July 2001
Technical Report: KUL/ESAT/PSI/0105
Retrospective correction
of the heel effect
in hand radiographs
G. Behiels, F. Maes,
D. Vandermeulen and P.
Suetens
Katholieke Universiteit Leuven - Center for Processing Speech
Kasteelpark Arenberg 10
B-3001 Heverlee, Belgium
Telephone: +32-(0)16-32.17.13 Fax: +32-(0)16-32.17.23
Retrospective correction of the heel effect in
hand radiographs
G. Behiels, F. Maes , D. Vandermeulen, and P. Suetens
Faculties of Medicine and Engineering,
Medical Image Computing (Radiology - ESAT/PSI),
University Hospital Gasthuisberg,
Herestraat 49, 3000 Leuven, Belgium
Gert.Behiels@uz.kuleuven.ac.be
Abstract. A method for retrospective correction of intensity inhomogeneities induced by the heel effect in digital radiographs is presented.
The method is based on a theoretical model for the heel effect which is
derived from the acquisition geometry. Because the heel effect is directly
measurable in the direct exposure area only, the image is partitioned
first to exclude collimation and diagnostic areas. The parameters of the
model are determined by fitting the model to the direct exposure area
and the correction is then applied to the whole image. The method iterates between background segmentation and heel effect correction until
convergence. We evaluate the suitability of the method on flat field and
phantom images and demonstrate its robustness on a database of 137
diagnostic hand radiographs.
1
Introduction
Digital radiography offers the possibility for computer aided diagnosis and quantitative analysis using image processing techniques such as contrast enhancement
[10] or segmentation [2–4, 8]. However, computer-based image interpretation is
hindered by the presence of the non-uniformities in X-ray exposure that are
inherent to the image formation and which can be largely attributed to the
heel effect. Although the intensity inhomogeneity induced by the heel effect is a
smoothly varying function of location and is easily corrected by the human visual
perception system, it complicates the use of automatic processing techniques because the brightness of an object within the image is position dependent. The
overall intensity range is unnecessarily enlarged by the presence of these slowly
varying shading components and hence the resolution available to represent diagnostic signal details is reduced. Because the image acquisition parameters that
affect intensity inhomogeneity vary from image to image (e.g. variable positioning of the recording device relative to the X-ray source) and can not be recovered
from the acquired image at read out, correction methods based on calibration
images are not feasible and retrospective methods are needed.
Frederik Maes is Postdoctoral Fellow of the Fund for Scientific Research - Flanders
(FWO-Vlaanderen, Belgium).
2
1
3
1
2
Fig. 1. A typical image of the hand with the heel effect clearly visible on the direct
exposure area (brighter on the left-side of the image and dark on the right). The image
consists of three important regions: (1) the collimation areas (2) the direct exposure
area (3) the hand or diagnostic region
In this paper, we present a fully automated method for intensity inhomogeneity correction of digital radiographs by fitting a mathematical model for the
heel effect derived from the acquisition geometry [5] to the image intensity data.
Because the inhomogeneities are only directly measurable in the background
or direct exposure areas of the image, we first extract the background region
and estimate the parameters of our model from this region only. Inhomogeneity
correction is then applied to the whole image, a new background region is extracted from the corrected data and the model parameters are re-estimated. This
is repeated until no significant changes in background and parameter estimation
occur. We demonstrate the performance of the method on a database of hand
radiographs that were acquired for bone age determination.
2
Modeling the heel effect
A typical hand radiograph is shown in Fig. 1. The background at the left side
of the image is clearly brighter than at the right side. This phenomenon can be
attributed to the so-called heel effect. It is only visible in the direct exposure
and diagnostic areas and not in the collimation area. The heel effect can be
understood from the construction of the X-ray tube as schematically depicted
in Fig. 2. Electrons originating from the cathode are attracted by the positively
charged anode. For better heat dissipation, the anode rotates and is inclined
by a small anode angle θ, which enlarges the area Sactual that is bombarded
by electrons while keeping the size of the focal spot Seff , from which rays are
projected downward to the object, fairly small. As shown in the diagram of
Fig. 2(b), this design makes the length of the path traveled by the X-rays through
the anode larger on the anode side of the field (Ta ) than on the cathode side
(a)
(b)
(c)
High
Voltage
p
Dave
θ
ω
Rotor
filament (cathode)
Ta
φ
e−
ξ= (p R,z R)
D is
Tc
θ
Sactual
target (anode)
S
D ave
ω
anode
y
θ
p
Intensity
S eff
S’eff
γ
pω
γ
Film
z
R
x
position
Fig. 2. (a)-(b) Schematic sides view of an x-ray tube. The anode angle allows the use
of a large focal spot (Sactual ) for heat-loading considerations and a small projected
focal spot (Seff ). X-rays are emitted at an average depth Dave ; the path length on the
anode side Ta is larger and causes a reduction in intensity. (c) X-ray coordinate system
where the X-ray originates at position (0, 0) and travels along R to the film at position
(p, Dis ).
