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Laminogram reconstruction through regularizing Fourier filtration
Laminogram reconstruction through regularizing Fourier filtration Z. Danovich and Y. Segal Quality Assurance and Reliability, Technion - Israel Institute of Technology, Haifa, 32000, Israel Received 19 May 1993; revised 11 February 1994
Coplanar rotational laminography (CRL) is a special case of laminography which is a tomographic technique used to image cross-sectional views through solid objects. In the case of CRL the film and object rotate synchronously, while the radiation source remains stationary. The image formed on the exposed film is highly blurred. This blurring may be described as a convolution of the ideal image with a radially symmetric point spread function (PSF). To deblur the laminographic image (laminogram) one has to perform a deconvolution of this image based on a laminographic PSF. The deconvolution is an ill-posed problem and therefore special methods are required for its realization. One of the possible methods, regularizing Fourier filtration, was tried and its most successful version was selected. A simple technique for the choice of filter parameters is proposed, based on an image quality evaluation method which was derived. Several laminograms (both simulated and experimental) were successfully reconstructed by this method. Keywords. laminography, regularizing Fourier filtration, deconvolution
Coplanar rotational laminography (CRL) is a special case of a tomographic technique tl] that creates a blurred image of a cross-sectional view through a solid body. A schematic of a CRL system unit is presented in Figure 1. A beam of X-rays illuminates a photographic plate after passing through the object. During the exposure, the radiation source is stationary, while both the body and photographic plate rotate synchronously. Their rotation axes are parallel, and the X-ray beam direction is almost perpendicular to them. Rotational motion of this kind ensures that there is a one-to-one cOrrespondence between points on the plate and points on the cross-section of the body. The points which are corresponding one to another are situated on the same radiation ray.
where ~ is a radius vector, 4~(f) the blurred image (laminogram), p(~) the PSF of the laminographic system, a(~) the real (desired) cross-sectional image and ** is the notation for two-dimensional convolution. This approach enables us to define the PSF as a laminogram of a special cross-sectional object representing one single point. In this case a(~) represents a delta-function 6(?). Following Equation (1) we get: ~b(~) = p(f)**6(?) = p(f)
(2)
There are different ways to show [2-41 that for the simplest laminographic system the PSF is radially symmetric and
The approach commonly adopted for the modelling of image formation in laminography is representing the image as a convolution of the cross-sectional image with a point spread function (PSF) of the imaging system t21. X X
According to this approach, we may write: =
0963-8695/94/03/0123-08 @ 1994 Butterworth-Heinemann Ltd
~
e
"'Q~
. I gilr~a plate
I
(1)
Figure
1
Geometry of the CRL apparatus
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Laminogram reconstruction by regu/artzation. Z. Danowch and Y. Sega/ has the form:
p(r)
=
I
(3)
r
where r is the radial distance from the origin of the coordinate system. The simplest laminographic system is produced when the X-ray radiation is parallel, the scattering in the object under examination is negligible and the response of the film is linear throughout its entire range. The deficiencies of the 'l/r'-like PSF derived for this simplest case have been discussed in a previous work TM. In that work we also derived and tested several more complicated versions of laminographic PSFs. The most successful one is the so-called truncated "rotating Gaussian'-like PSF (TRG-PSF). Its form is[g1: p(r) - {;i'(~ exp(-2rg)lot2r~)~r2\ / r2 \ ) exp(-,'::).,
(4,
where ro and r~ are parameters (G >> to) and I o is the modified Bessel function of zero order. The important advantage of the T R G - P S F is the existence of an analytically computed Fourier (Hankel) transform for it[4]:
f~z3/er
[-
(n G Io)2~ [(Trr~,p)2-])
[ _ (Trrop~Z~ x exp
\
fl / ]
15)
where .¢/' is the designation of the zero order Hankel transform in the form mentioned in Reference 4, p is a variable of the Hankel transform and corresponds to the radial distance from the origin in the Fourier domain and fl2=(ro/G)2 + 1 ~ 1. Expression (5) is used as a PSF in the Fourier domain for the reconstruction of the experimental laminograms.
Deconvolution-based laminogram reconstruction as an ill-posed problem As already mentioned, we base our work on the convolution model of image formation (1). Expression (l) may be considered as an equation in which p(f), the PSF of the laminographic system, and 4b(?) are known functions, while (r(~) is to be found. If the PSF is not known, it may be estimated through suitable modelling. In most cases deconvolution is a Hadamard sense ill-posed problemtSL This is also the situation in a iaminogram reconstruction. In fact, the laminogram reconstruction would have been called well-posed by Hadamard, if it had been uniquely solvable for every laminogram and the solution had depended continuously on the laminogram. It is obvious that for this problem
124 NDT&E International 1994 Volume 27, Number 3
even the most important requirement Isolvability)naa5 not be fulfilled, as the exact PSF of the particular laminography system is never known. This results m thc ill-posedness of the deconvolution-based laminogram reconstruction problem. If the convolution model (1) is intended to be an equation for a laminogram reconstruction, one must define a substitute for its solution, as it does not exist in the ordinary sense. First we choose the commonly adopted 'least squares' substitute, i.e. we define a solution of Equation (1 1which minimizes its integral square residual. In other words, we define a solution G(r) which minimizes the functional K[~r] defined as follows:
g[~(~)] = IIP(~)**~(~)
= f[p(~)**a(f)-
~b(r)N2 q~(~)]2 df
(6)
where the integration is performed over the whole image domain. Note that the above definition neither guarantees uniqueness nor continuity (with respect to qS(f)) of the solution. This is the main practical difficulty with an ill-posed problem, i.e. even if the meaning of its solution is defined, the two solutions need not be close one to another when the initial data (pictures) are close. To overcome this difficulty, i.e. to finally overcome the problem's ill-posedness, the regularization approach proposed by Tikhonov [61 is followed. The essence of this approach is the incorporation of an additional functional to the least squares functional K in Equation (6). The purpose of this addition is to achieve the uniqueness and the continuity mentioned above. The work of Tikhonov's group implies that such an additional (stabilizing) functional may be chosen in the following general form: L
n[~(p)] = ~ ~;II[¢(~)- c,(P)Y'II2
(7)
I=0
where [ . . . ] ~ ) designates the derivative of the /th order with respect to ~ (in the broad sense), :~i ~> 0 is the regularization coefficient (parameter) of the /th order while :t0 # 0, at(F) is the given /th constant image and I ! II2 designates the square norm as in Equation (6). Usually the order L is not more than 2. Now let us define a solution of Equation (1) in the Tikhonov sense. The image cr~(f) is called a solution of Equation (1) if it minimizes the functional:
rE~(p)] = K[~@)] + %f~[¢(~)]
(8)
where K[a(#)] is the 'least squares' functional (6), D[a(f)] is the stabilizing functional (7) and ao is the general scale coefficient. If the set of regularization parameters is chosen properly for the specific initial task, then, according to Tikhonov, minimization of R[a(f)] in Equation (8) is a well-posed problem in the Hadamard sense.
