a well-known property of Green’s functions, i.e. a connection between A, K and the δ-function, which is not restricted to operator A according to equation (2),

Dp (x) = A K(s, u − x)Dp (u) du = δ(x − u)Dp (u) du ⇒ AK(s, u − x) = δ(u − x). (4) With the help of the Fourier transform of the δ-function we obtain
−1 −1 −1 K(s, u − x) = A δ(u − x) = (2π) A exp(ik(u − x)) dk. The differentiations of exp(ik(u − x)) with respect to all the powers of A−1 yield
−1 K(s, u − x) = (2π) exp(−s 2 k 2 /4) exp(ik(u − x)) dk.

(5)

(6)

The integration boundaries of equations (1a), (3a), (4)–(6) are [−∞, +∞]. Equation (6) can readily be integrated and yields the normalized Gaussian kernel K(s, u − x) = exp(−(u − x)2 /s 2 )/(sπ 1/2 ). With regard to Dp (x) and Dm (x) this connection implies that if Dp (x) is recorded by a point detector, then the measured profile Dm (x) is broadened by a Gaussian convolution. It should be pointed out that there exists a different way to derive the kernel K(s, u − x) as a Green’s function K(A−1 ) of the operator A−1 via a spectral theorem of functional analysis (see e.g. von Neumann 1955). This way represents a specific case of the theory of reproducing kernels (Saitoh 2001). We consider the continuous spectrum of the eigenvalue equation A−1 ϕ(x) = exp(0.25s 2d2 /dx 2 )ϕ(x) = λϕ(x).

(7)

The solution function of equation (7) reads φ(x) = (2π)−1/2 exp(ikx) and the eigenvalue spectrum is given by λ(s, k) = exp(−0.25s 2 k 2 ). The application of the spectral theorem yields
K(A−1 ) = ϕ ∗ (x)ϕ(u)λ(s, k) dk
−1 exp(−0.25s 2k 2 ) exp(−ikx) exp(iku) dk = (2π) = exp(−(u − x)2 /s 2 )/(sπ 1/2 ).

(8)

710

W Ulmer and W Kaissl

2.2.1. Class C ∞ of smooth functions, integration of Lebesque-measurable functions and Banach space. Readers lacking interest in mathematical details may ignore this section. The application of equations (1), (1a), (2)–(6) has to take account of the definition areas, permitted classes of functions subjected to differentiations by A and A−1 and integral transforms by the kernel K(s, u − x). At first, we consider the class C ∞ (−∞, +∞) of smooth functions f (x) required by the differential operator expansions of A and A−1 . C ∞ implies the existence of derivations of arbitrary order 2n, n ∈ N. The definition area of D(A) = D(A−1 ) is the compact open set of the interval [−∞ < D < +∞]. At the margin M(D) each function f ∈ C ∞ (−∞, +∞) has to vanish, i.e. f (x) = 0, if |x| ⇒ ∞. C ∞ may also be bounded on any j finite domain j [a
nD
b], denoted by C (a, b). Therefore, the set of all finite polynomials Pj (x) = n=0 an x and convergent polynomial expansions (j ⇒ ∞, e.g. Taylor expansions, Bessel functions, hypergeometric series) of infinite order belong to this set of functions (the Weierstrass approximation theorem). With the help of the maximum norm f  = max |f | for each f ∈ C ∞ the class of smooth functions C ∞ satisfies the defining axioms of a Banach space LB . Thus, for f (x) and g(x) ∈ LB the operations Af = f1 and A−1 g = g1 imply that f1 and g1 ∈ LB , if max |f1 | and max |g1 | exist. We have examined by direct computation that by the mappings APj (x) = Tj (x) and A−1 P j (x) = T j (x) the relations hold: If P j (x) = Tj (x), then T j (x) = Pj (x) and max |Pj |
max |Tj | for arbitrary j and finite intervals [a, b]. The operators A and A−1 do not in every case require smooth functions. A rather trivial example is f (x) = |x| with A|x| = |x| and A−1 |x| = |x|, but a less trivial case is obtained by f (x) = exp(−|x|/β); A exp(−|x|/β) = exp(−|x|/β) exp(−s 2 /4β 2 ); A−1 exp(−|x|/β) = exp(−|x|/β) exp(s 2 /4β 2 ).

(9)

The latter example can be extended to f (x) = x 2n exp(−|x|/β), where operations with A and A−1 are applicable for arbitrary finite n (n = 0, 1, . . .), because both A and A−1 only contain derivations of even order, which provide continuity of the related derivatives at x = 0 throughout. Therefore, each specific determination of Dp (x) by Dp (x) = ADm (x) has to be examined, since the existence of ADm (x) or A−1 Dp (x) in the sense of the introduced Banach space LB is not ensured. The function f (x) = x 1/2 exp(−x 2 ) behavesin this manner. The question that arises is, whether A−1 f (x) is equivalent to f (u)K(s, u − x) du. This is, however, not true in general, because the integral operator formulation only requires integrability of the Lebesque-measurable functions. Some specific properties like smoothness, continuity, etc of f (x) may not hold. A practical example of dosimetry is given in section 5. The  set of all L-measurable functions f (x), denoted by L1 (M) and satisfying f (x) := |f (x)|  dx < ∞, also defines a Banach space. With regard to the integral transform g(x) = f (u)K(s, u − x) du the following relation holds: g
f  · K.

(10)

Since the Gaussian kernel K is normalized, i.e. K = 1, from relation (10) follows g
f . Some properties of relation (10) can be obtained by the inclusion of some special cases. Case 1: f (x) = β −1 (π)−1/2 exp(−x 2 /β 2 ) with f  = 1 ⇒ g(x) = (s 2 + β 2 )−1/2 (π)−1/2 exp(−x 2 /((s 2 + β 2 )) with g = 1 and hence g = f . If this condition (minimum norm) is satisfied, the whole situation is particularly convenient, because the existence of the inverse problem in every case exists and ill-posed tasks are excluded (Saitoh 1997, 1999, 2001). This minimum norm also exists, if the kernel K(s, subjected to the translation ∞u − x) is n n n n exp(−αd/dx)K(s, u − x) = K(s, u − x − α) = n=0 (−1) (α /n!)d K(s, u − x)/dx