(Tc ). Hence the incident X-ray intensity is smaller at the anode side than at the
cathode side of the recording device, which explains the inhomogeneity of the
background in Fig. 1.
A mathematical model for the heel effect can be derived from the simplified
one-dimensional model of the anode and beam geometry depicted in Fig. 2(c) [5].
In the coordinate system (p, z), with p along the anode-cathode axis and z along
the vertical direction, the X-rays can be taught off to originate within the anode
at point ω(0, 0), at a distance Dave from the anode surface S. Consider the ray
R at an angle φ from the vertical within the plane (ω, S) that hits the recording
device at point (p, Dis ) with Dis the distance between the X-ray source and the
recording device and tan φ = Dpis . The distance r traveled by R through the
anode is given by
r = |ξ − ω| =
2
p2R + zR
(1)
with ξ(pR , zR ) the intersection of R with S which can be found by solving the
system of equations:
S : pR = Dave − tan θ.zR
(2)
R : pR = tan φ.zR
Hence,
1+
r(p) = Dave
p
Dis
cos θ
= Dave
sin(φ + θ)
tan θ +
2
p
Dis
(3)
The radiation received on the recording device is
M (p) = I0 · e−µ·r(p)
(4)
with µ the attenuation coefficient of the anode material and I0 the radiation
originating at ω.
(a)
(b)
(c)
(d)
1
3
2
2
1
Fig. 3. (a) Collimator edge detection: the detected collimation area is painted black
and the rejected boundaries are displayed as white dotted lines. (b) Choice of the
boundary: the diagnostic region is discarded, together with some direct exposure pixels
at the bottom left of the image (c)-(d) first, intermediate and final step of the algorithm
after which no more seed points are left.
Model (4) predicts that the heel effect behaves exponentially along the anodecathode axis and assumes that it is constant perpendicular to this axis. This is
justified by flat field exposure experiments which show that the difference in
intensity perpendicular to the anode-cathode axis is relatively small compared
to the intensity differences along the anode-cathode axis (see Fig. 6).
3
Image partitioning
A typical hand radiograph, as shown in Fig. 1, consists of three regions: collimation area, direct exposure area and diagnostic regions. Because the heel effect is
not present in the collimation area and directly measurable in the direct exposure
area only, we need to partition the image to fit model (4) to the image intensity
data. We do this by first extracting the collimation area and then searching the
direct exposure area, the remaining areas being diagnostic regions.
We find the boundaries of the collimation area using the Hough transform [6],
assuming that these are rectilinear edges as is the case for all hand radiographs in
our database. To make this approach more robust, the contributions of each image point to the Hough accumulator are weighted by its gradient magnitude [1]
and, for each point, only the lines whose direction is within 10 degrees from
the normal to the local gradient direction are considered [7]. The 4 most salient
points in Hough space that represent a quadragon with inner angles between 80
and 100 degrees are selected as candidate boundaries of the collimation area. Because not all 4 collimation boundaries are always present in the image, candidate
boundaries along which the image intensity differs from the intensity expected
for the collimation region are rejected. A typical result is shown in Fig. 3(a).
To extract the background region B, a seed fill algorithm is used that starts
from the boundary of the collimation region as determined in the previous step.
