Relative Pose Estimation for the Femoral Component in Computer-
Assisted Total Hip Replacement Surgeries
Jiyang Gao 1, Hong Chen
1+, Shaojie Su
1 and Zhihua Wang
1
1 Institute of Microelectronics, Tsinghua University, Haidian, Beijing, China
Abstract. Numerous factors influence the rate of dislocation after total hip replacement (THR) surgeries
and malposition of the acetabular and femoral component has long been recognized as an important cause.
To help surgeons improve the accuracy of the positioning of the components, a computer-assisted system for
THR surgeries that estimates and displays the relative pose of femoral and acetabular component is proposed.
The system consists of a miniature camera that is fixed inside the femoral prosthesis trial and a set of
designated reference patterns that are printed on the internal surface of the acetabular prosthesis trial. In the
initialization process, the image, which contains reference patterns on the internal surface of the acetabular
cup, is captured and analyzed. As the femoral component moves, images are captured at different poses and
compared with the initial image to establish correspondences of feature points. The relative pose matrix of
femoral component is recovered from the fundamental matrix that is estimated by the correspondences of
feature points. The system has been evaluated under the simulation with rotation matrix and translation
vector and the experimental results have validated the effectiveness of the proposed pose estimation method.
Keywords: THR, fundamental matrix, reference patterns, feature points
1. Introduction
Numerous factors influence the rate of dislocation after THR such as greater age [1], previous surgery to
the affected hip [2], concomitant neurological deficiencies [1], excessive alcohol intake [3], and nonunion of
the greater trochanter [4]. Malposition has long been recognized as an important cause of dislocation [5], [6].
Accurate positioning of the acetabular component at the time of surgery is crucial in preventing post-
operative dislocation [7]. However the positioning of the components within the safe zone is not guaranteed
when using the freehand technique of operation [8]. If the placement is not in the safe zone, the dislocation
risk of the prostheses is highly increased and the range of the movement is restricted. The prevalence of
dislocation has been reported to be between 0.3% and 10% following primary total hip replacement and up
to 28% after revision arthroplasty [9].
To help surgeons improve the accuracy of the placement of the acetabular components, some navigation
system have been proposed and developed. The first kind of navigation system is based on CT. The accuracy
improvement is significant but CT-based navigation system has not become an established standard routine
due to the additional operation time, expense and exposure to radiation [10]. Several CT-free navigation
systems for THR surgeries are developed both in industry and academia. Orthopilot is a computer assisted,
CT-free navigation system that estimates the relative pose between femoral head and acetabular component
using kinematical calculations without pre-operative scanning [11]. The rotation and translation of the
femoral head are shown in Fig. 1. But extra body landmark equipments are needed in Orthopilot and the
operating process can cause extra damage to the patients’ body.
To overcome aforementioned problems, a visual aided system [12] for THR surgeries has been proposed.
That system estimate the pose of the femoral head using a miniature camera which is placed inside the
femoral prosthesis trial and a set of reference patterns printed on the internal surface of the femoral
Corresponding author. Tel.: +86-10-62795097; fax: +86-10-62771130.
E-mail address: [email protected].
2015 5th International Conference on Biomedical Engineering and Technology (ICBET 2015)
IPCBEE vol.81 (2015) © (2015) IACSIT Press, Singapore
DOI: 10.7763/IPCBEE. 2015. V81. 6
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prosthesis trial. The pose estimation problem is converted to a PNP problem [13]. Basically, the PNP
algorithm uses a set of correspondence of 3D points and their 2D projection to estimate the pose of the
camera. The feature points used in the visual aided system [12] are the vertexes of each reference pattern. It
means that the accurate 3D coordinates of each vertex should be known in advance. However, the problem is
that the accuracy of the position of the patterns can’t be guaranteed in the realistic printing process. The
inaccuracy of the positions of the feature points could lead to a worse pose estimation result.
Fig. 1: The relative pose between the acetabular cup and femoral component.
To get rid of the reliance of the accurate 3D position of each feature point, a more robust pose estimation
method for femoral head based on feature point matches between image pairs and the fundamental matrix is
proposed in this paper. In the remainder of the paper, the system design is introduced in Section II, and then
the algorithm design is discussed in Section III. In Section IV, the analyses of experiment results are given.
2. System Design
2.1. System Design
Fig. 2: The relative pose between the acetabular cup and femoral component.
As shown in Fig. 2, the system contains two major parts. One is for image processing and model
rendering which consists of a computing platform and a transceiver. The other part is to capture images and
transfer data which is placed inside of the femoral head. The image sensor captures an image of the internal
surface of the acetabular, which was covered by reference patterns, and the image was sent to computer
wirelessly by transceivers. Computer algorithms process the image and estimate the camera pose, which is
also the pose of the femoral head, because the camera was firmly fixed to the femoral component.