The inverse problem of a Gaussian convolution and its application

711

with |α| < ∞. The latter expression is equivalent to a Hermite polynomial expansion (see sections 2.3 and 5). Again, using f (x) = b −1 (π)−1/2 exp(−x 2 /b2 ) we obtain g(x − α) = (s 2 + β 2 )−1/2 (π)−1/2 exp(−(x − α)2 /((s 2 + β 2 )) and g = f  = 1. Case 2: The input function f (x) = 0.5[erf((a − x)/τ ) + erf((a + x)/τ )] already results from a convolution of a box with a Gaussian kernel (Ulmer and Harder 1996), where erf(qs ) refers to the Gaussian error function defined by erf(qs ) = (2/π 1/2 ) exp(−q 2 ) dq (taken from 0 to qs ) and f  = 1. The image function g(x) assumes the shape g(x) = 0.5[erf((a − x)/τc ) + erf((a + x)/τc )] with τc = (τ 2 + s 2 )1/2 and g = 1 (minimum norm).  Case 3: Assume an input function f (x) = (2π)−1/2  h(x) exp(ikx) dk resulting from a Fourier transform and satisfying f  = h = 1, i.e.  |h(k)| dk < ∞. The convolution of f (x) with K(s, u − x) provides g(x) = (2π)−1/2 h(k) exp(−s 2 k 2 /4) exp(ikx) dk, and g = (2π)−1/2 |h(k)| exp(−s 2 k 2 /4) dk < h ⇒ g < f . Thus, the desired minimum norm is not obtained. The same behaviour is also true for discrete Fourier expansions. With respect to equation (1a) and its wide field of physical applications it would be interesting to determine further classes, where the minimum norm is established. 2.2.2. Approximate computation of Dp (x) or Dm (x) by differential operators. According to equation (2) the inverse relation reads Dp (x) = ADm (x) = exp(−0.25s 2 d2 /dx 2 )Dm (x). If Dm (x) is not given by analytic functions, the application of equation (2) is connected to a serious problem, since Dm (x) has to be subjected to differentiations. If we expand A in terms of powers A = exp(−0.25s 2d2 /dx 2 ) = 1 − 0.25s 2 d2 /dx 2 + · · · − · · · , then the consideration of Dp (x) = Dm (x) − 0.25s 2 d2 /dx 2 Dm (x)

(11)

2

should provide a good approximation, as far as s is small and the powers of the higher order are negligible. However, relation (11) is only applicable, if Dm (x) is at least twice differentiable. The measured profile Dm (x), which also contains effects like noise, discontinuities, etc, has to be refined by proper spline functions with continuity of the second derivative at least as required by relation (11). If we write Dp (x) = (1 − 0.25s 2 d2 /dx 2 )Dm (x) according to relation (11), then there also exits a Green’s function representation, which is a two-point Yukawa kernel. But the discussion may go beyond the scope of this paper. The question that arises is, when is equation (11) an acceptable approximation of Dp (x). For this purpose, we introduce dimensionless variables σ = s/ud , ξ = x/ud (ud : unit length, e.g. 1 cm) and normalize Dm by Dm (ξ ) ⇒ Dm (ξ )/ max Dm (ξ ). The rescaled equation (2) now reads ∞  (−1)n 4−n (n!)−1 σ 2n d2n Dm (ξ )/dξ 2n . (12) Dp (ξ ) = exp(−0.25σ 2d2 /dξ 2 )Dm (ξ ) = n=0

With the help of the assumption that for each n ∈ N and sup |d2n Dm (ξ )/ dξ 2n |
4n n! is valid, we can estimate equation (12) by a geometric series. Assuming sup |d2n Dm (ξ )/dξ 2n | = 4n n! and σ < 1 we obtain S(σ ) = 1 − σ 2 + σ 4 − · · · = 1/(1 + σ 2 ). If we put σ = 0.25, then S(0.25) = 0.941 176 71 and 1 − σ 2 = 0.9375, whereas for σ = 0.5 we get S(0.5) = 0.8 and 1−σ 2 = 0.75. By keeping derivations of the fourth order, σ = 0.5 yields 1−σ 2 +σ 4 = 0.8125 instead of S(0.5) = 0.8. Assuming that Dm (ξ ) is very slowly bending/increasing, i.e. sup |d2n Dm (ξ )/dξ 2n | = 10−n 4n n! is lowered by 10−n , then the convergence of the series can be established by σ < 10. Thus, for measured profiles Dm (ξ ) in a finite interval D(a, b) and refined by spline functions the behaviour of the derivatives sup |d2n Dm (ξ )/dξ 2n | provides

712

W Ulmer and W Kaissl

the criterion, whether a finite order of differential operators is applicable for the calculation of Dp (ξ ). If a piecewise approximation of measured profiles Dm (x) by spline functions is performed, the non-vanishing derivatives indicate the applicability of the differential operator approach. With respect to the equation A−1 Dp (x) = Dm (x) we can perform a similar convergence estimation and define sup |d2n Dp (ξ )/dξ 2n | = 4n n!. Now the corresponding geometric series assumes S(σ ) = 1 + σ 2 + σ 4 + · · · = 1/(1 − σ 2 ) with σ < 1 to establish convergence. The essential difference is that S(σ ) behaves rather conversely with respect to equal conditions: 1
S(σ ) < ∞. Then the consideration of the maximum norm provides max|Dm (x)|
max|Dp (x)| in the sense of a Banach space. 2.2.3. Determination of the inverse kernel K −1 via Fourier transforms and use of FFT. In many problems one wants to perform calculations with the inverse relation of K, i.e.
Dp (x) = K −1 (s, u − x)Dm (u) du. (13) One solution valid for smooth functions and polynomials is given in equation (2). Using the Fourier transform of the δ-function and equation (2) we get


Dm (u) exp(ik(x − u)) du dk Dm (x) = Dm (u)δ(x − u) du = (2π)−1

Dp (x) = ADm (x) = (2π)−1 Dm (u) exp(−0.25s 2d2 /dx 2 ) exp(ik(x − u)) du dk. The evaluation of exp(−0.25s 2 d2 /dx 2 ) exp(ik(x − u)) yields exp(0.25s 2 k 2 ) exp(ik(x − u)). Therefore, we obtain

Dm (u) exp(−0.25s 2 d2 /dx 2 ) exp(ik(x − u)) du dk Dp (x) = (2π)−1 = (2π)−1



Dm (u) exp(0.25s 2 k 2 ) exp(ik(x − u)) du dk.