Laminogram reconstruction by regularization: Z. Danovich and Y. Segal Determining 'good' regularization parameters for a specific task is one of the crucial points in the application of the regularization method. In fact, the main topic of this paper is a method to find 'good' parameters for the iU'-posed deconvolution problem concerning the laminogram reconstruction task.
Fourier filtration based on Tikhonov's regularization method The Fourier transform is a very convenient tool to handle convolutions as they are turned to multiplications in the Fourier domain. Following the idea in References 6-9, let us convert the functional R[a(?)] in Equation (8) to the Fourier domain and minimize it there. For the sake of simplicity we assume at(J) in Equation (7) to be zero for all I. Let us choose the square norm of the gradient vector as the square of the image's first derivative: [a,(~)] 2
=
Lax j +L ay j Choose the Laplacian as the image's second derivative:
a"(~) -
~x 2 +
~y2
The filter Q(p) resembles the classical Wiener-Helstrom filterLlzl, where the spectral signal-to-noise ratio is represented by a polynomial. Let us call it the 'regularizing polynomial'. The form of the filter in Equation (13) seems to be more successful than many other Wiener-like filters. The role of each of the ;t coefficients in it can be understood on the basis of their origin in the stabilizing functional in Equation (7). It can be seen that the norm of the solution may be controlled by the value assigned to ;to- On the other hand, the smoothness of the filtered image may be controlled by ;tx and ;t2 as they are related to its gradient and Laplacian respectively. Another, more important, advantage of the filter in Equation (13) is the possibility of applying an almost objective method for determining the general scale coefficient ;to after choosing the internal structure of the regularizing polynomial, i.e. choosing the ratio between the different coefficients ;to, ;tx and ;t2. This method based on Tikhonov's regularization theory will be derived in the next section. We shall call the filter in Equation (13) the Wiener Tikhonov regularizing filter (WTR filter).
(10)
Using the linearity, differentiation and energy (Rayleigh) theorems for the two-dimensional Fourier transform [~°], on the basis in Equations (9), (10) and (6)-(8) we get for L=2:
Determining the values of the regularization coefficients Let us approach this problem using two aspects mentioned in the previous section.
R[a(?)] = Rv[ap(/5)]
Structure of the regularizing polynomial
= ~ [[PF(P)aF(P)-J
at)
+ %(;t~ + 4~z;t,1 p2 + 16~;t~p4)lar(/5)l= ] d~
(11) where ( )F designates the image or functional in the two-dimensional Fourier domain, /5 designates a coordinate vector in the Fourier domain, p is the radial distance from the origin in the Fourier domain, I'"12 designates the complex square norm and the integration is performed over the entire Fourier domain. Note that due to the radial symmetry of the PSF its Fourier image, Pv(P) is a real symmetrical function of the scalar p. In the Fourier space the minimizer a~(/5) of the functional R in Equation (11) is: a~(/5) =
Pr(P) q~v(/5) p2(p) + %(% + ;tlp2 + ;t2p4)
(12)
where ;to=;t~, ;tl =4n2;t'z and ;tz = 16~z4;t2• This minimizer is actually the Fourier image of Tikhonov's solution of Equation (1). On the other hand, Equation (12) can be looked at as a laminogram filtered by the filter:
O(p) = p2(p) + %(%Pv(P) + ;tlp2 + ;t2p4)
(13)
The first aspect is the internal structure of the regularizing polynomial, i.e. the ratio between the different coefficients ;t. Let us discuss this in detail. In most of the Tikhonov school work the zero order (% :~0, ;t~ = ;t2 = 0 ) or the mixed zero-first order (;to ~ 0, ;tl ~ 0, ;t2 = 0) regularizations are used. In several works dealing with the regularization approach to the tomographic problem, the authors t12'~31 use the mixed zero-second order ( % # 0 , ;tt = 0 , ; t 2 ¢ 0 ) regularization for the image domain reconstruction. None has used the regularization without a zero order part (;to = 0). The reason for always using a zero order part is due to the fact that the well-posedness of Tikhonov's functional (8) minimization problem exists only if ;to :~ 0 in a stabilizing functional (7). Let us now look at the WTR formula (Equation (13)). We can see that each part of the regularizing polynomial corrects frequencies in its way, but only the zero order coefficient % corrects the zero frequency (DC). If ;to is equal to zero (while other coefficients are not), we may say that the problem, having been regularized in such a way, remains formally ill-posed at the zero frequency. However, it is commonly known that the signal-to-noise ratio is usually maximal at the zero frequency. This last fact allows us to assume that the solution obtained
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Laminogram reconstruction by regularization. Z. Danowch and Y. Sega/ through a regularization without the zero order parl should be stable.
25O [i
~
On the other hand, we may reach the same conclusion by a physical analogy. One can interpret the stabilizing functional in Equation (7) as the expression for the m e m b r a n e generalized energy, where the zero order part relates to the average distance of a m e m b r a n e from the zero (flat) position, and the first-order part relates to the deformation energy. Physical reasons imply that minimization of the deformation energy will yield a stable solution of the m e m b r a n e position problem. The zero part does not exist in the m e m b r a n e analogue. Blake and Zisserman H41 applied the m e m b r a n e model to image processing. In the above work the authors use, in fact, regularization of the first order, without proving theoretically the well-posedness of their method.
2,5 o
4
200
,
150 t-
,
't i/
~:
I
{
2
lOO
j
i
tO0
5o
J
0 o
50
[
..... .....
SIMULATED RECONSTRUCTED " , ~ 1 . . . . I . . . . ~ . . . . i . . . . I . . . . I,t 50 tO0 150 200 250 300 PIXEL NUMBER
0
Figure 4 Reconstruction of the simulated laminogram by the zero order (~0 # O, ~1 = =2 = O) WTR filter. Value of ~-0:0.0158 (as Iog(~0) = - 1 . 8 )
In order to test the different versions of the W T R filters, we have simulated the laminogram by convolving the specifically built 'source' image with one of the '1/r'-like PSFs from our previous w o r k [~]. This PSF is called "rotating rectangular' (or RR-PSF).
250 -
250
r.D
~2oo "
200 4
The scheme of the 'source' image is presented in Figure 2. This image represents a set of five strips of 'density' 230, between which there are four slits of 'density' 20 having different widths,
I
150 lO0 -
Profiles passing t h r o u g h the 75th line of the different images obtained from the image of Figure 2 are now shown. Figure 3 presents such a profile for the 'source'
50
150 100 50
-
SIMULATED
0
RECONSTRUCTED
I .... I .... I .... I , ~.~_, I . . . . ~ J 50 i00 150 200 250 300 PIXEL NUMBER
.... 0
F i g u r e 5 Reconstruction of the simulated laminogram by the first-order (~1 # O, z0 = z2 = O) WTR filter. Value of ,~: 0.316 (as log (:~0) = 0.5)
_•
260
/ ' ~
250
2o0 Figure 2 Scheme of the 'initial" image for a laminogram simulation. The arrows indicate the location of the profiles shown in Figures 3 - 5
150
/~ k , ~
100
~ 4 2
/
q 150
0
i /
l~°°
-42
.... 0
J .... , .... , ....... 100 200 300 400
- J 50 500 PIXEL NUNBER
~;
~200
200
150
p~O.1
.
