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Theoretical aspects and practical applications of Moire topography
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1979 Phys. Med. Biol. 24 250 (http://iopscience.iop.org/0031-9155/24/2/002) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 130.237.29.138 This content was downloaded on 05/09/2015 at 11:41
Please note that terms and conditions apply.
PHYS. MED. BIOL., 1979, Vol. 24, No. 2, 250-261.
Printed in Great Britain
Theoretical Aspects and Practical Applications of Moir6 Topography S. S. XENOFOS,
PH.D.
and C. H. JONES, PH.D.,
F.INST.P.
Physics Department, The Institute of Cancer Research, Royal Marsden Hospital, Fulham Road, London SW3 6JJ, U.K.
Received 15 March 1978, infcnal form 20 July 1978 ABSTRACT. When a surface is observed a t a distance through a parallel line grating which is illuminated by a source of light laterally displaced to the viewing axis, the grating is superimposed on its shadow and this gives rise to a moire fringe pattern. The fringes represent cross-sections of the surface parallel to the grating plane. The principle of the formation of the contour map has been studied assuming a grating represented by a Fourier series, a source of lightrepresented by a 3D intensity distribution and a viewing aperture of finite dimensions. The experimental system used for illumination and observation of the moire fringes consists of an equispaced parallel-lined glass grating andtwo small lightsources positioned symmetrically about the imaging device. Thearrangement constitutes a 3D imaging system withan accuracy of the order of 1 mm.Thetechnique has been appliedinradiotherapy problems such as the design of contour compensators and the measurement of body contours. Moire patterns have been analysed with the aid of a PDP-8 computer, and have been used to calculate surface areas and volumes such as in breast studies and superficial tumour measurements.
1. Introduction
The need for accurate measurement of 3D objects arises in many fields of science. I n medicine the measurement of surface topography of the human changes body and its subsequent change is of importance.Surfacecontour are sometimesassociatedwithcertain diseases or abnormalities and can be related to the treatment scheme applied. The measurement of body contours is also important in fitting prostheses and in radiotherapy. Three dimensional measurements can be carried out using stereophotographic methods (Miskin 1960),photogrammetrictechniques of the gridprojection type (Lovesey 1973), holographic stereomethods (Mikhail, Kurz and Stevenson 1972) and coherent moire interferometrictechniques(Brooksand Heflinger 1969). Incoherent moire techniques can also be used. They have the advantage of being accurate, inexpensive and simple and are the subject of this paper. The principles of moire topographyhave been discussed by a number of authors (Meadows, Johnson and Allen 1970, Takasaki 1970, Hovanesian and Hung 1971). In the first part of this paper we present a more general theory of the formation of the contour map assumingalight source and viewing a Fourier series. aperture of finite dimensions and a grating represented by Thisapproachcanbe used to evaluate the moire imagingsystem and is 0031-9155/79/020250+ 12 $01.00
@ 1979 The Institute of Physics
Moire’ Topography
251
considered in more detail in another paper (Xenofos, Jones and Dance 1978). The second part of this paper describes the application of the technique to a variety of clinical problems. Theory The principle of moire surface topography is illustrated in fig. 1. An equispaced parallel-line grating is projected on the surface S by a light source a t A with an intensity distribution I ( X , Y , Z ) . When the surface is observed from B the grating lines are superimposed on their shadows and this gives rise t o a (Ip)is given bythe moire fringe pattern.Theresultantintensitypattern product of the superimposed intensity distributions. 2.
Fig. 1. Principle of moiresurfacetopography. &(x,, ys, 0) and T(z,,yt, 0) are points y, z ) on the where the rays from the light source A directed towards the pointS(%, surface intersect the grating plane. The dark line through point S represents the shadow of the grating around Q on the surface around S. The dark line through point P represents the shadow around S as seen around point P on the grating when viewed through an aperture at B.
In this section we will deriveageneral expression describing the overall intensitypatternas seen from point B (fig. 1). Fromthis expression the contour information required can then be obtained. Provided that the grating spacing is not very small, diffraction at the grating may be neglected even for z. Due tothe finite dimensions of the source and relativelylargedepths viewing aperture, however, the moire pattern willbe blurred. These effects are considered below.