The pose estimation problem is formulated in the following coordinate systems. Femoral coordinate
frame (with three axes Xc Yc Zc shown in Fig. 2) is attached to the femoral head and its origin located in the 26
camera center. The camera is fixed inside the femoral head so it is rigidly aligned and attached to the femoral
component. As a result, camera coordinate frame is also the femoral coordinate frame. Acetabular coordinate
frame (with three axes Xa Ya Za shown in Fig. 2) is attached to the patient’s body and it is the reference
frame to the femoral frame. Its origin is located in the center of the acetabular cup that is approximately a
hemisphere.
2.2. Reference Patterns Design
To estimate the pose of the camera, two kinds of reference patterns are designed and printed on the
internal surface of the acetabular prosthesis trial liner during the production process. The surface
development of it is shown in Fig. 3.
Fig. 3: Surface development of the reference patterns printed on the internal surface of the liner.
The first kind of pattern is designed as a circular ring, as shown in Fig. 3, which is used as a landmark to
adjust the femur to a preset pose in the initialization step. The projection of a circular ring onto the image
plane is an elliptical ring due to the pinhole camera model [14]. When the camera is right above the pattern I,
the elliptical ring turns to a circular ring on the image plane and the distance between the camera and the
pattern influences the ratio.
The second kind of patterns (as shown in Fig. 3) are designed by the following rules: (1) Pattern II is a
square with black border; (2) Four vertexes of it are used as feature points; (3) Nine small blocks are set
inside the border with three columns and three rows and their colors can be white or black, which represents
1 or 0. Samplings at each block from top left corner to bottom right corner make up a binary sequence that is
the ID of the pattern. In order to make the feature points distinguishable, each reference pattern must have
four different IDs in four directions. For example, the ID sequence of the first pattern in Fig. 3 is
“011111111”. After a clockwise rotation of 90, 180 and 270 degrees, the ID sequences are “110111111”,
“111111110” and “111111011”. So the four vertexes of it are distinguishable. However, the fifth pattern and
the sixth pattern are indistinguishable because the ID sequences are the same “011111110”. So these kinds of
patterns should be avoided.
3. Algorithm Design
Fig. 4: System initialization and pose estimation software pipeline.
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The algorithm contains two major parts. The first part is system initialization and the second part is pose
estimation. The software pipeline is shown in Fig. 4. An initial image is captured and analyzed in system
initialization stage. To estimate the pose, system captures a current image and matches the feature points of it
with the initial image to establish point correspondences. Then fundamental matrix between these two views
is estimated and the pose matrix is recovered from the fundamental matrix.
3.1. System Initialization
Please acknowledge collaborators or anyone who has helped with the paper at the end of the text. Two
problems are solved in the initialization process. The first one is to adjust the femoral head to the preset pose
and the second problem is to initialize the coordination system by the image taken at the preset pose.
When femoral head is inside the acetabular cup, pattern I is in the visual range of the camera. If the
camera’s axis is not perpendicular to the circular ring’s plane and the projection of it is an elliptical ring as
shown in left picture of Fig. 5. When the femoral head is adjusted to the preset position where the camera’s
axis is perpendicular to the circular ring’s plane and the distance between the center of camera and the center
of the ring is exactly (a preset value), a pink circle will appear as shown in right picture of Fig. 5. The pink
circle means that the femoral head is at the preset position inside the acetabular cup and then the motion of
femur could be tracked from that initial position.
Fig. 5: A random pose of the camera and the ellipse in the center (left); the preset pose of the camera and the pink circle
projection of the circle (right).
At the preset pose, an initial image of the reference patterns is captured by image sensors and then is
analyzed to pick out every single reference pattern, as shown in Fig. 6. To search and identify the patterns,
the image is processed in the following steps: (1). Convert the RGB image to gray scale image. (2). Conduct
the adaptive threshold processing to the image. (3). Find the contour in the image. (4). Conduct polygonal
approximation. (5). Pick out the quadrilateral and eliminate the error.
Fig. 6: The analysis of the initial image.
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After picking out the reference patterns in the image, samples are taken at nine points in each reference
pattern to determine nine small blocks’ colors. White represents “1” and black represents “0”. After sampling,
the 9-bit binary sequence is converted to a decimal number that is the ID of the reference pattern. Then the
information of the patterns is stored for further use.
3.2. Pose Estimation
After initialization, the system starts tracking the motion of the femoral head. Suppose we capture an
image at current pose, the following algorithms are used to estimate the pose of the camera, which is also the
pose of the femoral head.
A. Searching and matching feature points
An image of the liner’s internal surface contains reference patterns and is analyzed in the same steps as
shown in the former section. The feature points in the current image are matched with those in the initial
image as shown in Fig. 7. As we can see, the vertexes of the same reference pattern are matched together and
there is no outlier.
Fig. 7: Match feature points between initial image (left) and current image (right).