(14)

Thus, the desired result would be: the measured signal profile Dm (x) obtained by the finite size of an ionization chamber/detector, which is rather different from the behaviour of a point detector, results in a Gaussian convolution (1a). The inverse problem, basically the calculation of a signal distribution, which a point detector should register, should be given by equation (14). If one compares both these relations (1a) and (14), one finds that with respect to a Gaussian convolution the inequality valid for L[−∞, +∞] integrable functions Dm (x)
Dp (x) · K(s, u − x) always holds, if the norm of Dp (x) exists. The converse need not be true, if the integration boundaries of the integral (14) are infinite [−∞, +∞]. This difficulty results from the term exp(0.25 s 2 k 2 ) in equation (14), which has the consequence that Dm (x) has to vanish sufficiently fast so that the inequality (10) can be taken into account. If lim s ⇒ 0 in equation (14) is performed, then we obtain the identity Dp (x) = Dm (x). A good example is Dm (u) = exp(−u2 /β 2 ). The integration of equation (14) over u yields
√ exp(−u2 /β 2 ) exp(iku) du = (β/2 (π)) exp(−β 2 k 2 /4).  In order to perform the further integration exp(−k 2 /4β 2 ) exp(0.25s 2 k 2 ) exp(−ikx) dk according to equation (14), the inequality β 2 − s 2 > 0 must be satisfied. The formulation of the inverse kernel K −1 by Fourier transforms in equation (14) is a typical case of a

The inverse problem of a Gaussian convolution and its application

713

potentially ill-posed problem (Saitoh 1997, 1999), because the existence of the integral (14) requires enormous restrictions. However, only for finite integration boundaries [a, b] with a  −∞ and b  ∞ of the function Dm (u), the reverse problem Dm (x) ⇒ Dp (x) according to equation (14) can be established. This fact may enable and simplify considerably the applicability of equation (14), but it remains rather unclear, what happens with regard to the expression exp(0.25s 2 k 2 ) exp(ik(x −u)), when Dm (x) does not vanish sufficiently fast and oscillations might be amplified and therefore the applicability of equation (14) is questionable. Since the application of FFT has become a common tool in image processing, the practical use of equation (14) can be considerably simplified with regard to computational methods. Therefore, equation (14) may also be read as sums of Fourier series. In all practical cases it is useful to replace the complex Fourier transforms by discrete sums with finite boundaries. If we consider symmetric profiles then all summations can be reduced to the corresponding half-plane:    Dm (un ) cos(kn un ) cos(kj xj ) exp 0.25s 2 kn 2 un kj . (15) Dp (x) = 2(2π)−1 n

j

In the following section we will consider an example of a transverse photon profile to demonstrate FFT with regard to the reverse transform of a Gaussian convolution. At present,  we do not know any threshold values s or upper bounds for kn of the expression exp 0.25s 2 kn 2 , which turn equation (15) into an ill-posed problem. A further problem is the inverse transform of polynomial functions Dm (u) = P (u), which are neither absolutely integrable nor square-integrable within infinite intervals [−∞, +∞]. A simple example is Dm (u) = u and

1/2 −1 (16) u exp(−(u − x)2 /s 2 ) du = x. uK(s, u − x) du = (sπ ) Thus the reverse transform of Dm (x) is easy to obtain via the relation exp(−0.25s 2 d2 /dx 2 )x = [1 − 0.25s 2 d2 /dx 2 + · · · −]x = x

(16a)

since the second and higher derivatives of Dm (x) = x vanish. However, of the evaluation u exp(0.25s 2 k 2 ) the inverse transform according to formula (14) Dp (x) = (2π)−1 exp(ik(x − u)) du dk implies the consideration of the residuum theorem (see relations (18) and (19) to obtain x (or a polynomial P (x)) again on the left-hand side. In the case of the polynomials, i.e. Dm (x) = Pj (x) is a polynomial of the j th order: Pj ∈ C j (a, b), the immediate use of relation (3a) is much more apparent, since according to this relation we obtain Dp (x) = Pj (x) − 0.25s 2 d2 Pj (x)/dx 2 + · · · −  (−1)n (1/4)n(1/n!)s 2n d2n Pj (x)/dx 2n = Pj (x) +

(n = 1, . . . , j + 1).

(17)

n

In relation (17), the sum has to be taken from 1 to that order, where the 2n-derivative will vanish: 2n = j + 1. The use of the residuum theorem arises, if we evaluate relation (14) with regard to the polynomials and partial integration. Let Dm (u) = uj , then the partial integration over u within a finite interval [−L
u
+L] leads to
uj exp(−iku) du = (−ik)−1 [Lj exp(−ikL) − (−L)j exp(ikL)]
− j (−ik)−1 uj −1 exp(iku) du. (18) The partial integration has to be recursively repeated until the power of u vanishes. This procedure will provide poles from k −1 to k −j −1 . Lim L ⇒ ∞ may be applied after integrations

714

W Ulmer and W Kaissl

with the residuum theorem have been carried out, but it has to be assumed that |exp(−ikl)| = 1 and |exp(ikl)| = 1 by taking k ⇒ 0 and L ⇒ ∞:  
 (k −j aj (L) exp(−ikL) − k −j bj (L) exp(ikL) dk (1/2π)−1 exp(ikx) exp(0.25s 2 k 2 ) j

(j = 1, . . . , j + 1).

(18a)

The coefficients aj and bj result from partial integrations of relation (18). Since we have poles k −j at k = 0 the residuum theorem (see e.g. Withaker and Watson 1952) assumes the shape
 Re sj (f (k ⇒ 0)) f (k) dk = 2πi j





Re sj f (k ⇒ 0) = lim(k ⇒ 0)dj −1 (k j f (k))/dk j −1 ((j − 1)!)−1 .

(18b)

The differentiations according to k will provide polynomials in x. The function f (k) according to equation (18) is given by    f (k) = exp(ikx) exp(0.25s 2 k 2 ) (k −j aj (L) exp(−ikL) − k −j bj (L) exp(ikL) 2π. j

(19) In practical situations, where Dm (x) is non-zero only within finite intervals, the application of the residuum theorem can be avoided and, moreover, the measured profiles Dm (x) are, in general, not given as analytic polynomials. If s is a small parameter (s
0.25 cm) and the function Dm (x) is only non-vanishing in a very narrow domain, then by the expansion exp(0.25k 2s 2 ) = 1 + 0.25k 2s 2 + 0 (terms of higher order) the approximation may be considered:

−1 Dp (x) ≈ (2π) Dm (u)(1 + 0.25s 2 k 2 ) exp(ik(x − u)) du dk = Dm (x) + (2π)−1



Dm (u)(0.25s 2 k 2 ) exp(ik(x − u)) du dk. (20)

Since the criterion is unclear when Dm (u) is sufficiently smooth within a given narrow interval and oscillations by increasing the values of k may become awkward, we have not applied this approximation. 2.3. Formulation of the inverse problem (deconvolution) by Hermite polynomials If K −1 (s, x − u) can be constructed, the inverse formulation of formulae (1) and (1a) is determined by equation (13). As already pointed out the differential operator relation Dp (x) = ADm (x) and the Fourier transform (14) represent possible methods of solving the inverse task (13). It is also possible to construct K −1 (s, x − u) by means of two different Hermite expansions. Using relation (4), i.e. AK(x − u) = δ(x − u), we are able to perform an operator iteration A2 K(s, x − u) = Aδ(x − u).