Appropriate seed points for B are found by considering a small band along each
of the collimator edges and retaining all pixels whose intensity is smaller than
the mean of the band. This approach avoids choosing pixels that belong to the
diagnostic region as candidate seed pixels. B is then grown by considering all
neighboring pixels ni , i = 1, . . . , 8 of each pixel p ∈ B and adding qi to B if the
No
Image
Collimation
Detection
Direct Exposure
n=0Estimate a n
n=0
Yes
Determine AnodeCathode Axis
Parameter b n
Estimation
Model
Correction
|an - a n-1|<e 1
or
|bn - b n-1|<e 2
Yes
Corrected
Image
Gradient
Image
n=n+1
No
Fig. 4. Flowchart of the algorithm for heel effect correction. We iterate between model
correction and background estimation until there are no significant changes.
intensity difference between p and qi is smaller than some specified threshold. A
few snapshots of the progress are shown in Fig. 3(b)-(d).
4
Heel effect estimation
To fit the model (4) to the image data N (x, y) we have to find the direction γ
of the anode-cathode axis and the parameters α = [I0 , µ, θ, Dis , Dave , pω ] such
that the model best fits the image data within the direct exposure area extracted
above. pω is a parameter introduced to map point ω where the X-ray originates
to the correct image coordinates (see Fig. 2(c)).
Assuming that γ is known, the average image profile Pγ (p) along this direction in the direct exposure region B is given by
Pγ (p) = N (x, y)(x,y)∈B|x·cos γ+y·sin γ=p
with x and y the image coordinates as defined in Fig. 2(c) and · the averaging
operator. We can then find the optimal model parameters α∗ by fitting the
expected profile M (p, α) to the measured profile Pγ (p):
α∗ (γ) = arg min Pγ (p) − M (p, α)
α
(5)
The fitted one-dimensional model M (p, α∗ (γ)) is then back projected perpendicular to the projection axis γ to obtain a reconstruction R(x, y, γ, α∗ (γ))
for the whole image:
R(x, y, γ, α∗ (γ)) = M (x · cos γ + y · sin γ, α∗ (γ))
The direction of the anode-cathode axis γ is then determined such that this
reconstruction best fits the actual image data within the direct exposure region
using
γ ∗ = arg min N (x, y) − R(x, y, γ, α∗ (γ))(x,y)∈B
(6)
γ
or
γ ∗ = arg min γ
N (x, y)
− 1(x,y)∈B
R(x, y, γ, α∗ (γ))
(7)
depending on whether we wish to use additive or multiplicative correction. The
estimated heel effect is R(x, y, γ ∗ , α∗ (γ ∗ )) and the corrected image is respectively
N̂ (x, y) = N (x, y) − R(x, y, γ ∗ , α∗ (γ ∗ ))
(8)
1
2
3
4
5
6
7
Fig. 5. Segmentations of the first 14 images of our database. The direct exposure area
is painted white and the collimation area is filled with black. The dark lines are the
rejected lines which were candidate collimator edges.
or
N̂ (x, y) =
N (x, y)
.
R(x, y, γ ∗ , α∗ (γ ∗ ))
(9)
The optimal parameters α∗ and γ ∗ are found by multidimensional downhill simplex search [9]. We noticed that the anode-cathode axis in our setup is almost
always parallel to the image or collimation edges. This reduces the number of
orientations which have to be evaluated in (6-7) and reduces computation time.
After inhomogeneity correction of the image using (8-9), the direct exposure area B is updated by thresholding, using a threshold derived from the histogram of the corrected image intensities N̂ . Keeping the previously determined
anode-cathode orientation γ, new values for the optimal model parameters α∗
are determined using (5) taking the newly selected direct exposure region into
account. We thus iterate three or four times between background segmentation
and heel effect correction until convergence. The whole algorithm is summarized
in Fig. 4.
5
Results
The method was tested on 137 digital hand radiographs, recorded with Agfa
ADC cassettes and Agfa ADC-MD10 & ADC-MD30 imaging plates and irradiated by X-ray tubes Philips SRM 06 12 - ROT 500 or Siemens Bi 125/40 RL.
Results of the image partitioning procedure are shown in Fig. 5. Visual inspection showed that the algorithm was able to correctly extract direct exposure,
collimation and diagnostic areas for all images in our database. The time required to perform the partitioning was about 1 to 2 seconds using a Pentium III
800MHz on images resized to about 512 × 512 pixels.