3.2.1. Estimating fundamental matrix
To estimate the fundamental matrix, a method similar to [14] is used in the computer-assisted system.
Suppose that there is a point correspondence x '« x between two images, one point is x = (u,v,1)T in the
initial image, the other point is x ' = (u ',v ',1)T in the current image, then the fundamental matrix is defined
by the equation
x 'T Fx = 0 (1)
From a set of n point matches, a set of linear equation can be obtained
1 1 1 1 1 1 1 1 1 1 1 1' ' ' ' ' ' 1
: : : : : : : : : 0
' ' ' ' ' ' 1n n n n n n n n n n n n
u u u v u v u v v v u v
Af f
u u u v u u v u v v u v
(2)
11 12 13 21 22 23 31 32 33
Tf f f f f f f f f f (3)
where A is a 9n matrix. Given at least 8 pairs of matches, which means at least 2 reference patterns should
be found in the image, equation (2) could be used to calculate the fundamental matrix F.
3.2.2. Recovering pose matrix
If the camera calibration matrices K andK ' are known, the essential matrix can be defined as
' [ ]TE K FK T R (4)
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which could be decomposed into a skew-symmetric matrix corresponding to translation and a orthonormal
matrix corresponding to rotation between the two views. On the other hand, the single value decomposition
of the essential matrix is TE U V (5)
where 1 2 3( , , )diag and the matrices U and V are orthonormal. According to equation (4) and
equation (5), there are four possible camera pose matrices relative to the initial camera pose.
P ' = R ' | t '[ ] = [UWVT | +u3] or [UWV T | -u3] or [UW TV T | +u3] or [UW TV T | -u3] (6)
where
0 1 0
1 0 0
0 0 1
W
(7)
The correct pose matrix can be distinguished by ensuring that the reconstructed points lie in front of the
camera. After confirming the correct relative pose matrix, the absolute pose matrix can be calculated by
P = P 'P0 (8)
where P0 is the initial camera matrix. As shown in Fig. 8, the green tetrahedron represents the initial pose of
the camera and the blue one represents the current pose of the camera.
Fig. 8: The initial pose and the current pose of the camera.
4. Experiment Result Analysis
An experimental platform as shown in Fig. 9 was set up to simulate the relative movement between
femoral head and acetabular cup. The camera is fixed to a handle that can rotate around x-axis and z-axis and
it faces the acetabular cup that has reference patterns on its internal surface. The camera and the handle are a
test model of the femoral component and their rotations simulate the motion between femoral head and
acetabular cup. The images are transmitted to computer through USB and then the relative rotation matrix R
and the relative translation vector T of the camera are calculated by the algorithm on computer.
The true camera rotation and translation are measured using the scales on the experimental
platform and the relative error of the estimated rotation R is computed the by
(%)ture
rot
R RE
R
(9)
where and R are the normalized rotation matrices. Similarly, the relative error of the estimated
translation T is determined by
(%)ture
trans
T TE
T
(10)
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The relative error of rotation and translation by rotating the femur about x-axis from -35 degrees to 35
degrees and about z-axis from 0 degree to 170 degrees is illustrated in Fig. 10 and Fig. 11 respectively.
Fig. 9: The experimental platform.
Fig. 10: Relative error of rotation and translation by rotating the femur about x-axis from -35 degrees to 35 degrees.
Fig. 11: Relative error of rotation and translation by rotating the femur about z-axis from 0 degree to 170 degrees.
The experimental results show that the relative error of rotation is less than 6% and the actual relative
error of translation is less than 9%, which is more accurate and robust than the PnP-based algorithm in [12].
In addition, the pose estimation algorithm does not rely on the 3D coordinate of feature points which means
that the printing error introduced in the production process would not influence the accuracy of pose
estimation.
5. Conclusion
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This paper presents a method based on feature points between image pairs to estimate relative position
between the femoral and acetabular component. A miniature camera is placed inside the femoral prosthesis
trial and two kinds of reference patterns are designed and printed on the internal surface of the acetabular
prosthesis trial liner. The system acquires an image of at the preset pose and analyzes the reference patterns
to initialize the system. Images captured in the running stage are compared with the initial image to establish
matches of feature points. In the next step, the pose matrix is recovered from the fundamental matrix that is
calculated by matches of feature points. Finally, the relative pose of the acetabular cup and femoral head is
displayed in 3D graphics in vivo, which is very helpful for surgeons to make decision of the position of the
prosthesis during the surgery. The method proposed in this paper does not need any extra landmarks and
does not rely on accurate 3D positions of the feature points. The experimental results show that the relative
error of rotation is less than 6% and the actual relative error of translation is less than 9%, which is more
accurate and robust than the PnP-based algorithm. In future, inertial sensors will also be set inside the
femoral head to improve the accuracy of the estimation and clinic experiments will be carried on.
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