S. S . Xenofos and C . H . Jones
252
2.1. Light Source of jinite dimensions 2.1.1. The overall intensity pattern. For the time being we will assume that the
aperture at B is of very small dimensions so that it can be considered as a point for ray optics, but not small enough for diffraction to be important. The blurring in the moire is due to the fact that rays from the source towards a particularpointon the surfaceintersect thegrating a t points of different transmission. Consider a symmetric grating represented by Fourier series m
G(xp) = E + C 4cos ar(e + xp) r=l
where E is a constant, F, aretheFourier coefficients, e the phase, and a = 277/s0 and so is the grating spacing. The intensity pattern Il around S on the surface is obtained by integrating over the source a t A (fig. 1 )
Il = c o / / / [ E +
1
gF,cosar(e+x,) I(X, Y,Z)dXdYdZ. r=l
The factor c. will be discussed below. grating observedfrom the The overallintensity patternIparound pointPon the point B will then be described by the product of the gratingitself and itsshadow Il
x
/ S[.+
1
gF,cosar(e+x,) I(X, Y,Z)dXdYdZ. r=l
(1)
The factorc. depends uponthe geometry andphysical propertiesof the reflecting surface and that of the imagingsystem. By neglecting variationsin solid angle and assuming that the surface is a diffuse scatterer, this factor can be taken as a constant. For simplicity let c. = 1. In order to pick out the moire term in eqn ( l ) ,it is necessary to carry out the multiplication and express the cosine productsas cosine sumsand differences. The result is cosar(x,-x,)I(X, Y,Z)dXdYdZ r=l
5
+:
~Fm&///cosa(me-re+mxp-rxq)
m#r r=l
xI(X, Y,Z)dXdYdZ
+-l
* Zm~ m & S / / o o s a ( m e + r e + m x p + r x q )
2 m = l r=1
xI(X, Y,.Z)dXdYdZ
W
+ I,E WC1Fmcos am(e +xp)
Moirk Topography
253
where
I,
=/
/ / I ( X , Y,Z)dXdYdZ.
This expressiondescribes the overall intensitypatternas seenfrom the that the second term in the above viewing point. It willbeshownbelow eqn (2) provides contour information (fig. 2). The third term represents the
Fig. 2. Moire patterns of a plaster model of t~ human face with embryosarcoma. Depth difference between fringes A z r 2 mm, L = 100 cm, d = 50 cm, q, = 1 mm. In (a)the moir6 contour lines, the ‘aliasing’ moire on the sides of the face and the grating lines superimposed on the image can be identified. These correspond to the second, third and sixthterms of eqn (2) respectively. The moire pattern of the same object taken with a moving grating is also shown in ( b ) . The ‘aliasing’ and grating lines have been removed whereas the contour moir6 has remained unchanged.
result of superimposition of higher harmonics and when the surface under examination presents a steep inclination to theilluminating beam, an ‘aliasing’ moire is formed which doesnot correspond to thecontour lines (Takasaki 1970, Xenofos 1977) (fig. 2(a)). The fourth, fifth and sixth terms represent the sum of shadow and grating, the shadow, and the grating respectively (fig. 2(a)). All terms except the first two are phase dependent, and can beaveraged out by moving the grating in its plane. Movement of the grating therefore decreases the noise of the system (fig. 2(b)). The pattern observed in such a case will be described by the first two terms in eqn (2) which, as shown below, corresponds to a contour map of the surface behind the grating (fig. 2(b)).
S. X. Xenofos and C . H . Jones
254
2.1.2. The contour moire'. If we express xp,xq as a function of the x coordinate of the point on the surface (fig. 1) the pattern observed (fig. 3) will be given by
x
[?:z:dL-(L+z-z)(L+z)xxz
] I ( X , Y, 2)d X d Yd Z .
(&h+
source
Fig. 3. Experimental arrangement for the observation and recording of the moire fringe pattern of an object.
To facilitate the interpretation of this result we now expand to first order in the quantities X(L+x ) and Z / ( L+ x ) and assume that I ( X , Y,Z) is a symmetric function. Using elementary trigonometric formulae we obtain
where
I, = /&(X,Z)coskrXdX,
S
TV = I(X, Y,Z)dY, k = az/(L+z).
I n a practical system the maximum value of x will be a small fraction of d, and/or L and the x variation in eqn (3) can be neglected. For the systems we have used L 2 100 cm and d 2 L12 cm. To a good approximation, therefore, the moire pattern M ( z ) isindependent of the (x,y) positionon the object. The relative importance of the three cosine terms in eqn (3) can be assessed by comparing their arguments. It then becomes clear that the patternis due to a
Moirk Topography
255
rapidly varying cosine function (cos krd) which is modulated by two slowly varying cosine functions of z (cosine functions within the integral). If these two slowly varying terms are compared for a typical system with d = 0*5L, the term cos krX is the more important but nevertheless the z variation cannot alwaysbeneglected.The moire pattern M ( z ) therefore provides contour information via the function cos krd. 2.2. Aperture of finite dimensions
When the image is viewed through a finite aperture, the rays of light which reach the aperture from a particular point on the surface, will intersect the grating a t points which do not have the same transmission (points P and P’ in fig. 1). I n this case that part of the expression in eqn ( 1 ) before the integral becomes
I’
=
l/[..