The model (4) of the heel effect was verified using a flat field image and
an image of a hand phantom (Fig. 6). The heel effect is clearly visible in the
flat field image by inspection of intensity traces along both image axes, showing
a smooth degradation along the anode-cathode axis and an almost constant
behavior perpendicular to this axis. A similar pattern is visible in the traces of
the phantom image. Inspection of the traces of the corrected images shows that
most of the background intensity variation is indeed eliminated using model (4).
Physical models of X-ray production, recording and read-out predict a Gaussian
distribution for the intensity of background pixels. This is not the case for the
original flat field and phantom images (see histograms of Fig. 6 a,d). However, the
(a)
Traces
(b)
(c)
Traces
Histogram
Histogram
2650
2650
2600
Surface
Received intensity
2550
2600
2500
2450
50
100
120 100
150
40 20
80 60
Surface
2550
2500
2450
1D model
Measured profile
20
40
60
80
100
Position on film
120
140
160
2680
Traces
Traces
2660
2700
2650
2640
2600
Received intensity
2620
Histogram
2550
2500
2600
2450
2580
50
100
150
120100
Histogram
40 20
80 60
2560
2540
2520
2500
2480
1D model
Measured profile
20
(d)
40
60
80
100
Position on film
(e)
120
140
160
(f)
Fig. 6. (a) Flat field image (background) rendered as a surface (bottom) with histogram
(top-right) and traces (top-left) along the anode-cathode axis (long solid line) and
perpendicular to this axis (shorter dotted line). (b) Average projected data of the
flat field (dotted line) and the fitted model (solid line). The sub-image contains back
projected data (white grid) on top of the rendered background. (c) Same as (a) for
the corrected image. (d) Same as (a) for the phantom image without the background
surface and traces taken along the white lines. (e) Same as (b) for the phantom image
(f) Same as (d) for the corrected phantom image.
distribution of the background pixels of the corrected images is almost perfectly
Gaussian (Fig. 6 c,f), which is another indication that our model performs very
good.
The possibly disturbing effect on the estimation of the model parameters α
of the presence of diagnostic areas (that may not have been properly removed by
the image partitioning procedure) was tested by correcting the flat field image
once with parameters computed from the flat field image itself and once with
parameters computed from the phantom image. The reconstruction errors for
both cases are summarized in table 1. The values for both cases are very similar,
which indicates that the algorithm can cope well with the presence of diagnostic
regions when estimating the background inhomogeneity.
Some selected results on real hand radiographs are shown in Fig. 7. In the first
image, the heel effect is clearly visible, while in the second and third example its
presence can be detected from the increased background intensity at the bottom
right of the images. For all cases the heel effect smears out the histogram of the
image intensity of background. The corrected images have better contrast and
Table 1. Correction errors = N − R of the flat field image (dynamic range = 800)
with model parameters α estimated from either the flat field image itself or from the
hand phantom image.
α estimated from
flat field
phantom image
-3.10
-5.50
σ
12.42
27.11
min -46.73
-63.93
max 60.78
49.28
their background histogram is Gaussian distributed. After correction, specifying
a histogram-derived threshold is sufficient to properly segment the hand, which
is not possible for the original images due to the overlap in intensity between
diagnostic and background regions.
6
Discussion
In this paper we presented a fully automated method for heel effect correction of
digital radiographs. We have evaluated the method on a database of hand radiographs on which we were capable of segmenting the hand very accurately after
inhomogeneity correction. The method is fast and reliable enough to be used
for standardized image display on diagnostic workstations. It can also provide
segmentation techniques with properly normalized images whose relative intensity differences are not disturbed by image inhomogeneities. We are currently
investigating how this affects the specificity of intensity models constructed for
Active Shape Model-based segmentation of the hand bones [2, 3].
The relatively simple theoretical model (4) is able to correct most of the variation present in the direct exposure area of the images that can be attributed to
the heel effect. However, in some images in our experiments intensity inhomogeneities could also be perceived in the direction orthogonal to the anode-cathode
axis. Future work includes extension of model (4) to two-dimensions.
7
Acknowledgments
This work was partly supported by a grant of the Research Fund KU Leuven
GOA/99/05 (Variability in Human Shape and Speech).
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