.
.
.
:
B
-6
6
-8 100
200 b:
300 400 500 PIXEL NUMBER p>O.i
Figure 6 Profiles passing through the same place at: (a) 'inside' image, p ~< 0.1; (b) 'outside' image, p > 0.1
150 i
,oo
s
,oo
t
i ............... ;_5 .~ 0 ..... SIMUI.WI'ND CONVOLVED+NOISED :.... .... i .... i .... i .... i .... i .... i 0 50 100 150 200 250 300 PIXNL NUMBER
image Figure 2 and for its convolution with R R - P S F i'u plus the addition of noise. The average deviation of the noise is 1.5% of the m a x i m u m laminogram density value. This noisy convolution served as the simulated testing laminogram.
50 0
O u r laminogram reconstruction procedure is: (a) performing a direct F F T obtaining Or(P).
3 Horizontal profiles of the initial simulated image and its 1.5%-noised convolution with RR-PSF Figure
126
~
.
i 6 ,i
200
/-/ 50
.
i-
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of a laminogram,
i.e.
Laminogram reconstruction by regularization: Z. Danovich and Y. Segal (b) computing a~,(/5) through Equation (12), where Pr(P) is a Fourier (Hankel) transform of the PSF of the laminographic system; (c) performing an inverse F F T of a}(tS), i.e. obtaining the desired solution. Under the above steps Pr(P) is the F F T of RR-PSF t41. Figures 4 and 5 show the reconstructed laminograms, where the zero order (So # 0, sl = Sz = 0) and the first order (s 1 :~ 0, % = s2 = 0) WTR filters were used for reconstruction respectively. The way the s values were chosen is discussed in the next section. Some features of the above examples are presented later. It may be seen that the reconstruction quality is better for the first-order reconstruction (Figure 5) than for the zero order one (Figure 4). This can be seen in at least two ways: 1. The corners on the profile of Figure 5 are more rectangular than in Figure 4. 2. The profile of Figure 5 enters three of the four slits, while the profile of Figure 4 hardly enters the widest slit.
Noise is a major factor determining the quality of an image. Therefore finding a method to evaluate image quality taking into account the nature of the noise may be useful. A significant effort has been invested in finding a quantitative and objective image quality measure for a given image, but up to now there has not been a generally adopted definition of 'image quality'. In the present work we use the following criterion for image quality: the fiat parts of a 'good' image must have no micro-jumps from pixel to pixel ('saw'). If a 'saw' pattern appears on a flat part of an image it may be interpreted as a noise. We would like to assume that the noise is 'white', i.e. all the frequencies are represented in it with equal amplitude. On the other hand, we assume that any laminogram is frequency bounded, i.e. almost all its Fourier image is situated inside the specific circle near the origin of the Fourier domain. On the basis of these two assumptions, we propose the following method for the estimation of the relative noise 6: Let us estimate 6 as the normalized average value of the image's high frequency (p > Po) component:
It is worthwhile to note that changing s o does not improve the shape of the 'zero order reconstructed' laminogram. This brings us to the conclusion that the first-order regularization is preferred over the zero order one. Let us call the WTR filter in Equation (13) (with S 0 = S 2 = 0 in the regularizing polynomial) WTR1. This specific WTR filter with the first-order regularizing polynomial will be used below as the best WTR filter.
General scale of the regularizing polynomial If the kind of regularizing polynomial is already chosen, it is necessary to choose its general scale % in Equation (13) or (12). Our approach is based on Tikhonov's 'deviation principle'. For our case, let us formulate it in the following form: Choose the general scale coefficient sg so that the following equality will apply: [ Ilas:~,(~)**p(r) -- ~b(~)H2] 1/2
iq4~(~)llZ
= 6
(14)
where most of the designations are as in Equation (1), a .... (~) is the solution obtained according to the procedure mentioned above with a specific %, and 6 represents the relative noise of the laminographic image. The main difficulty in the above method is the inadequacy of information regarding the noise of a real complicated experimental system. In most cases, both the theoretical and experimental approaches to the estimation of the noise yield unreliable results. In order to utilize Equation (14) we have to choose 6, preferably on an objective basis.
fP ~4a(pO)=
-
f 'Jp>PO
>po
IIq~r(~)ll 2
dSF/("11 HSF
(15)
where dSF is the element of a Fourier domain area and Po is the bound frequency. Let us consider this value as an image quality parameter and call it the 'saw' parameter. This dimensionless parameter allows comparison of significantly different images. If the 'saw' parameter is smaller, the image quality is better. Many numerical tests were performed in order to find a suitable value of Po for experimentally obtained laminograms. For these tests we computed the inverse F F T of the two parts of the same laminogram Fourier image: once inside the circle Po and secondly outside it. The value Po is considered to be 'not too small' if: (a) the 'inside' reconstructed image does not seem smoothed in important details compared with the original image; (b) the 'outside' reconstructed image may be considered as noise. In Figure 6 two profiles are presented, both passing through the same places in the 'inside' and 'outside' pair of images. It was found that for all cases the desired boundary is smaller than 0.I (if the maximal frequency available by F F T is regarded as 0.5). For greater confidence, the value Po = 0.15 was chosen. If, as was mentioned above, 1. the noise is 'white' (the amplitude as a function of frequency is constant), 2. the image information is bounded in the frequency range p < 0.1 then estimation of the noise using any range outside the
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Laminogram reconstruction by regu/arization: Z. Danovich and Y. Sega/ circle p = 0.1 has to yield the same value. In practice, we found that for values of Po in the range 0.15 0.3 the variation in ¢~ was smaller than 20%. We tried to reconstruct laminograms by the WTR 1 filter (see above) using criterion (14) and 6~ estimated by Equation (15) while go = 0.15, but not every trial was a success. In certain cases personal subjective observation of the laminograms showed that the reconstruction was not good. Therefore it was decided to append to criterion (14) an additional criterion based on the 'saw" parameter
of the resulting (reconstructed) laminogr'am
il~v(i})ii-' dSr
L.
~d :tl1
I
The additional criterion is that the 0~ value for a reconstructed laminogram must not be greater than 54, from Equation (15). In other words, the "saw" quality of a reconstructed laminogram must not be worse than the 'saw' quality of the original laminogram. All the reasons mentioned above in this section allow us to formulate the method for the determination of the general scale ~,: (a) find % which satisfies criterion (14); (b) if 6~ ~< 6~ (defined in Equations (15) and (16)), the determination is finished, otherwise: (c) enlarge % until equality 5~ = ~o is reached. If still one is not satisfied with the resulting quality, one may enlarge % until a more satisfactory quality is obtained.