5 F,cosam(e+x,,)
m=l
1
&(X’, Y’)dX’dY’
(4)
where &(X‘,Y’) is the aperture function and x*., is shown in fig. 1. By simple geometry we have
If we assume that Q is symmetric, eqns (4)and (5) give m
I‘
= E&,+
C F,Q,cosam(e+x,) m=l
where
Q, = s&JX’) cos kmX’dX’, m = 0 , 1 , 2 ,m,
=
!Q’,(.
Y’)dY’
If we now substitute the first expression in eqn ( 1 ) by that given by eqn ( 6 ) , we obtain the moire pattern due to a combination of a finite source and a finite aperture
M ( z ) = IoQoE2+ - C F?&,COS krd 1 m
2 r=l
Once again we see that M ( z ) provides contour information due to the factor cos krd which is a rapidly varying functionof z. The effect of the finite dimensions of both aperture and source is to degrade the moire pattern as thedepth z is increased. Themagnitude of the degradationdependsuponthedetailed form of Iu(X,Z) and &,(X’) which represent the projections of the light source distribution on the X , Z plane and that of the aperture function on the X’ axis respectively. When the source and viewing aperture become very small I and Q tend to delta functions, and M ( z ) becomes a periodic functionwithconstantamplitude of oscillation. When therefore these distributions are narrow the moire image is improved. If we assume that the integration can be performed independently then M ( z ) is independent of the light source and aperture distribution along the Y, Y‘
256
S . S.Xenofos and C . H . Jones
dimension respectively. If the X and 2 light source distributions have similar form and the latter is not wider than the former the 2 dimension will be also of little consequence provided that d < L. We therefore conclude that themost important dimensions in the light source distribution and theviewing aperture are those along X and X’.An evaluation of the image quality of a moire can thus be adequately assessed by studying a one dimensional (X,X’) system. 3. Depth of thecontourfringes Thedepth of the brightcontourfringescanbeobtained cos krd = 1 (in eqn ( 7 ) ) and this gives
byputting
where n is a positive integer called ‘fringe number’. Though dark fringes produced by different harmonics arenot always spatially coincident, those of the fundamental frequency ( r = 1) are much stronger in intensity. The depth of the dark fringes can thus be obtained from eqn (8) by putting n = n - 0.5. I n order that the depth z can be calculated, the fringe number n must be known. The measured distance of a reference point on the surface can be used to assign fringe numbers to the moire fringes through eqn (8) solved for n. Some knowledge of the sign of the gradient of the surface contour is necessary in order that fringe numbers n are assigned correctly. The system, however, can be calibratedso that thefringe number can be obtained from measurements on the image instead of by measurements on the object (Takasaki 1973). Experimentalconfiguration Though the theory developed in section 2 is for one light source, the same results can be obtained when two light sources are used. We have used two light sources arranged symmetrically about the point of observation for shadowfreeillumination,infront of a 50.8 cm x 61 cm (20”x 24”) glass grating with approximately 10 lines per cm. This is supported vertically in a metal frame of adjustable height which allows the grating to slide laterally across the field of view (fig. 3). A camera is used at the viewing point to photograph the moire pattern at a distance of 1 m from the grid. Continuous illumination is used (15 V, 60 W bulbs enclosed in a light shield : the moire image is photographed using a 35 mm with an aperture in the front) camera with a 55 mm Asahi-Pentax lens, and is recorded on Ilford FP4 film with an exposure of 1 S a t f / l 6 . For depth differences between fringes of the order of 4 mm or more a television camera can be used. A permanent record of the image is then obtained by photographing the television screen using a Polaroid camera loaded with4 x 5 inch Polaroid film, type 55 positive/negative, with an exposure of Q S a t f/8. The use of the above combination offers high contrast and small exposure times. The television camera is not used a t small 4.