(21)

This iteration is permitted, since the derivations of δ(x − u) are defined as distributions. Then

The inverse problem of a Gaussian convolution and its application

we compare the following identities:

715




ADm (x) = Dp (x) =

Aδ(x − u)Dm (u) du


=

A2 K(x − u)Dm (u) du


=

K −1 (s, x − u)Dm (u) du.

(22)

The iterated operator A2 is given by A2 = exp(−0.25s 2d2 /dx 2 ) × exp(−0.25s 2 d2 /dx 2 ) = exp(−0.5s 2 d2 /dx 2 ).

(23)

Thus the reversal transform to the Gaussian convolution kernel assumes the shape K −1 (x − u) = A2 K(s, x − u) = Aδ(x − u).

(24)

We may summarize the both results K(s, u − x) = (sπ 1/2 )−1 exp(−(u − x)2 /s 2 )  (−1)n s 2n (n!)−1 2−n d2n K −1 (s, u − x) = (sπ 1/2 )−1

(25)

n

× exp(−(u − x)2 /s 2 )/dx 2n

(n = 0, . . . , ∞).

The first term (n = 0) is the normalized Gaussian itself, the second term (n = 1) is given by the second derivative of a Gaussian, etc. According to the definition of Hermite polynomials (see Abramowitz and Stegun 1970) and their application in generalized convolutions in nonlinear problems of quantum mechanics (Ulmer 1980) the order of the derivative of a Gaussian provides a corresponding Hermite polynomial. Since only derivations of even order appear in equation (25), we can summarize the whole inverse problem by  K −1 (s, x − u) = cn (s)H2n ((x − u)/s)K(s, x − u) (n = 0, . . . , ∞). n


Dp (x) =

K

−1

(26)

(s, x − u)Dm (u) du.

The coefficients cn are determined by cn = (−1)n s 2n /(2n n!). This formulation has the advantage that Hermite polynomials represent an orthogonal set of functions and the minimal norm according to the inequality (10) can also be satisfied by the inverse relation, if Dm (x) of the function Dm (x) exists. Thus, the deconvolution of polynomial functions Pj (u) = Dm (u) with the help of equation (26) is not connected to any serious problem, and the use of the residuum theorem should  not have to be considered.  Please  note the following property: If Dm (u) = H0 =1, then H0 K −1 (s, x − u) du = H0 n cn (s)H2n ((x − u)/s)K(s, x − u) du = 1, hence H0 H2n ((x − u)/s)K(s, x − u) du = 0 (if n = 0). A modified notation of formula (26) is obtained in the following way. Based on equation (4), we are able to write AK(s, u − x) = δ(u − x) ⇒ A−1 K −1 (s, u − x) = δ(u − x). The relation AK(s, u − x) = δ(u − x) provides the correspondence  s 2n 22n (n!)−1 H2n ((x − u)/s)K(s, x − u) = δ(x − u).

(27)

(27a)

n

According to equation (24) K −1 results from K −1 = A2 K. By multiplication of K −1 with

716

W Ulmer and W Kaissl

A−1 we can verify that relation (4) is also valid for K −1 . Now we decompose the inverse operator A−1 by A−1 = 1 + A−1 rest and replace K −1 by the expansion (25). Then we obtain K −1 (s, u − x) = δ(u − x) − A−1 rest (K −1 (s, u − x)) = δ(u − x) − A−1 rest (A2 K(s, u − x)). −1



−1

(28) −1 2n

A rest is determined by A rest = n s (4 n!) d /dx (n = 1, . . . , ∞). The evaluation of relation (28) requires the collection of all terms with an identical order of the related derivation. The result is  (−1)n s 2n (2n − 1)(4n n!)−1 d2n K(s, u − x)/dx 2n K −1 (s, u − x) = δ(u − x) + 2n

n

2n

n

(n = 1, . . . , ∞).

(29) −1

If we use this representation of K (s, u − x), the reverse transform is given by
 Dp (x) = Dm (x) + (−1)n s 2n (2n − 1)(4n n!)−1 d2n K(s, u − x)/dx 2n Dm (u) du n

(n = 1, . . . , ∞)
 = Dm (x) + (−1)n s 2n (2n − 1)(4n n!)−1 H2n ((x − u)/s)K(s, x − u)Dm (u) du n

(n = 1, . . . , ∞).

(30)

Since by the differentiations of the Gaussian kernel K(s, u−x) the already introduced Hermite polynomials result, formula (30) may also be written in the same manner as equation (26). The advantage of formula (30) is that Dp (x) is identical with Dm (x) and the additional correction terms resulting from the convolutions of Dm (x). An application of the expansion (30) will be given in section 3.3. 3. Results: application to photon profiles and Bragg curves of protons In the previous section we have worked out three possible ways of performing deconvolutions: 1. Differential operator expansion (Lie series). This method is only applicable, if the input function Dm (x) is a smooth function or a polynomial. Measured curves Dm (x) have to be subjected to fits by complete analytical and smooth functions or at least by spline functions. 2. Discrete Fourier expansions (FFT ). This method may be connected to serious problems, if the measured curves contain noise and other discontinuities. 3. Hermite expansions of the inverse kernel K −1 (s, x − u) according equations (26) and (30). These methods are not connected to the difficulties of point 2. They work with respect to the analytical functions, smooth functions and numerical input functions. The computational effort is significantly reduced. We will present examples of deconvolutions with respect to the three methods in photon and proton dosimetry. 3.1. Deconvolution of analytical curves Dm (x) In one paper (Garcia-Vicente et al 2000) a Gaussian convolution of a Heaviside unit step function and a constant to fit some profiles of a 4 × 4 cm2 field have been used. The measured transverse profile is then adapted by a sum of two error functions and a constant c. Such an optimization is, in general, possible only for rather small fields sizes, e.g.
5 cm2 . Larger