¸11%1I1!:
R e c o n s t r u c t i o n e x a m p l e s and discussion Two examples were mentioned in the previous section. The formulation of their details in the terms defined previously are presented below. The 'saw' parameter (15) for the noisy simulated image (Figure 3) is 2.25%. It is interesting to note that this value was obtained for the image made noisier by 1.5% of the maximum value. The reason for the difference may be due to the fact that the 'saw" parameter refers somehow to the average value of the laminogram. Two images (one before and the other after reconstruction) are presented in Figure 7, The first image is a laminogram of object 4 from Reference 15. Profiles passing through the same horizontal lines of those images are presented in Figure 8. Similar information regarding test object 2 tl5] is shown in Figures 9 and 10.
Figure 7 Laminogram of test object 4 I15]: (a) before; and (b) after reconstruction
The information related to the above reconstructions is summarized in Table 1. We can see that in all cases it was necessary to keep a residual deviation greater than an initial laminogram 'saw" parameter 5~. This situation
Table 1. Information related to reconstructions Initial data Figure 2 cross-section Figure 7a, laminogram Figure 9a, laminogram
Reconstructed laminogram L
Figure 7b Figure 9b
0 1 1 1
Iog(c~g)
Profile examples
-1.8 -0.5 -0.95 0.25
Figure Figure Figure Figure
4, Figure 3 5, Figure 3 8 10
6~ (%)
Deviation 6~ (%) (%)
2.25
3.77 2.31 0.72 1.35
0.42 1.1 8
2.33 2.23 0.42 0.07
L the regularization order; 5~ -- the 'saw' parameter of an initial laminogram (15); 5, - the 'saw' parameter of a reconstructed laminogram (16); deviation - the normalized residual deviation for the case considered, computed using the left-hand side of equation (14)
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Laminogram reconstruction by regularization."Z. Danovich and Y. Segal ~250
~250
Restored CRL data
~200
Restored CRL data
225 200
" " al CRL d a t a
175 -\1
150
Original CRL dat~~a
i50 i25
100
iO0 75
50
,
/
::
5O
0
25
0
100
200
300
400 500 PIXEL NUMBER
Figure 8 Profiles passing through the 272nd horizontal line (designated by arrows) in the images of Figure 7
(a)
0
100
200
300
400 500 PIXEL NUMBER
Figure 10 Profiles passing through the 257th vertical column (designated by arrows) in the images of Figure 9
may be explained in the framework of the more detailed Tikhonov theory. The theory says that we must add to the right-hand side of criterion (14) one additional positive term. This term is proportional to the error in the PSF being used, and we cannot evaluate it objectively. This means that a method based on criterion (14) may be helpful for choosing only on a coarse general scale. The fine tuning must be performed on the basis of comparison between the 'saw' quality parameters of the initial and the reconstructed laminograms, as was explained previously. Test object 2 (see Figures 9 and 10) is an example of a case for which the quality of the initial laminogram is relatively bad. Therefore it is necessary to choose the parameters % by subjective reasoning of 'sufficient quality'.
Conclusions
(b)
1 Figure 9 Laminogram of test object 2 o 5]: (a) before; and (b) after reconstruction
Reconstruction (enhancement) of images through deconvolution is a complicated problem that may be separated into two parts: first study and modelling of the point spread function (PSF), and second the creation of a deconvolution algorithm, which is suitable for the specific PSF and for the problem in general. In Reference 4 we have derived the PSF models for our specific problem reconstructions of laminograms. The present paper is devoted to deconvolution based on the use of a special form of Fourier filtration and applying Tikhonov's regularization theory. Different versions of the WienerTikhonov regularizing filter (WTR filter) were tried, and the first order WTR filter (WTR1) was chosen as the most successful. An approach for image quality evaluation based on assumptions of 'white' noise and image frequency boundedness was proposed. Using this approach, the technique for filter parameter selection was derived. All the proposed methods were tried for the reconstruction of both simulated and experimental laminograms.
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Laminogram reconstruction by regularization. Z. Danovich and Y. Segal The important advantage of the derived approach is its semi-objectivity in filter parameter selection. It means that in most cases using our approach, an investigator, relaying only on numerical analysis of an initial laminogram, obtains a filter that delivers the reconstructed laminogram with a desired quality. If nevertheless the reconstructed laminogram is not of sufficient quality, our approach makes it possible to improve it under control. Another advantage of our approach is its universality. In the present work the method was developed for laminogram reconstruction based on the WTR filter. But if one has any other deconvolution problem (with another PSF), and wants to solve it using the Fourier filtration with any filter, our approach may be used in most cases. All examples presented in this work were prepared using the image processing package 'SARIL' developed by ourselves.
2
4
5
6 7 8
9
10 1I 12
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References 1
Herman, G.T. lmaoe Reconstruction fiom Prqjections Academic Press, New York (1980)
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14 15
Barrett, HAt. and Swindell, W. Radioor~;/?;o /m,a/;;;~; ' . , Academic Press, New York ilgSl) Gore, J.S. and Orr, J.C. "Image formation by hack-p,~uecU~,n ,~ reappraisal' Phw,~ ,~lcd flio/24 (1979) pp 793 ~;0l Danovich, Z. and Segal, I¢. "Point spread t'uncthm ol coplanar rotational laminography' in P;',ceeding,~ ol H;e S'et'eHwemh ('om'~,~iHon a*M Evhitfflimz ¢?t E/c~ t;'i~al and L'/ccgrmtic~ L';z,!i*tcc; ~ m LvraHI1991)pp 159 162 Hadamard, J. Le ProhR'me &" ('auch.r ct h'.s Equ~zion.s a~,v ¢a,t;,&,.~ partiHh's liJE'aires InT~erholiques Herman, Paris (1932) Tikhono,~, A.N. and Goncharsky, A.V. llI-Po~cd Pr,h/ems m Ha' Natural Science,s. M it, Moscox~ {1987l Bertero, M., DeMol, C. and Pike, E.R. 'Linear inverse problems with discrete data" lm'erw Prohl 4 (1988)pp 573 594 Louis, A.K. "Tikhonov-Phillips regularization of the radon transform' in Int. Serie~ 0/ Numerical ,14alhematic,s. Berkhouser, Basel, w)l 73 (1985) pp 211 223 Natterer, F. The Mathematic.~ ~!1 Computerized Tomography John Wiley, New York 11986) pp 8~; 92 Bracewell, R.N. l'hc Fourier Tran,~/brm and its Applicalion McGraw Hill, New York (1986! Helstrom, C.W. 'Image restoration by the method of least squares' J Opt So~ Am 57 (1967) pp 297 303 Artzy, E., Elfving, T. and Herman, G.T. "Quadratic optimization for image reconstruction ll' ('omput Graph bna~le Proc It (1979) pp 242 2,q 1 Kashyap, R.L and Mittal, M.C. "Picture reconstruction from projections' IEEE Trans. Compul. C-24 (1975) pp 915 923 Blake, A. and Zisserman, A. Visual Reconstruction The MIT Press, Cambridge, MA (1987) Segal, Y. and Cohen, B. 'Computerized laminography' NDTlntern 23 3 (1990) pp 137 146
Coplanar rotational laminography (CRL) is a special case of laminography which is a tomographic technique used to image cross-sectional views through solid objects. In the case of CRL the film and object rotate synchronously, while the radiation source remains stationary. The image formed on the exposed film is highly blurred. This blurring may be described as a convolution of the ideal image with a radially symmetric point spread function (PSF). To deblur the laminographic image (laminogram) one has to perform a deconvolution of this image based on a laminographic PSF. The deconvolution is an ill-posed problem and therefore special methods are required for its realization. One of the possible methods, regularizing Fourier filtration, was tried and its most successful version was selected. A simple technique for the choice of filter parameters is proposed, based on an image quality evaluation method which was derived. Several laminograms (both simulated and experimental) were successfully reconstructed by this method. Keywords. laminography, regularizing Fourier filtration, deconvolution
Coplanar rotational laminography (CRL) is a special case of a tomographic technique tl] that creates a blurred image of a cross-sectional view through a solid body. A schematic of a CRL system unit is presented in Figure 1. A beam of X-rays illuminates a photographic plate after passing through the object. During the exposure, the radiation source is stationary, while both the body and photographic plate rotate synchronously. Their rotation axes are parallel, and the X-ray beam direction is almost perpendicular to them. Rotational motion of this kind ensures that there is a one-to-one cOrrespondence between points on the plate and points on the cross-section of the body. The points which are corresponding one to another are situated on the same radiation ray.
