Recovering Elastic Property of Soft Tissues Using 2D Image Sequences With Limited Range Data Yong Zhang, Dmitry B. Goldgof and Sudeep Sarkar Department of Computer Science & Engineering University of South Florida Tampa, FL 33620 zhang, or goldgof, or
[email protected]
Abstract
Min C. Shin Department of Computer Science University of North Carolina, Charlotte 9201 University City Blvd Charlotte, NC 28223
Frame-1
Detecting abnormal material properties associated with diseased tissues from images requires accurate displacement data. We examined three affine model-based methods for measuring 3D displacements in regular image sequences. The advantage of these methods is that only one or no range image is needed for the whole sequence. We demonstrate, using images of both synthetic material and real burn patients, that the affine tracking method is suitable for many image based medical analyses.
Frame-2
Frame-n
Affine Model 3D displacement Forward Physical Model Regularized Parameter Estimation Recovered Elastic Property
Figure 1. Framework for recovering material properties.
1
Introduction
Determining actual material properties of soft tissues from images has important scientific as well as clinical consequences. It has been known that many pathological conditions can cause changes in material properties such as elasticity and electric conductivity [2, 1, 4] Mathematical models used for recovering material properties includes three components: (1) a method that measures 3D displacements from images, (2) a forward model that defines the physical behavior of biological objects, and (3) an optimization algorithm that estimates the material properties from the displacement data (Figure 1). This paper focus on the 3D displacement measurement using affine model-based methods. Because the computed material property is very sensitive to data errors, it is critical to have a method that can measure the displacement as accurately as possible. Specially designed ultrasonic devices have been used to generate vibrations and thus deformations in a small area of soft tissue, from which strain and elasticity can be further computed [6]. The advantage of ultrasonic displacement is that it can be measured in both internal organs and external surface of tissues. The main drawback is the strong artifacts associated with unstable speckle patterns. In our previous studies on burn scar assessment [10, 11], we used active
contour and natural features to measure displacements on regular and range images. The resulting displacement field is complete and reliable, which enables physical models to be constructed for the whole object rather than for a local region, as in the case of ultrasonic approach. Physical models that cover the entire body allow more accurate boundary assignment and hence material property computation. The drawback of this method is that range data is needed for each frame of the image sequence, which restrains its applications. In this paper, we have experimented with three new methods to measure 3D displacements. These methods are derived from a affine motion model and have less reliance on range images. Specifically, we have been able to compute good displacements from an image sequence using only one or no range image, which expands the applicability of the proposed method to a wider range of domains.
2
Physical Model and Property Recovery
Before addressing the three methods for measuring displacements, we give a brief discussion on how to recover material property from displacement data by minimizing a Tikhonov functional. The elastic deformation of an object
1051-4651/02 $17.00 (c) 2002 IEEE
under the influence of external forces can be numerically approximated by a finite element model:
where M is the mass matrix, D is the damping matrix, and K is the stiffness matrix with functions of Young’s modulus and Poisson ratio as its coefficients. We are interested in estimating the Young’s modulus as a m-dimensional vector (m is the number of elements) from a n-dimensional displacement vector measured on n selected locations (n is usually less than m). We map the parameter vector to the data vector by a nonlinear opthat represents the forward model: erator
Methods for measuring 3D Displacements Tracking Alignment Align+1TrueZ
(1)
Table 1. Range Data Requirements Range Data Needed one range frame none one range point
DE
F
G
G
F
K F
G
G
K
K F
F
K
G MN
K F
(4) .
C
G F
K F
F
.
.
. .
.
transforms a 3D point ( ) in frame to the next frame ( ) by applying to as . The 2D point and its corresponding 3D point in the same frame are related by perspective projection. The intrinsic calibration parameters are used to convert between (image coordinate) and (sensor plane coordinate). Then, the perspective projection equation relates with (3D coordinate). The error function is the sum of distance between the motion-driven structure sequence and the structuredriven sequence . Let be the guessed motion and be the guessed structure. is computed by applying to all 3D points in the initial frame ( ) for ( ) times. is computed by estimating 3D location of all points using the corresponding estimated depth ( ) and the senand sor plane coordinate ( ). Let . For each pair of and , we compute which is the sum of the absolute difference between and in each dimension. The error function is the sum of all and the penalty function . K
K
P
C
B
O
Q
P
(2)
This is a typical ill-posed parameter estimation problem, because the real data are always noisy and the dependence of the parameter on the data is not continuous. We have to use the regularization technique to obtain a stable estimate of [7]:
P
P
B
O
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C
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,
6
8
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