The inverse problem of a Gaussian convolution and its application

717

Relative dose

1

z 0 = 5 cm

0.5

z 0 =10 cm

0 -3

-2

-1

0

1

2

3

x/cm Figure 1. Transverse profiles of 4 × 4 cm2 and 2 × 2 cm2 of 6 MV photon beams at z0 = 5 cm (solid curves) and z0 = 10 cm (dashes) fitted by equation (31). Table 1. Parameters of a pencil beam model for 6 MV photons used in equations (23) and (23a) (c1 = 0.84, c2 = 0.13, c3 = 0.03, s = 0.24 cm). z0 (cm)

σ 1 (cm)

σ 2 (cm)

σ 3 (cm)

5 10

0.252 1244 0.292 1061

1.963 4487 2.660 9857

5.207 5451 6.208 3149

field sizes cannot be fitted in such a manner, since the spatial fluence distribution contains ‘horns’and an energy phase space due to the flattening filter. In the following, we will also take into account an analytical access to the deconvolutions of profiles by using a pencil beam model with three Gaussians (Ulmer and Harder 1996). A transverse profile at depth z0 and y = 0 is represented by

  D(x, z0 ) = D(z0 ) ck [erf((ap − x)/σk (z0 ) + erf((ap + x)/σk (z0 ))]/(erf(ap /σk (z0 )) k

(k = 1, 3).

(31)

D(x, z0 ) provides the correct profile height, since it results from a normalized profile multiplied with the depth dose value D(z0 ). The beam divergence is included by ap = a(1+z0 /SSD), and a is half of the field-size at surface. In the following we will always use SSD = 100 cm. The parameters used for the evaluation for equation (31) are stated in table 1. The measured profiles (VARIAN, Palo Alto) of six MV photons at z0 = 5 cm and z0 = 10 cm have been recorded by a cylindrical ionization chamber (rch = 0.28 cm); s should be somewhat smaller than rch and we assume s = 0.24 cm. All profiles of the 2×2 cm2 and 4×4 cm2 fields according to figure 1 have been fitted by equation (31). The mean standard deviation stdev is smaller than 0.22%. The profile structure is a model of the application of equation (2) Dp (x) = ADm (x), but due to the extremely small contributions of the higher order we consider the approximation Dp (x) = Dm (x) − (s 2 /4)d2 Dm (x)/dx 2 according to equation (11); the succeeding correction is of the order s 4 /32. Then the profile function (31) assumes the shape  ck [erf((ap − x)/σk (z0 ) + erf((ap + x)/σk (z0 ))] Dp (x, z0 ) = D(z0 ) k

 

 − (2/π 1/2 )s 2 (ap − x) σk3 (z0 ) exp(−(ap − x)2 /σk (z0 )2 )    + (ap + x)/σk3 (z0 ) exp(−(ap + x)2 /σk (z0 )2 ) erf(ap /σk (z0 )).

(31a)

718

W Ulmer and W Kaissl 1

z0 = 5 cm

Relative dose

0.8

0.6

0.4

0.2

0 0

1

2

3

x/cm

Figure 2. Transverse profiles of 2 × 2 cm2 and 4 × 4 cm2 , one half side, 6 MV, z0 = 5 cm. Solid curve: fit by formula (31); dashed curve: calculated by formula (31a), s = 0.24 cm. The samples result from FFT (15) and formulae (31)–(33).

0.8 z 0 = 10 cm Relative dose

0.6

0.4

0.2

0 0

1

x/cm

2

3

Figure 3. Transverse profiles of a 2 × 2 cm2 and a 4 × 4 cm2 field, one half side, 6 MV, z0 = 10 cm. Solid curve: fit by formula (31); dashed curve: calculated by formula (31a), s = 0.24 cm.

The corrected dose profiles of figure 1 are presented in figures 2 and 3 (one half side). The order of the chosen s seems to provide reasonable results, but nevertheless accurate measurements with extremely small detectors are necessary in order to determine the relationship between chamber/detector size and the convolution parameter s. 3.2. Application of FFT to the analytical profile Equation (15) provides a comparison with the results obtained by equations (31), (31a), and as an example we consider the profile (31) in the case of the 4 × 4 cm2 field at z0 = 5 cm. Since equation (15) represents a numerical integration, we use the most precise representation of the error function (Abramowitz and Stegun 1970) erf(u) = (2/π 1/2 ) exp(−u2 )[u + 2u3 /(1 × 3) + 22 u5 /(1 × 3 × 5) + · · · + 2n u2n+1 /(1 × 3 × 5 × 7 · · · (2n + 1))].

(32)

The inverse problem of a Gaussian convolution and its application

719

1 z 0 = 20 cm Relative dose

0.8

z 0 = 5 cm

0.6 0.4 0.2 0 0

1

3

2

4

5

x /cm Figure 4. Measured transverse profiles (one half side, grid: 0.35 cm) of a 6 × 6 cm2 field of an 18 MV photon beam at z0 = 5 cm and z0 = 20 cm (solid curves). Normalization: D = 1, if x = 0 cm for both curves. The dashed curves are deconvolutions (Hermite polynomials up to order n = 2) according to formulae (25) and (26), s = 0.266 cm.

This expansion converges uniformly. The complete task may also be solved analytically  the Fourier transform of the expressions (see e.g. Garcia-Vicente et al 2000), because erf((ap + u)/σk (z0 )) exp(−iku) du can be erf((ap − u)/σk (z0 )) exp(−iku) du and evaluated analytically. However, a numerical evaluation provides a check with regard to the number of intervals necessary to reach the same precision as via equation (31a). With the help of the relation (32) and un = 0.0025 cm, un = un−1 + un , kn = 0.001 cm−1 , kn = kn−1 + kn , (cut-off: xmax = 3 cm, kmax = 3000 cm−1 ), the oscillating behaviour can no more verified and the results reached the same precision as via equation (31a), but the computational effort is comparably enormous. The whole computation procedure had to be performed by double precision operations. In figure 2, the samples represent results of FFT calculations. Thus the reason for the slow convergence and vanishing of undesired oscillations is the numerical handling with the Fourier representation of the δ-function
δ(x − u) = (2π)−1 exp(ik(x − u)) dk = lim(k0 ⇒ ∞) sin(k0 (x − u))/(π(x − u)). (33) Relation (33) provides awkward contributions (oscillations) for finite values of k0 and |x −u|  0, because in FFT lim k0 (or kmax ) ⇒ ∞ and lim kn ⇒ dk are not carried out. This difficulty does not arise in the Gaussian kernel K, since K converges uniformly against the δ-function by taking s ⇒ 0 and oscillations do not appear. Equation (15) contains a discrete forward and inverse transform, and therefore we agree with the remarks of Garcia-Vicente et al (2000) that non-smoothed profiles containing noise and measurement uncertainties and lacking precision of the computational procedures, which may even be amplified by FFT, should not be subjected to deconvolution by pure numerical methods. 3.3. Numerical deconvolution of photon profiles by Hermite polynomial expansions As already pointed out, analytical methods to perform deconvolutions of Dm (x) are only possible if the measured profile functions are fitted by analytical models or at least by spline functions. In the following, we apply equations (26) and (30) to the measured profiles without any refining of Dm (x). Figure 4 presents profiles of an 18 MV photon beam (field size:

720

W Ulmer and W Kaissl

1

z 0 = 5 cm

Relative dose

0.8 z 0 = 10 cm

0.6 0.4 0.2 0 0

2

4

6 x /cm

8

10

12

Figure 5. Measured transverse profile (one half side, grid: 0.25 cm) of a 20 × 20 cm2 field of a 6 MV photon beam at z0 = 5 cm and z0 = 10 cm (solid curves). The dashed curves are deconvolutions (Hermite polynomials up to order n = 2) according to formula (30), s = 0.24 cm.

6 × 6 cm2 , z0 = 5 cm and z0 = 20 cm), recorded by a cylindrical Farmer chamber. The sensitive volume is 0.7 cm3 (estimated deconvolution parameter s = 0.266 cm due to the assumption that s should be of the order of ≈0.75rch ; for plate chambers/detectors we estimate s ≈ 0.45 × thickness). Since this small uncertainty should be most widely removed by comparisons with measurements recorded by detectors, whose thickness is less than 1 mm, performing integrations of equation (26) by taking into account Hermite polynomials beyond order 2 did not appear to be useful. A comparison of the measured input profiles Dm (x) with the corrected profiles Dp(x) indicates that the Hermite expansion provides good corrections via numerical integration, because we have only used the contributions provided by H2 ((x −u)/s) and H4 ((x − u)/s). Since the available measured profiles have been represented in terms of a 0.35 cm grid, which implies lack of profile roundings in the penumbra region, the correction profiles reflect this behaviour at least partially. These results also confirm that measurements with rather thin detectors can support the decision with respect to the order of Hermite polynomial appropriate for deconvolution and the connection between detector thickness and deconvolution parameter s occurring in all formulae of the previous section. According to equation (30) we have derived a further Hermite expansion of the kernel K −1 . We apply this expansion to experimentally determined transverse profiles of a 20 × 20 cm2 field of 6 MV photons at z0 = 5 cm and z0 = 10 cm by taking into account the orders H2 and H4 . Figure 5 shows the original profiles Dm (x) and the corrected output functions Dp (x). Since the measurement conditions completely agree with those in section 3.1, the parameter s is also assumed to be 0.24 cm. An adaptation of these profiles with the help of error functions discussed by equation (31) is impossible without a suitable modification, because the ‘horns’ produced by the flattening filter are not included. These ‘horns’ are subjected to scatter processes and therefore their maximum decreases with increasing depth yielding an additional broadening of the penumbra region. An application of the analytical method according to section 3.1 is only possible, if the scatter influence of the horns is modelled by a further analytical function and added to the error function profiles described in a previous study (Ulmer and Harder 1996).

The inverse problem of a Gaussian convolution and its application

721

1 80 MeV

Relative dose

0.8 0.6 0.4 0.2 0 0

1

2

3 z /cm

4

5

6

Figure 6. Depth dose curve of an 80 MeV monoenergetic proton beam, calculated by the Monte Carlo code GEANT-Fluka and its measurement by a detector (thickness: 0.5 cm) modelled by a Gaussian convolution (s = 0.23 cm). The Bragg peak is normalized to 1.

3.4. Shape of Bragg curves of protons and influence of the detector size It is well known that there may be a significant discrepancy between the shape of recorded Bragg curves (diamond or diode detectors/ionization chambers) and Monte Carlo calculations (Noon 1998). This discrepancy increases corresponding to the increasing chamber thickness. Therefore, with regard to protons and heavily charged particles, e.g. 6 C, the dosimetry with gels and CCD camera has been put forward. Besides the relative dosimetry of the Bragg curves the determination of the absolute doses appears to be a particular problem, if the absolute height of a Bragg curve bears some uncertainties. With regard to the irradiation of targets the spread out Bragg peaks (SOBPs) consisting of a superposition of Bragg curves with appropriate proton energies have to be taken into account, and the mentioned error may imply incorrect superposition and dose deposition. It has been pointed out (Noon 1998) that the role of detector thickness with regard to the shape of the Bragg curve is noteworthy. The author considered monoenergetic proton beams of 80 MeV (Louvain) and 180 MeV (Uppsala) calculated by an analytical model and the influence of the detector size has been estimated by the stopping power concept (details are given by Noon (1998)). Thus, the influence of the thickness of a measurement detector on the Bragg curve is extremely high for proton energies E
100 MeV and decreases for increasing proton energy, since due to energy straggling the Bragg curves are broadened in the environment of the Bragg peaks. The energy-straggling effect itself increases corresponding to the increasing proton energy E. With respect to a comparison and a reconstruction of the results obtained by Noon (1998), we present in figures 6 and 8 the corresponding Monte Carlo calculations of the mentioned monoenergies (Ulmer et al 2000). In order to reach an excellent agreement with the results of Noon (1998) by referring to a detector thickness of 0.5 cm, we have subjected the monoenergetic Bragg curves to a Gaussian convolution with s = 0.23 cm. Thus, for 80 MeV protons the peak height is reduced to 62.5% (figure 6), whereas for 180 MeV protons the effect is significantly smaller owing to the increasing importance of energy straggling (figure 8). The peak height now amounts to 91%. The dashed curves represent the back transform with the help of equation (26) by taking into account the Hermite

722

W Ulmer and W Kaissl

1 80 MeV

Relative dose

0.8 0.6 0.4 0.2 0 4.5

4.7

4.9

5.1

5.3

5.5

z /cm Figure 7. Environment of the Bragg peak of 80 MeV protons according to figure 6. The dashed curve is referred to a deconvolution via formula (26) using Hermite polynomials up to order 4.