where ~ is a radius vector, 4~(f) the blurred image (laminogram), p(~) the PSF of the laminographic system, a(~) the real (desired) cross-sectional image and ** is the notation for two-dimensional convolution. This approach enables us to define the PSF as a laminogram of a special cross-sectional object representing one single point. In this case a(~) represents a delta-function 6(?). Following Equation (1) we get: ~b(~) = p(f)**6(?) = p(f)
(2)
There are different ways to show [2-41 that for the simplest laminographic system the PSF is radially symmetric and
The approach commonly adopted for the modelling of image formation in laminography is representing the image as a convolution of the cross-sectional image with a point spread function (PSF) of the imaging system t21. X X
According to this approach, we may write: =
0963-8695/94/03/0123-08 @ 1994 Butterworth-Heinemann Ltd
~
e
"'Q~
. I gilr~a plate
I
(1)
Figure
1
Geometry of the CRL apparatus
N D T & E International 1994 Volume 27, N u m b e r 3
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Laminogram reconstruction by regu/artzation. Z. Danowch and Y. Sega/ has the form:
p(r)
=
I
(3)
r
where r is the radial distance from the origin of the coordinate system. The simplest laminographic system is produced when the X-ray radiation is parallel, the scattering in the object under examination is negligible and the response of the film is linear throughout its entire range. The deficiencies of the 'l/r'-like PSF derived for this simplest case have been discussed in a previous work TM. In that work we also derived and tested several more complicated versions of laminographic PSFs. The most successful one is the so-called truncated "rotating Gaussian'-like PSF (TRG-PSF). Its form is[g1: p(r) - {;i'(~ exp(-2rg)lot2r~)~r2\ / r2 \ ) exp(-,'::).,
(4,
where ro and r~ are parameters (G >> to) and I o is the modified Bessel function of zero order. The important advantage of the T R G - P S F is the existence of an analytically computed Fourier (Hankel) transform for it[4]:
f~z3/er
[-
(n G Io)2~ [(Trr~,p)2-])
[ _ (Trrop~Z~ x exp
\
fl / ]
15)
where .¢/' is the designation of the zero order Hankel transform in the form mentioned in Reference 4, p is a variable of the Hankel transform and corresponds to the radial distance from the origin in the Fourier domain and fl2=(ro/G)2 + 1 ~ 1. Expression (5) is used as a PSF in the Fourier domain for the reconstruction of the experimental laminograms.
Deconvolution-based laminogram reconstruction as an ill-posed problem As already mentioned, we base our work on the convolution model of image formation (1). Expression (l) may be considered as an equation in which p(f), the PSF of the laminographic system, and 4b(?) are known functions, while (r(~) is to be found. If the PSF is not known, it may be estimated through suitable modelling. In most cases deconvolution is a Hadamard sense ill-posed problemtSL This is also the situation in a iaminogram reconstruction. In fact, the laminogram reconstruction would have been called well-posed by Hadamard, if it had been uniquely solvable for every laminogram and the solution had depended continuously on the laminogram. It is obvious that for this problem
124 NDT&E International 1994 Volume 27, Number 3
even the most important requirement Isolvability)naa5 not be fulfilled, as the exact PSF of the particular laminography system is never known. This results m thc ill-posedness of the deconvolution-based laminogram reconstruction problem. If the convolution model (1) is intended to be an equation for a laminogram reconstruction, one must define a substitute for its solution, as it does not exist in the ordinary sense. First we choose the commonly adopted 'least squares' substitute, i.e. we define a solution of Equation (1 1which minimizes its integral square residual. In other words, we define a solution G(r) which minimizes the functional K[~r] defined as follows:
g[~(~)] = IIP(~)**~(~)
= f[p(~)**a(f)-
~b(r)N2 q~(~)]2 df
(6)
where the integration is performed over the whole image domain. Note that the above definition neither guarantees uniqueness nor continuity (with respect to qS(f)) of the solution. This is the main practical difficulty with an ill-posed problem, i.e. even if the meaning of its solution is defined, the two solutions need not be close one to another when the initial data (pictures) are close. To overcome this difficulty, i.e. to finally overcome the problem's ill-posedness, the regularization approach proposed by Tikhonov [61 is followed. The essence of this approach is the incorporation of an additional functional to the least squares functional K in Equation (6). The purpose of this addition is to achieve the uniqueness and the continuity mentioned above. The work of Tikhonov's group implies that such an additional (stabilizing) functional may be chosen in the following general form: L
n[~(p)] = ~ ~;II[¢(~)- c,(P)Y'II2
(7)
I=0
where [ . . . ] ~ ) designates the derivative of the /th order with respect to ~ (in the broad sense), :~i ~> 0 is the regularization coefficient (parameter) of the /th order while :t0 # 0, at(F) is the given /th constant image and I ! II2 designates the square norm as in Equation (6). Usually the order L is not more than 2. Now let us define a solution of Equation (1) in the Tikhonov sense. The image cr~(f) is called a solution of Equation (1) if it minimizes the functional:
rE~(p)] = K[~@)] + %f~[¢(~)]
(8)
where K[a(#)] is the 'least squares' functional (6), D[a(f)] is the stabilizing functional (7) and ao is the general scale coefficient. If the set of regularization parameters is chosen properly for the specific initial task, then, according to Tikhonov, minimization of R[a(f)] in Equation (8) is a well-posed problem in the Hadamard sense.