Moire’ Topography
257
depth differences for two reasons; firstly, because of its poorer resolution and secondly, because when the moire is a t a small angle to the television lines an ‘aliasing’ or (sampling) moire may be formed. 5. 3D analysis of moire patterns
The three dimensional reconstruction of a surface canbe calculated from the corresponding moire image. With reference to fig. 3 the z coordinate is given by eqn (8) and the x and y coordinates are given by
x = m,(L+z)x,/c,
y
= m,(L+z)y,/c
(9)
where m, is the magnification factor corresponding to distance c from the fist nodal point of the camera lens, and xi,yi the coordinates of the point on the nth fringe on the photograph. The magnification m, can be calculated from the dimensions on the final image of a scale of known length. 6. Accuracy From eqns (8) and (9) we see that the systematic errors in x, y and z will depend upon the accuracy with which the parameters L, d, S, and c have been measured. Another error is introduced in x and y when the optical axis is not perpendicular tothegrating plane. For errors of the order of a few per thousand in the measurement of lengths the error in the z coordinate (which depends only on uncertainties in thegeometry of the system) is of the order of 0.5% of the value of x . The percentage error in the x,y coordinates is due to both systematic errors and errors introduced during the analysis procedure and depends upon the value of x,y . The order of magnitude of the overall error is +_ 1 mm, I n order that eqn (8) can be used when a glass grating (3 mm thick) is incorporated, the grid lines should be on the surface of the glass plate facing the object. The error introduced in the calculation of the coordinates by using eqn (8) is in this case negligible. When, however, the plate is placed the other way round a small correction should be applied (Xenofos 1977).
Clinicalapplications A number of clinical applications have been investigated and are discussed below.
7.
7 . 1 . Patients’ outlines required for radiotherapy
Cross-sections of the objectthroughthepoint 0 (fig. 3) correspond to straight lines on the recorded image. These can be obtained by measuring the distance x between the point in the image corresponding to 0 and the centre of each fringe which intersects the straight line. Eqns (8) and (9) can then be used to calculate the z and x coordinates. The moire patterns areanalysed from an enlarged photograph with the aid of a PDP-s/E computer used routinely for radiotherapy treatment planning. One of the peripherals of this machine is a p, 0 device which may be used to make on-line measurements of polar
258
Xenofos and C. H . Jones
S. S.
coordinates. This device is used to trace the fringes along the given section and the fringe numbers n are fed automatically at the time of the tracing procedure.Thecontouris then drawn full scale on an incrementalplotter. Measurements on models of various body parts have shown that the overall accuracy of the method is within 2 mm. A useful application of the method would be the measurement of posterior outlines when the patient is irradiated from under the couch. 7.2. Contour compensators required for radiotherapy
By using the appropriate geometry the moire technique can be used for the design of compensators in radiotherapy (Keck, Binder,Gabaj,Klaring and Windischbauer 1976, Xenofos and Jones 1976, 1977). With a camera placed at theposition of the radiation source and the patient set as for treatment (fig. 4) the corresponding moire pattern under a magnification of m, = b/c constitutes a projectionof the contour map of the patient’s Light
sourceyv.
L
l
t
l
b
C
1_;I Compensator tray support
Scaie
--t
______ __
Radiatlon field
Clratln~
Contour
Fig. 4. Application of the moire technique in the design of radiotherapycontour compensators.
skin surface onto the compensator plane (the tray on which the compensator is supported). The missing tissue z, = z - t (fig. 4) corresponding t o each fringe, can then be converted to the equivalent thickness of the compensator material, and the various layers of the compensator cut to the shape of the fringes. Themethodhas been successfully tested using atissue-equivalent phantom irradiated on a 6OCo treatment unit. The technique should be very useful for large field compensation such as in radiotherapytreatment of Hodgkins disease (mantletechnique),theupper third oesophagus and for whole body irradiation techniques. The method is
Moirk Topography
259
particularly useful for contour mapping for radiotherapy of the head and neck region. For some applications i t is best to have the camera-light assembly mounted on the head of a simulator (or treatment unit). I n this way effective useof the light beam device for aligning the camera with the radiation beam can be made with the patient set in the treatment position. 7.3. Calculation of surface area and volumes
A computer program has been devised by which surface areas and volumes can be calculated from a moire pattern of a curved surface by tracingthe fringes along their contour using the p , 0 device of our computer. Fringes are digitised with a separation between points which varies with object size and is at most 3 mm in real space. Planeareas enclosed by the fringes are calculated by polar integration. Volumes arethenobtainedby using the prismoidal formula and/orthe trapezoidal rule when necessary. Surface areas are obtained by point-by-point numerical integration between successive contour fringes. The technique has been applied in the calculation of the surface area and volume of superficial tumours and thefemale breast (fig. 5). By this means the response of superficial
Fig. 5. ?tIoir
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My IOPscience
Theoretical aspects and practical applications of Moire topography
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1979 Phys. Med. Biol. 24 250 (http://iopscience.iop.org/0031-9155/24/2/002) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 130.237.29.138 This content was downloaded on 05/09/2015 at 11:41
Please note that terms and conditions apply.