1

180 MeV

Relative dose

0.8

0.6

0.4

0.2

0 0

4

8

12

16

20

z /cm

Figure 8. Depth dose curve of a 180 MeV monoenergetic proton beam, calculated by the Monte Carlo code GEANT-Fluka and its measurement by a detector (thickness: 0.5 cm) modelled by a Gaussian convolution (s = 0.23cm). The Bragg peak is normalized to 1.

polynomials up to the order n = 4. Numerical calculations have been performed using a grid (=0.025 cm), since the determination of the narrow Bragg peak does not permit a grid size of a few millimeters as is usual in photon dose calculations. The slight difference between the Gaussian convolution (solid curves) and deconvolution (dashed curves, restriction to order 4) can be verified in figures 7 (80 MeV) and 9 (180 MeV) referring only to the environment of the Bragg peaks. With regard to the complete Bragg curves the differences are too small to be clearly distinguished. The influence of the detector size on the shape of measured Bragg curves can obviously be verified by figures 6–9.

The inverse problem of a Gaussian convolution and its application

723

1

180 MeV

Relative dose

0.8

0.6

0.4

0.2

0 21

21.5

22

z/cm

Figure 9. Environment of the Bragg peak of 180 MeV protons according to figure 8. The dashed curve is referred to a deconvolution via formula (26) using Hermite polynomials up to order 4.

4. Conclusions We have obtained three different versions of the inverse problem of a Gaussian convolution K(s, x − u). The relation Dp (x) = ADm (x) is only applicable if the function Dm (x) belongs to the class of smooth functions, where either differentiations of the infinite order exist (e.g. trigonometric, exponential and related functions) or the derivations vanish identically after a specific order (polynomials behave in this manner). An approximate application of the measured dose distributions up to order 2 according to equation (11) is in general only possible if Dm (x) is refined by the spline functions, as it may also contain noise and other discontinuities (a finite grid size may already lead to noteworthy problems). The application of Fourier transforms, in particular FFT, is possible, if Dm (x) is only non-zero within a finite domain. The pitfalls of FFT are similar as mentioned above, because noise and other uncertainties of the input functions should be removed by smoothing procedures, and oscillations vanish only by extremely small calculation grids. If Dm (x) is given by analytical polynomials, then the formulation of the deconvolution by Fourier transform involves the application of the residuum theorem, although the original problem, namely a Gaussian convolution of a polynomial, is rather trivial. A further solution of the inverse problem, working without any restriction, is given by Hermite expansions according to equations (26) and (30). The question that may also arise is whether the formulation of the inverse problem by differential operators may only work if Dm (x) is differentiable till an infinite order or if the derivations vanish identically from a fixed finite order j to ∞, and equations (26) and (30) do not make this presumption. In the case of equations (26) and (30) the differentiations are mapped to the Gaussian convolution. If Dm (x) contains discontinuities, relations (26) and (30) represent integrations of L1-measurable functions. Therefore, even numerical data are rather convenient input functions. With regard to the dosimetric impact of the deconvolution of transverse profiles one may adopt the viewpoint that the calculation of Dp (x) from Dm (x) only has a cosmetic importance. However, this viewpoint is not acceptable because of the rather small field sizes employed in radiosurgery and IMRT. With respect to the measurement and calculation (Monte Carlo) of proton dose profiles (e.g. Bragg curves, SOBP), the influence of the detector size is much more important in both relative and absolute dosimetry. In

724

W Ulmer and W Kaissl

recent time, macro Monte Carlo (MMC) calculations have become of increasing importance with respect to electron and photon beams. The connection between a macroscopic event distribution simulated by MMC and the corresponding local statistical distribution provided by Monte Carlo calculations would be a quite natural domain of the deconvolution of Gaussian kernels. Acknowledgments The authors would like to thank Prof. Saitoh for valuable comments on the mathematical foundation of the integral transforms in Hilbert space and a referee of Physics in Medicine and Biology for some improvements. Appendix A. Properties of Hermite polynomials and some mathematical aspects A.1. Some properties of Hermite polynomials The orthogonal Hermite polynomials are defined by Hn (ρ) = (−1)n exp(ρ 2 )dn exp(−ρ 2 )/ dρ n . By the introduction of a normalization factor αn = 2−n/2 (n!)−1/2 π −1/4 we can write fn (ρ) = αn Hn (ρ). The orthogonality relation now reads (see e.g. Schiff 1955):
(34) fm (ρ)fn (ρ) exp(−ρ 2 ) dρ = δmn . The expansion (26) can readily be redefined by normalized Hermite polynomials. In practical applications, where Dm (u) may contain the usual polynomials like un , it is convenient to represent un by Hermite polynomials in order to exploit the preferences of the orthogonality. According to Abramowitz and Stegun (1970) there exists a correspondence between Hn (u)  and un : un = j dj Hj (u) (j = 0, n). If one performs deconvolutions via formulae (26) and (30), then by taking account of the orthogonality all evaluations are considerably simplified. Using the Hermite polynomials Hn (ρ) of odd order n the coefficients of the powers ρ n can be represented by the recursion formula K i   Hn = 2n ρ n + n (−1)i (n − j )2n−2i ρ n−2i (where K = (n − 1)/2). (35) I =1

j =1

In order to obtain the coefficients of even-order Hermite polynomials, we make use of the differential equation Hn+1 (ρ) = 2ρHn (ρ) − Hn (ρ). This relation provides polynomials of even-order m = n + 1(n odd, K = (n − 1)/2 = m/2 − 1): K i   m m m−1 m−2 i + (m − 1) (−1) (m − 1 − j )2m−i ρ m−2i Hm = 2 ρ − (m − 1)2 ρ j =1

i=1

− (m − 1)

K  i=1

(−1)i (m − 1 − 2j )

i 

(m − 1 − j )2m−2i−1 ρ m−2i−2 .

(36)

j =1

We have checked by complete induction the correctness of formulae (35) and (36). A.2. Conditional equivalence between K(s, u − x) and A−1 The transverse profiles of collimated proton beams can be described by a box, which is broadened by a medium scatter (one Gaussian convolution). The input box function of the fluence is represented by unit step functions :

(x, y) = ((a + x) − (a − x))((b + y) − (b − y)). (37)

The inverse problem of a Gaussian convolution and its application

725

The convolution with a Gaussian kernel K(τ (z), x − x , y − y ) yields the transverse profile Dtransprof of the depth dose curve

Dtransprof (x, y, z) = K(τ (z), x − x ) (x , y ) dx dy = 0.25[erf((a + x)/τ (z)) + erf((a − x)/τ (z))][erf((a + x)/τ (z)) + erf((a − x)/τ (z))].