Laminogram reconstruction by regularization: Z. Danovich and Y. Segal Determining 'good' regularization parameters for a specific task is one of the crucial points in the application of the regularization method. In fact, the main topic of this paper is a method to find 'good' parameters for the iU'-posed deconvolution problem concerning the laminogram reconstruction task.
Fourier filtration based on Tikhonov's regularization method The Fourier transform is a very convenient tool to handle convolutions as they are turned to multiplications in the Fourier domain. Following the idea in References 6-9, let us convert the functional R[a(?)] in Equation (8) to the Fourier domain and minimize it there. For the sake of simplicity we assume at(J) in Equation (7) to be zero for all I. Let us choose the square norm of the gradient vector as the square of the image's first derivative: [a,(~)] 2
=
Lax j +L ay j Choose the Laplacian as the image's second derivative:
a"(~) -
~x 2 +
~y2
The filter Q(p) resembles the classical Wiener-Helstrom filterLlzl, where the spectral signal-to-noise ratio is represented by a polynomial. Let us call it the 'regularizing polynomial'. The form of the filter in Equation (13) seems to be more successful than many other Wiener-like filters. The role of each of the ;t coefficients in it can be understood on the basis of their origin in the stabilizing functional in Equation (7). It can be seen that the norm of the solution may be controlled by the value assigned to ;to- On the other hand, the smoothness of the filtered image may be controlled by ;tx and ;t2 as they are related to its gradient and Laplacian respectively. Another, more important, advantage of the filter in Equation (13) is the possibility of applying an almost objective method for determining the general scale coefficient ;to after choosing the internal structure of the regularizing polynomial, i.e. choosing the ratio between the different coefficients ;to, ;tx and ;t2. This method based on Tikhonov's regularization theory will be derived in the next section. We shall call the filter in Equation (13) the Wiener Tikhonov regularizing filter (WTR filter).
(10)
Using the linearity, differentiation and energy (Rayleigh) theorems for the two-dimensional Fourier transform [~°], on the basis in Equations (9), (10) and (6)-(8) we get for L=2:
Determining the values of the regularization coefficients Let us approach this problem using two aspects mentioned in the previous section.
R[a(?)] = Rv[ap(/5)]
Structure of the regularizing polynomial
= ~ [[PF(P)aF(P)-J
at)
+ %(;t~ + 4~z;t,1 p2 + 16~;t~p4)lar(/5)l= ] d~
(11) where ( )F designates the image or functional in the two-dimensional Fourier domain, /5 designates a coordinate vector in the Fourier domain, p is the radial distance from the origin in the Fourier domain, I'"12 designates the complex square norm and the integration is performed over the entire Fourier domain. Note that due to the radial symmetry of the PSF its Fourier image, Pv(P) is a real symmetrical function of the scalar p. In the Fourier space the minimizer a~(/5) of the functional R in Equation (11) is: a~(/5) =
Pr(P) q~v(/5) p2(p) + %(% + ;tlp2 + ;t2p4)
(12)
where ;to=;t~, ;tl =4n2;t'z and ;tz = 16~z4;t2• This minimizer is actually the Fourier image of Tikhonov's solution of Equation (1). On the other hand, Equation (12) can be looked at as a laminogram filtered by the filter:
O(p) = p2(p) + %(%Pv(P) + ;tlp2 + ;t2p4)
(13)
The first aspect is the internal structure of the regularizing polynomial, i.e. the ratio between the different coefficients ;t. Let us discuss this in detail. In most of the Tikhonov school work the zero order (% :~0, ;t~ = ;t2 = 0 ) or the mixed zero-first order (;to ~ 0, ;tl ~ 0, ;t2 = 0) regularizations are used. In several works dealing with the regularization approach to the tomographic problem, the authors t12'~31 use the mixed zero-second order ( % # 0 , ;tt = 0 , ; t 2 ¢ 0 ) regularization for the image domain reconstruction. None has used the regularization without a zero order part (;to = 0). The reason for always using a zero order part is due to the fact that the well-posedness of Tikhonov's functional (8) minimization problem exists only if ;to :~ 0 in a stabilizing functional (7). Let us now look at the WTR formula (Equation (13)). We can see that each part of the regularizing polynomial corrects frequencies in its way, but only the zero order coefficient % corrects the zero frequency (DC). If ;to is equal to zero (while other coefficients are not), we may say that the problem, having been regularized in such a way, remains formally ill-posed at the zero frequency. However, it is commonly known that the signal-to-noise ratio is usually maximal at the zero frequency. This last fact allows us to assume that the solution obtained
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Laminogram reconstruction by regularization. Z. Danowch and Y. Sega/ through a regularization without the zero order parl should be stable.
25O [i
~
On the other hand, we may reach the same conclusion by a physical analogy. One can interpret the stabilizing functional in Equation (7) as the expression for the m e m b r a n e generalized energy, where the zero order part relates to the average distance of a m e m b r a n e from the zero (flat) position, and the first-order part relates to the deformation energy. Physical reasons imply that minimization of the deformation energy will yield a stable solution of the m e m b r a n e position problem. The zero part does not exist in the m e m b r a n e analogue. Blake and Zisserman H41 applied the m e m b r a n e model to image processing. In the above work the authors use, in fact, regularization of the first order, without proving theoretically the well-posedness of their method.
2,5 o
4
200
,
150 t-
,
't i/
~:
I
{
2
lOO
j
i
tO0
5o
J
0 o
50
[
..... .....
SIMULATED RECONSTRUCTED " , ~ 1 . . . . I . . . . ~ . . . . i . . . . I . . . . I,t 50 tO0 150 200 250 300 PIXEL NUMBER
0
Figure 4 Reconstruction of the simulated laminogram by the zero order (~0 # O, ~1 = =2 = O) WTR filter. Value of ~-0:0.0158 (as Iog(~0) = - 1 . 8 )
In order to test the different versions of the W T R filters, we have simulated the laminogram by convolving the specifically built 'source' image with one of the '1/r'-like PSFs from our previous w o r k [~]. This PSF is called "rotating rectangular' (or RR-PSF).
250 -
250
r.D
~2oo "
200 4
The scheme of the 'source' image is presented in Figure 2. This image represents a set of five strips of 'density' 230, between which there are four slits of 'density' 20 having different widths,
I
150 lO0 -
Profiles passing t h r o u g h the 75th line of the different images obtained from the image of Figure 2 are now shown. Figure 3 presents such a profile for the 'source'
50
150 100 50
-
SIMULATED
0
RECONSTRUCTED
I .... I .... I .... I , ~.~_, I . . . . ~ J 50 i00 150 200 250 300 PIXEL NUMBER
.... 0
F i g u r e 5 Reconstruction of the simulated laminogram by the first-order (~1 # O, z0 = z2 = O) WTR filter. Value of ,~: 0.316 (as log (:~0) = 0.5)
_•
260
/ ' ~
250
2o0 Figure 2 Scheme of the 'initial" image for a laminogram simulation. The arrows indicate the location of the profiles shown in Figures 3 - 5
150
/~ k , ~
100
~ 4 2
/
q 150
0
i /
l~°°
-42
.... 0
J .... , .... , ....... 100 200 300 400
- J 50 500 PIXEL NUNBER
~;
~200
200
150
p~O.1
.