PHYS. MED. BIOL., 1979, Vol. 24, No. 2, 250-261.
Printed in Great Britain
Theoretical Aspects and Practical Applications of Moir6 Topography S. S. XENOFOS,
PH.D.
and C. H. JONES, PH.D.,
F.INST.P.
Physics Department, The Institute of Cancer Research, Royal Marsden Hospital, Fulham Road, London SW3 6JJ, U.K.
Received 15 March 1978, infcnal form 20 July 1978 ABSTRACT. When a surface is observed a t a distance through a parallel line grating which is illuminated by a source of light laterally displaced to the viewing axis, the grating is superimposed on its shadow and this gives rise to a moire fringe pattern. The fringes represent cross-sections of the surface parallel to the grating plane. The principle of the formation of the contour map has been studied assuming a grating represented by a Fourier series, a source of lightrepresented by a 3D intensity distribution and a viewing aperture of finite dimensions. The experimental system used for illumination and observation of the moire fringes consists of an equispaced parallel-lined glass grating andtwo small lightsources positioned symmetrically about the imaging device. Thearrangement constitutes a 3D imaging system withan accuracy of the order of 1 mm.Thetechnique has been appliedinradiotherapy problems such as the design of contour compensators and the measurement of body contours. Moire patterns have been analysed with the aid of a PDP-8 computer, and have been used to calculate surface areas and volumes such as in breast studies and superficial tumour measurements.
1. Introduction
The need for accurate measurement of 3D objects arises in many fields of science. I n medicine the measurement of surface topography of the human changes body and its subsequent change is of importance.Surfacecontour are sometimesassociatedwithcertain diseases or abnormalities and can be related to the treatment scheme applied. The measurement of body contours is also important in fitting prostheses and in radiotherapy. Three dimensional measurements can be carried out using stereophotographic methods (Miskin 1960),photogrammetrictechniques of the gridprojection type (Lovesey 1973), holographic stereomethods (Mikhail, Kurz and Stevenson 1972) and coherent moire interferometrictechniques(Brooksand Heflinger 1969). Incoherent moire techniques can also be used. They have the advantage of being accurate, inexpensive and simple and are the subject of this paper. The principles of moire topographyhave been discussed by a number of authors (Meadows, Johnson and Allen 1970, Takasaki 1970, Hovanesian and Hung 1971). In the first part of this paper we present a more general theory of the formation of the contour map assumingalight source and viewing a Fourier series. aperture of finite dimensions and a grating represented by Thisapproachcanbe used to evaluate the moire imagingsystem and is 0031-9155/79/020250+ 12 $01.00
@ 1979 The Institute of Physics
Moire’ Topography
251
considered in more detail in another paper (Xenofos, Jones and Dance 1978). The second part of this paper describes the application of the technique to a variety of clinical problems. Theory The principle of moire surface topography is illustrated in fig. 1. An equispaced parallel-line grating is projected on the surface S by a light source a t A with an intensity distribution I ( X , Y , Z ) . When the surface is observed from B the grating lines are superimposed on their shadows and this gives rise t o a (Ip)is given bythe moire fringe pattern.Theresultantintensitypattern product of the superimposed intensity distributions. 2.
Fig. 1. Principle of moiresurfacetopography. &(x,, ys, 0) and T(z,,yt, 0) are points y, z ) on the where the rays from the light source A directed towards the pointS(%, surface intersect the grating plane. The dark line through point S represents the shadow of the grating around Q on the surface around S. The dark line through point P represents the shadow around S as seen around point P on the grating when viewed through an aperture at B.
In this section we will deriveageneral expression describing the overall intensitypatternas seen from point B (fig. 1). Fromthis expression the contour information required can then be obtained. Provided that the grating spacing is not very small, diffraction at the grating may be neglected even for z. Due tothe finite dimensions of the source and relativelylargedepths viewing aperture, however, the moire pattern willbe blurred. These effects are considered below.
