Solution of Inverse Problems in Biomechanical Imaging
In this set of notes we consider inverse problems in biomechanical imaging. We briefly describe the clinical relevance of these problems and how the measured data is acquired. We focus on two distinct strategies for solving these problems. These include a
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Matrix Inverse Eigenvalue Problems
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; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;
Figure 4. The graph underlying A is a tree.
x ∈ Rn . The stiffness matrix K of an (undamped) anchored system is PD; xT Kx is related to the potential, or strain, energy of the system. It is well-known that xT Ax > 0 if and only if all the principal minors of A are positive. A principal minor of A is a minor taken from a set of columns and the same set of rows. A ∈ Sn is PD if and only if its eigenvalues (λi )n1 are all positive. Now consider the concept of a positive matrix. If A ∈ Mn , then A≥0 means that each entry in A is non-negative A>0 means A ≥ 0 and at least one entry is positive A >> 0 means that each entry in A is positive. Let Zn = {A : A ∈ Mn ;
aij ≤ 0,
i = j}.
(20)
If A ∈ Zn , then it may be expressed in the form A = sI − B,
s > 0,
B ≥ 0.
(21)
The spectral radius of B ∈ Mn is ρ(B) = max{|λ|;
λ is an eigenvalue of B}.
A matrix A ∈ Mn of the form (21) with s ≥ ρ(B) is called an M -matrix ; if s > ρ(B) it is a non-singular M -matrix. Berman and Plemmons (1994) construct an inference tree of properties of non-singular M -matrices; one of the most important properties is that A−1 > 0; each entry in A−1 is non-negative and at least one is positive. A symmetric non-singular M -matrix is called a Stieltjes matrix, and importantly A ∈ Sn ∩ Zn is a Stieltjes matrix iff it is PD. Moreover, if A
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G.M.L. Gladwell
is an irreducible Stieltjes matrix then A−1 >> 0: each entry in A−1 is strictly positive. This result has important consequences for some physical systems. For a physical system, like an on-line spring mass system, or a finite element model of a membrane in out-of-plane movement, the relation between static displacement and applied force has the form Ku = f . For an anchored spring-mass system, the stiffness matrix is an irreducible non-singular M -matrix, so that K−1 >> 0. This agrees with our intuition that a positive force applied to any mass will displace each mass in the positive direction. Gladwell et al. (2009) showed that the analogue of this result holds for a FE model of a membrane. For in-line spring-mass systems, and for the model of a beam in flexure, the stiffness matrix satisfies more stringent conditions which ensure that the eigenvalues are distinct, that the first eigenmode has no node, that the second mode has one node, etc. The matrix property is so-called t
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