.
.
.
:
B
-6
6
-8 100
200 b:
300 400 500 PIXEL NUMBER p>O.i
Figure 6 Profiles passing through the same place at: (a) 'inside' image, p ~< 0.1; (b) 'outside' image, p > 0.1
150 i
,oo
s
,oo
t
i ............... ;_5 .~ 0 ..... SIMUI.WI'ND CONVOLVED+NOISED :.... .... i .... i .... i .... i .... i .... i 0 50 100 150 200 250 300 PIXNL NUMBER
image Figure 2 and for its convolution with R R - P S F i'u plus the addition of noise. The average deviation of the noise is 1.5% of the m a x i m u m laminogram density value. This noisy convolution served as the simulated testing laminogram.
50 0
O u r laminogram reconstruction procedure is: (a) performing a direct F F T obtaining Or(P).
3 Horizontal profiles of the initial simulated image and its 1.5%-noised convolution with RR-PSF Figure
126
~
.
i 6 ,i
200
/-/ 50
.
i-
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1994 Volume
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of a laminogram,
i.e.
Laminogram reconstruction by regularization: Z. Danovich and Y. Segal (b) computing a~,(/5) through Equation (12), where Pr(P) is a Fourier (Hankel) transform of the PSF of the laminographic system; (c) performing an inverse F F T of a}(tS), i.e. obtaining the desired solution. Under the above steps Pr(P) is the F F T of RR-PSF t41. Figures 4 and 5 show the reconstructed laminograms, where the zero order (So # 0, sl = Sz = 0) and the first order (s 1 :~ 0, % = s2 = 0) WTR filters were used for reconstruction respectively. The way the s values were chosen is discussed in the next section. Some features of the above examples are presented later. It may be seen that the reconstruction quality is better for the first-order reconstruction (Figure 5) than for the zero order one (Figure 4). This can be seen in at least two ways: 1. The corners on the profile of Figure 5 are more rectangular than in Figure 4. 2. The profile of Figure 5 enters three of the four slits, while the profile of Figure 4 hardly enters the widest slit.
Noise is a major factor determining the quality of an image. Therefore finding a method to evaluate image quality taking into account the nature of the noise may be useful. A significant effort has been invested in finding a quantitative and objective image quality measure for a given image, but up to now there has not been a generally adopted definition of 'image quality'. In the present work we use the following criterion for image quality: the fiat parts of a 'good' image must have no micro-jumps from pixel to pixel ('saw'). If a 'saw' pattern appears on a flat part of an image it may be interpreted as a noise. We would like to assume that the noise is 'white', i.e. all the frequencies are represented in it with equal amplitude. On the other hand, we assume that any laminogram is frequency bounded, i.e. almost all its Fourier image is situated inside the specific circle near the origin of the Fourier domain. On the basis of these two assumptions, we propose the following method for the estimation of the relative noise 6: Let us estimate 6 as the normalized average value of the image's high frequency (p > Po) component:
It is worthwhile to note that changing s o does not improve the shape of the 'zero order reconstructed' laminogram. This brings us to the conclusion that the first-order regularization is preferred over the zero order one. Let us call the WTR filter in Equation (13) (with S 0 = S 2 = 0 in the regularizing polynomial) WTR1. This specific WTR filter with the first-order regularizing polynomial will be used below as the best WTR filter.
General scale of the regularizing polynomial If the kind of regularizing polynomial is already chosen, it is necessary to choose its general scale % in Equation (13) or (12). Our approach is based on Tikhonov's 'deviation principle'. For our case, let us formulate it in the following form: Choose the general scale coefficient sg so that the following equality will apply: [ Ilas:~,(~)**p(r) -- ~b(~)H2] 1/2
iq4~(~)llZ
= 6
(14)
where most of the designations are as in Equation (1), a .... (~) is the solution obtained according to the procedure mentioned above with a specific %, and 6 represents the relative noise of the laminographic image. The main difficulty in the above method is the inadequacy of information regarding the noise of a real complicated experimental system. In most cases, both the theoretical and experimental approaches to the estimation of the noise yield unreliable results. In order to utilize Equation (14) we have to choose 6, preferably on an objective basis.
fP ~4a(pO)=
-
f 'Jp>PO
>po
IIq~r(~)ll 2
dSF/("11 HSF
(15)
where dSF is the element of a Fourier domain area and Po is the bound frequency. Let us consider this value as an image quality parameter and call it the 'saw' parameter. This dimensionless parameter allows comparison of significantly different images. If the 'saw' parameter is smaller, the image quality is better. Many numerical tests were performed in order to find a suitable value of Po for experimentally obtained laminograms. For these tests we computed the inverse F F T of the two parts of the same laminogram Fourier image: once inside the circle Po and secondly outside it. The value Po is considered to be 'not too small' if: (a) the 'inside' reconstructed image does not seem smoothed in important details compared with the original image; (b) the 'outside' reconstructed image may be considered as noise. In Figure 6 two profiles are presented, both passing through the same places in the 'inside' and 'outside' pair of images. It was found that for all cases the desired boundary is smaller than 0.I (if the maximal frequency available by F F T is regarded as 0.5). For greater confidence, the value Po = 0.15 was chosen. If, as was mentioned above, 1. the noise is 'white' (the amplitude as a function of frequency is constant), 2. the image information is bounded in the frequency range p < 0.1 then estimation of the noise using any range outside the
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Laminogram reconstruction by regu/arization: Z. Danovich and Y. Sega/ circle p = 0.1 has to yield the same value. In practice, we found that for values of Po in the range 0.15 0.3 the variation in ¢~ was smaller than 20%. We tried to reconstruct laminograms by the WTR 1 filter (see above) using criterion (14) and 6~ estimated by Equation (15) while go = 0.15, but not every trial was a success. In certain cases personal subjective observation of the laminograms showed that the reconstruction was not good. Therefore it was decided to append to criterion (14) an additional criterion based on the 'saw" parameter
of the resulting (reconstructed) laminogr'am
il~v(i})ii-' dSr
L.
~d :tl1
I
The additional criterion is that the 0~ value for a reconstructed laminogram must not be greater than 54, from Equation (15). In other words, the "saw" quality of a reconstructed laminogram must not be worse than the 'saw' quality of the original laminogram. All the reasons mentioned above in this section allow us to formulate the method for the determination of the general scale ~,: (a) find % which satisfies criterion (14); (b) if 6~ ~< 6~ (defined in Equations (15) and (16)), the determination is finished, otherwise: (c) enlarge % until equality 5~ = ~o is reached. If still one is not satisfied with the resulting quality, one may enlarge % until a more satisfactory quality is obtained.