S. S . Xenofos and C . H . Jones
252
2.1. Light Source of jinite dimensions 2.1.1. The overall intensity pattern. For the time being we will assume that the
aperture at B is of very small dimensions so that it can be considered as a point for ray optics, but not small enough for diffraction to be important. The blurring in the moire is due to the fact that rays from the source towards a particularpointon the surfaceintersect thegrating a t points of different transmission. Consider a symmetric grating represented by Fourier series m
G(xp) = E + C 4cos ar(e + xp) r=l
where E is a constant, F, aretheFourier coefficients, e the phase, and a = 277/s0 and so is the grating spacing. The intensity pattern Il around S on the surface is obtained by integrating over the source a t A (fig. 1 )
Il = c o / / / [ E +
1
gF,cosar(e+x,) I(X, Y,Z)dXdYdZ. r=l
The factor c. will be discussed below. grating observedfrom the The overallintensity patternIparound pointPon the point B will then be described by the product of the gratingitself and itsshadow Il
x
/ S[.+
1
gF,cosar(e+x,) I(X, Y,Z)dXdYdZ. r=l
(1)
The factorc. depends uponthe geometry andphysical propertiesof the reflecting surface and that of the imagingsystem. By neglecting variationsin solid angle and assuming that the surface is a diffuse scatterer, this factor can be taken as a constant. For simplicity let c. = 1. In order to pick out the moire term in eqn ( l ) ,it is necessary to carry out the multiplication and express the cosine productsas cosine sumsand differences. The result is cosar(x,-x,)I(X, Y,Z)dXdYdZ r=l
5
+:
~Fm&///cosa(me-re+mxp-rxq)
m#r r=l
xI(X, Y,Z)dXdYdZ
+-l
* Zm~ m & S / / o o s a ( m e + r e + m x p + r x q )
2 m = l r=1
xI(X, Y,.Z)dXdYdZ
W
+ I,E WC1Fmcos am(e +xp)
Moirk Topography
253
where
I,
=/
/ / I ( X , Y,Z)dXdYdZ.
This expressiondescribes the overall intensitypatternas seenfrom the that the second term in the above viewing point. It willbeshownbelow eqn (2) provides contour information (fig. 2). The third term represents the
Fig. 2. Moire patterns of a plaster model of t~ human face with embryosarcoma. Depth difference between fringes A z r 2 mm, L = 100 cm, d = 50 cm, q, = 1 mm. In (a)the moir6 contour lines, the ‘aliasing’ moire on the sides of the face and the grating lines superimposed on the image can be identified. These correspond to the second, third and sixthterms of eqn (2) respectively. The moire pattern of the same object taken with a moving grating is also shown in ( b ) . The ‘aliasing’ and grating lines have been removed whereas the contour moir6 has remained unchanged.
result of superimposition of higher harmonics and when the surface under examination presents a steep inclination to theilluminating beam, an ‘aliasing’ moire is formed which doesnot correspond to thecontour lines (Takasaki 1970, Xenofos 1977) (fig. 2(a)). The fourth, fifth and sixth terms represent the sum of shadow and grating, the shadow, and the grating respectively (fig. 2(a)). All terms except the first two are phase dependent, and can beaveraged out by moving the grating in its plane. Movement of the grating therefore decreases the noise of the system (fig. 2(b)). The pattern observed in such a case will be described by the first two terms in eqn (2) which, as shown below, corresponds to a contour map of the surface behind the grating (fig. 2(b)).
S. X. Xenofos and C . H . Jones
254
2.1.2. The contour moire'. If we express xp,xq as a function of the x coordinate of the point on the surface (fig. 1) the pattern observed (fig. 3) will be given by
x
[?:z:dL-(L+z-z)(L+z)xxz
] I ( X , Y, 2)d X d Yd Z .
(&h+
source
Fig. 3. Experimental arrangement for the observation and recording of the moire fringe pattern of an object.
To facilitate the interpretation of this result we now expand to first order in the quantities X(L+x ) and Z / ( L+ x ) and assume that I ( X , Y,Z) is a symmetric function. Using elementary trigonometric formulae we obtain
where
I, = /&(X,Z)coskrXdX,
S
TV = I(X, Y,Z)dY, k = az/(L+z).
I n a practical system the maximum value of x will be a small fraction of d, and/or L and the x variation in eqn (3) can be neglected. For the systems we have used L 2 100 cm and d 2 L12 cm. To a good approximation, therefore, the moire pattern M ( z ) isindependent of the (x,y) positionon the object. The relative importance of the three cosine terms in eqn (3) can be assessed by comparing their arguments. It then becomes clear that the patternis due to a
Moirk Topography
255
rapidly varying cosine function (cos krd) which is modulated by two slowly varying cosine functions of z (cosine functions within the integral). If these two slowly varying terms are compared for a typical system with d = 0*5L, the term cos krX is the more important but nevertheless the z variation cannot alwaysbeneglected.The moire pattern M ( z ) therefore provides contour information via the function cos krd. 2.2. Aperture of finite dimensions
When the image is viewed through a finite aperture, the rays of light which reach the aperture from a particular point on the surface, will intersect the grating a t points which do not have the same transmission (points P and P’ in fig. 1). I n this case that part of the expression in eqn ( 1 ) before the integral becomes
I’
=
l/[..