(38)

The scatter function τ (z) is a monotonously increasing function that can be computed by the Moli`ere scatter theory for protons or its Highland approximation (order of τ : 0.1–1.2 cm). According to section 2.2.1, equations (37) and (38) represent an example of the conditional equivalence between the convolution operator K defined on L1 (M) and A−1 defined on C ∞ . Thus, the operator A−1 cannot be defined on (x) = ((a + x) − (a − x)); hence the operator transform A−1 (x, y) = Dtransprof does not exist. In order to modify relation A−1 (x, y) = Dtransprof in an appropriate sense, it would be necessary to approximate (x, y) by smooth functions. As an example, we regard (x) = (a + x) − (a − x), which can be approximated by an expansion with a Gaussian and Hermite polynomials of even order:
∞  2 2 a2n H2n (x/a) exp(−x /a ) | (x)|2 dx

(x) = (a + x) − (a − x) ≈ smooth (x) = n=0


=

| smooth (x)| dx. 2

(39)

 The coefficients a2n are calculated by a well-known procedure a2n = (x)H2n (x/a) exp(−x 2 /a 2 ) dx (taken from −a to a). This approximation implies convergence in the mean at the discontinuities x = ±a. If (x) is approximated by smooth functions smooth (x), we are able to perform A−1 smooth (x). Thus, we may conclude that the integral transform calculus given by K is often much simpler to handle than to perform differential operator transforms according to equation (2). A.3. Relationship to Weierstrass transform and other integral transforms in Hilbert space H In a series of studies Saitoh (1997, 1999, 2001 and references therein) has developed the theory of reproducing kernels and their inverse problems of the Weierstrass transform, Fourier transform and Laplace transform in Hilbert space H. The Weierstrass transform provides the solution manifold of the heat transport, which reads ∂u(x, t)/∂t = D∂ 2 u(x, t)/∂x 2 .

(40)

If F (ξ ) represents an input function, the image function u(x, t) obeying equation (40) results from the Weierstrass transform
u(x, t) = (4πDt)−1/2 F (ξ ) exp(−(x − ξ )2 /4Dt) dξ. (41) R

The solutions given by equation (41) also satisfy the necessary initial condition u(x, t ⇒ +0). Since there are some further transport equations with an identical structure as in equation (40), e.g. diffusion equation, the Fermi–Eyges age theory of energy transport of neutrons, the Kolmogorov equation of probability distributions, the Weierstrass transform delivers a wide field of possible applications. If we set s 2 = 4Dt, the integral transform (1a) completely agrees with the relation (41). However, there is a principal difference between the two kernels, resulting from the associated differential equations. Thus the kernel (1) follows from the operator A−1 . If we keep in mind the substitution s 2 = 4Dt, the operator would contain the

726

W Ulmer and W Kaissl

time t as a parameter, whereas in equation (40) the time is handled in quite a different manner, namely as a differential operator referring to the time evolution of u(x, t). A further interesting aspect is the use of a Hilbert space H for the description of reproducing kernels. In quantum mechanics, the Hilbert space H is the quite natural statespace. The Fourier transform is a  unitary transform in H, since the L2 -norm is left invariant: |ψ(x)|2 dx = |ψF t (k)|2 dk = 1 (see textbooks of quantum mechanics). The formulation of the theory of reproducing kernels in H and the Hayashi identities used in the papers of Saitoh seem to have an outstanding importance with regard to scatter theory problems in quantum mechanics. In the framework of that theory, the Weierstrass transform assumes the shape of Feynman path integrals (nonrelativistic). The Banach space as a state space in quantum mechanics only appears in the density-matrix formulation (von Neumann 1955). By that, the situation is comparable to the Kolmogorov equation of probability distributions P (x, t), which describes the space–time evolution: ∂P (x, t)/∂t = DP ∂ 2 P (x, t)/∂x 2 .

(42)

Since the probability distribution always has to satisfy P (x, t)  0, it is assumed that a Banach space is the appropriate representation space for the description of the solutions of equation (42). In the density-matrix formulation one considers a corresponding view, because the wave function (x) has been replaced by the density-matrix ρ(x). Finally, we want to note that we have performed some extensions of the functional Lie operators A and A−1 in such a way that they also obey differential equations of the first order with respect to time. However, this analysis goes beyond the present scope.

References Abramowitz M and Stegun I A 1970 Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Washington, DC: National Bureau of Standards) Chang K-S, Yin F-F and Nie K-W 1996 The effect of detector size to the broadening of the penumbra—a computer simulated study Med. Phys. 23 1407–11 Charland P, El-Khatib E and Wolters J 1998 The use of deconvolution and total least squares in recovering a detector line spread function Med. Phys. 25 152–60 Garcia-Vicente F, Delgado J M and Perazza C 1998 Experimental determination of the convolution kernel for the study of spatial response of a detector Med. Phys. 25 202–7 Garcia-Vicente F, Delgado J M and Rodriguez C 2000 Exact analytical solution of the convolution integral equation for a general profile fitting function and Gaussian detector kernel Phys. Med. Biol. 45 645–50 Higgins P, Sibata C, Siskind L and Sohn J W 1995 Deconvolution of detector size effect for small field measurement Med. Phys. 22 1663–6 Metcalfe P, Kron T, Elliot A and Wong T 1993 Dosimetry of 6 MV X-ray beam penumbra Med. Phys. 20 1439–45 Noon S 1998 Dosimetry and quality control of scanning proton beam Dissertation Rijksuniversiteit Groningen Saitoh S 1997 Integral Transforms: Reproducing Kernels and their Application (Pitman Research Notes in Mathematics Series) vol 369 (London: Addison-Wesley, Longman) Saitoh S 1999 Integral Transforms Involving Smooth Functions: Reproducing Kernels and their Applications (Dordrecht: Kluwer) Saitoh S 2001 Applications of the reproducing kernel theory to inverse problems Commun. Korean Math. Society 16 371–83 Schiff L I 1955 Quantum Mechanics 2nd edn (New York: McGraw-Hill) pp 60–9 Sibata C, Higgins P, Mota H C, Beddar A S and Shin K H 1991 Influence of the detector size in photon beam profile measurements Phys. Med. Biol. 36 621–31 Ulmer W 1980 On the representation of atoms and molecules as self-interacting field with internal structure Theor. Chim. Acta 55 179–203

The inverse problem of a Gaussian convolution and its application

727

Ulmer W and Harder D 1996 Applications of a triple Gaussian pencil beam model for photon beam treatment Z. Med. Phys. 6 68–74 Ulmer W, Schaffner B and Kaissl W 2000 A Monte Carlo based 3D pencil beam algorithm for dose calculation of proton beams Med. Phys. 27 51761–2 Von Neumann J 1955 Mathematical Foundations of Quantum Mechanics (Princeton, NJ: Princeton University Press) Withaker E T and Watson G N 1952 A Course of Modern Analysis (Cambridge: Cambridge University Press)