¸11%1I1!:
R e c o n s t r u c t i o n e x a m p l e s and discussion Two examples were mentioned in the previous section. The formulation of their details in the terms defined previously are presented below. The 'saw' parameter (15) for the noisy simulated image (Figure 3) is 2.25%. It is interesting to note that this value was obtained for the image made noisier by 1.5% of the maximum value. The reason for the difference may be due to the fact that the 'saw" parameter refers somehow to the average value of the laminogram. Two images (one before and the other after reconstruction) are presented in Figure 7, The first image is a laminogram of object 4 from Reference 15. Profiles passing through the same horizontal lines of those images are presented in Figure 8. Similar information regarding test object 2 tl5] is shown in Figures 9 and 10.
Figure 7 Laminogram of test object 4 I15]: (a) before; and (b) after reconstruction
The information related to the above reconstructions is summarized in Table 1. We can see that in all cases it was necessary to keep a residual deviation greater than an initial laminogram 'saw" parameter 5~. This situation
Table 1. Information related to reconstructions Initial data Figure 2 cross-section Figure 7a, laminogram Figure 9a, laminogram
Reconstructed laminogram L
Figure 7b Figure 9b
0 1 1 1
Iog(c~g)
Profile examples
-1.8 -0.5 -0.95 0.25
Figure Figure Figure Figure
4, Figure 3 5, Figure 3 8 10
6~ (%)
Deviation 6~ (%) (%)
2.25
3.77 2.31 0.72 1.35
0.42 1.1 8
2.33 2.23 0.42 0.07
L the regularization order; 5~ -- the 'saw' parameter of an initial laminogram (15); 5, - the 'saw' parameter of a reconstructed laminogram (16); deviation - the normalized residual deviation for the case considered, computed using the left-hand side of equation (14)
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Laminogram reconstruction by regularization."Z. Danovich and Y. Segal ~250
~250
Restored CRL data
~200
Restored CRL data
225 200
" " al CRL d a t a
175 -\1
150
Original CRL dat~~a
i50 i25
100
iO0 75
50
,
/
::
5O
0
25
0
100
200
300
400 500 PIXEL NUMBER
Figure 8 Profiles passing through the 272nd horizontal line (designated by arrows) in the images of Figure 7
(a)
0
100
200
300
400 500 PIXEL NUMBER
Figure 10 Profiles passing through the 257th vertical column (designated by arrows) in the images of Figure 9
may be explained in the framework of the more detailed Tikhonov theory. The theory says that we must add to the right-hand side of criterion (14) one additional positive term. This term is proportional to the error in the PSF being used, and we cannot evaluate it objectively. This means that a method based on criterion (14) may be helpful for choosing only on a coarse general scale. The fine tuning must be performed on the basis of comparison between the 'saw' quality parameters of the initial and the reconstructed laminograms, as was explained previously. Test object 2 (see Figures 9 and 10) is an example of a case for which the quality of the initial laminogram is relatively bad. Therefore it is necessary to choose the parameters % by subjective reasoning of 'sufficient quality'.
Conclusions
(b)
1 Figure 9 Laminogram of test object 2 o 5]: (a) before; and (b) after reconstruction
Reconstruction (enhancement) of images through deconvolution is a complicated problem that may be separated into two parts: first study and modelling of the point spread function (PSF), and second the creation of a deconvolution algorithm, which is suitable for the specific PSF and for the problem in general. In Reference 4 we have derived the PSF models for our specific problem reconstructions of laminograms. The present paper is devoted to deconvolution based on the use of a special form of Fourier filtration and applying Tikhonov's regularization theory. Different versions of the WienerTikhonov regularizing filter (WTR filter) were tried, and the first order WTR filter (WTR1) was chosen as the most successful. An approach for image quality evaluation based on assumptions of 'white' noise and image frequency boundedness was proposed. Using this approach, the technique for filter parameter selection was derived. All the proposed methods were tried for the reconstruction of both simulated and experimental laminograms.
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Laminogram reconstruction by regularization. Z. Danovich and Y. Segal The important advantage of the derived approach is its semi-objectivity in filter parameter selection. It means that in most cases using our approach, an investigator, relaying only on numerical analysis of an initial laminogram, obtains a filter that delivers the reconstructed laminogram with a desired quality. If nevertheless the reconstructed laminogram is not of sufficient quality, our approach makes it possible to improve it under control. Another advantage of our approach is its universality. In the present work the method was developed for laminogram reconstruction based on the WTR filter. But if one has any other deconvolution problem (with another PSF), and wants to solve it using the Fourier filtration with any filter, our approach may be used in most cases. All examples presented in this work were prepared using the image processing package 'SARIL' developed by ourselves.
2
4
5
6 7 8
9
10 1I 12
13
References 1
Herman, G.T. lmaoe Reconstruction fiom Prqjections Academic Press, New York (1980)
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Barrett, HAt. and Swindell, W. Radioor~;/?;o /m,a/;;;~; ' . , Academic Press, New York ilgSl) Gore, J.S. and Orr, J.C. "Image formation by hack-p,~uecU~,n ,~ reappraisal' Phw,~ ,~lcd flio/24 (1979) pp 793 ~;0l Danovich, Z. and Segal, I¢. "Point spread t'uncthm ol coplanar rotational laminography' in P;',ceeding,~ ol H;e S'et'eHwemh ('om'~,~iHon a*M Evhitfflimz ¢?t E/c~ t;'i~al and L'/ccgrmtic~ L';z,!i*tcc; ~ m LvraHI1991)pp 159 162 Hadamard, J. Le ProhR'me &" ('auch.r ct h'.s Equ~zion.s a~,v ¢a,t;,&,.~ partiHh's liJE'aires InT~erholiques Herman, Paris (1932) Tikhono,~, A.N. and Goncharsky, A.V. llI-Po~cd Pr,h/ems m Ha' Natural Science,s. M it, Moscox~ {1987l Bertero, M., DeMol, C. and Pike, E.R. 'Linear inverse problems with discrete data" lm'erw Prohl 4 (1988)pp 573 594 Louis, A.K. "Tikhonov-Phillips regularization of the radon transform' in Int. Serie~ 0/ Numerical ,14alhematic,s. Berkhouser, Basel, w)l 73 (1985) pp 211 223 Natterer, F. The Mathematic.~ ~!1 Computerized Tomography John Wiley, New York 11986) pp 8~; 92 Bracewell, R.N. l'hc Fourier Tran,~/brm and its Applicalion McGraw Hill, New York (1986! Helstrom, C.W. 'Image restoration by the method of least squares' J Opt So~ Am 57 (1967) pp 297 303 Artzy, E., Elfving, T. and Herman, G.T. "Quadratic optimization for image reconstruction ll' ('omput Graph bna~le Proc It (1979) pp 242 2,q 1 Kashyap, R.L and Mittal, M.C. "Picture reconstruction from projections' IEEE Trans. Compul. C-24 (1975) pp 915 923 Blake, A. and Zisserman, A. Visual Reconstruction The MIT Press, Cambridge, MA (1987) Segal, Y. and Cohen, B. 'Computerized laminography' NDTlntern 23 3 (1990) pp 137 146