5 F,cosam(e+x,,)
m=l
1
&(X’, Y’)dX’dY’
(4)
where &(X‘,Y’) is the aperture function and x*., is shown in fig. 1. By simple geometry we have
If we assume that Q is symmetric, eqns (4)and (5) give m
I‘
= E&,+
C F,Q,cosam(e+x,) m=l
where
Q, = s&JX’) cos kmX’dX’, m = 0 , 1 , 2 ,m,
=
!Q’,(.
Y’)dY’
If we now substitute the first expression in eqn ( 1 ) by that given by eqn ( 6 ) , we obtain the moire pattern due to a combination of a finite source and a finite aperture
M ( z ) = IoQoE2+ - C F?&,COS krd 1 m
2 r=l
Once again we see that M ( z ) provides contour information due to the factor cos krd which is a rapidly varying functionof z. The effect of the finite dimensions of both aperture and source is to degrade the moire pattern as thedepth z is increased. Themagnitude of the degradationdependsuponthedetailed form of Iu(X,Z) and &,(X’) which represent the projections of the light source distribution on the X , Z plane and that of the aperture function on the X’ axis respectively. When the source and viewing aperture become very small I and Q tend to delta functions, and M ( z ) becomes a periodic functionwithconstantamplitude of oscillation. When therefore these distributions are narrow the moire image is improved. If we assume that the integration can be performed independently then M ( z ) is independent of the light source and aperture distribution along the Y, Y‘
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dimension respectively. If the X and 2 light source distributions have similar form and the latter is not wider than the former the 2 dimension will be also of little consequence provided that d < L. We therefore conclude that themost important dimensions in the light source distribution and theviewing aperture are those along X and X’.An evaluation of the image quality of a moire can thus be adequately assessed by studying a one dimensional (X,X’) system. 3. Depth of thecontourfringes Thedepth of the brightcontourfringescanbeobtained cos krd = 1 (in eqn ( 7 ) ) and this gives
byputting
where n is a positive integer called ‘fringe number’. Though dark fringes produced by different harmonics arenot always spatially coincident, those of the fundamental frequency ( r = 1) are much stronger in intensity. The depth of the dark fringes can thus be obtained from eqn (8) by putting n = n - 0.5. I n order that the depth z can be calculated, the fringe number n must be known. The measured distance of a reference point on the surface can be used to assign fringe numbers to the moire fringes through eqn (8) solved for n. Some knowledge of the sign of the gradient of the surface contour is necessary in order that fringe numbers n are assigned correctly. The system, however, can be calibratedso that thefringe number can be obtained from measurements on the image instead of by measurements on the object (Takasaki 1973). Experimentalconfiguration Though the theory developed in section 2 is for one light source, the same results can be obtained when two light sources are used. We have used two light sources arranged symmetrically about the point of observation for shadowfreeillumination,infront of a 50.8 cm x 61 cm (20”x 24”) glass grating with approximately 10 lines per cm. This is supported vertically in a metal frame of adjustable height which allows the grating to slide laterally across the field of view (fig. 3). A camera is used at the viewing point to photograph the moire pattern at a distance of 1 m from the grid. Continuous illumination is used (15 V, 60 W bulbs enclosed in a light shield : the moire image is photographed using a 35 mm with an aperture in the front) camera with a 55 mm Asahi-Pentax lens, and is recorded on Ilford FP4 film with an exposure of 1 S a t f / l 6 . For depth differences between fringes of the order of 4 mm or more a television camera can be used. A permanent record of the image is then obtained by photographing the television screen using a Polaroid camera loaded with4 x 5 inch Polaroid film, type 55 positive/negative, with an exposure of Q S a t f/8. The use of the above combination offers high contrast and small exposure times. The television camera is not used a t small 4.
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depth differences for two reasons; firstly, because of its poorer resolution and secondly, because when the moire is a t a small angle to the television lines an ‘aliasing’ or (sampling) moire may be formed. 5. 3D analysis of moire patterns
The three dimensional reconstruction of a surface canbe calculated from the corresponding moire image. With reference to fig. 3 the z coordinate is given by eqn (8) and the x and y coordinates are given by
x = m,(L+z)x,/c,
y
= m,(L+z)y,/c
(9)
where m, is the magnification factor corresponding to distance c from the fist nodal point of the camera lens, and xi,yi the coordinates of the point on the nth fringe on the photograph. The magnification m, can be calculated from the dimensions on the final image of a scale of known length. 6. Accuracy From eqns (8) and (9) we see that the systematic errors in x, y and z will depend upon the accuracy with which the parameters L, d, S, and c have been measured. Another error is introduced in x and y when the optical axis is not perpendicular tothegrating plane. For errors of the order of a few per thousand in the measurement of lengths the error in the z coordinate (which depends only on uncertainties in thegeometry of the system) is of the order of 0.5% of the value of x . The percentage error in the x,y coordinates is due to both systematic errors and errors introduced during the analysis procedure and depends upon the value of x,y . The order of magnitude of the overall error is +_ 1 mm, I n order that eqn (8) can be used when a glass grating (3 mm thick) is incorporated, the grid lines should be on the surface of the glass plate facing the object. The error introduced in the calculation of the coordinates by using eqn (8) is in this case negligible. When, however, the plate is placed the other way round a small correction should be applied (Xenofos 1977).
Clinicalapplications A number of clinical applications have been investigated and are discussed below.
7.
7 . 1 . Patients’ outlines required for radiotherapy
Cross-sections of the objectthroughthepoint 0 (fig. 3) correspond to straight lines on the recorded image. These can be obtained by measuring the distance x between the point in the image corresponding to 0 and the centre of each fringe which intersects the straight line. Eqns (8) and (9) can then be used to calculate the z and x coordinates. The moire patterns areanalysed from an enlarged photograph with the aid of a PDP-s/E computer used routinely for radiotherapy treatment planning. One of the peripherals of this machine is a p, 0 device which may be used to make on-line measurements of polar
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S. S.
coordinates. This device is used to trace the fringes along the given section and the fringe numbers n are fed automatically at the time of the tracing procedure.Thecontouris then drawn full scale on an incrementalplotter. Measurements on models of various body parts have shown that the overall accuracy of the method is within 2 mm. A useful application of the method would be the measurement of posterior outlines when the patient is irradiated from under the couch. 7.2. Contour compensators required for radiotherapy
By using the appropriate geometry the moire technique can be used for the design of compensators in radiotherapy (Keck, Binder,Gabaj,Klaring and Windischbauer 1976, Xenofos and Jones 1976, 1977). With a camera placed at theposition of the radiation source and the patient set as for treatment (fig. 4) the corresponding moire pattern under a magnification of m, = b/c constitutes a projectionof the contour map of the patient’s Light
sourceyv.
L
l
t
l
b
C
1_;I Compensator tray support
Scaie
--t
______ __
Radiatlon field
Clratln~
Contour
Fig. 4. Application of the moire technique in the design of radiotherapycontour compensators.
skin surface onto the compensator plane (the tray on which the compensator is supported). The missing tissue z, = z - t (fig. 4) corresponding t o each fringe, can then be converted to the equivalent thickness of the compensator material, and the various layers of the compensator cut to the shape of the fringes. Themethodhas been successfully tested using atissue-equivalent phantom irradiated on a 6OCo treatment unit. The technique should be very useful for large field compensation such as in radiotherapytreatment of Hodgkins disease (mantletechnique),theupper third oesophagus and for whole body irradiation techniques. The method is
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particularly useful for contour mapping for radiotherapy of the head and neck region. For some applications i t is best to have the camera-light assembly mounted on the head of a simulator (or treatment unit). I n this way effective useof the light beam device for aligning the camera with the radiation beam can be made with the patient set in the treatment position. 7.3. Calculation of surface area and volumes
A computer program has been devised by which surface areas and volumes can be calculated from a moire pattern of a curved surface by tracingthe fringes along their contour using the p , 0 device of our computer. Fringes are digitised with a separation between points which varies with object size and is at most 3 mm in real space. Planeareas enclosed by the fringes are calculated by polar integration. Volumes arethenobtainedby using the prismoidal formula and/orthe trapezoidal rule when necessary. Surface areas are obtained by point-by-point numerical integration between successive contour fringes. The technique has been applied in the calculation of the surface area and volume of superficial tumours and thefemale breast (fig. 5). By this means the response of superficial
Fig. 5. ?tIoir
