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**Newton’s method on Riemannian manifolds and a geometric model for the human spine.**
*(English)*
Zbl 1056.92002

From the introduction: We are motivated by a geometric model of the spine to study a certain optimization problem. Since the orientation of vertebrae can be specified by a frame of three orthogonal vectors in Euclidean three-space, we are led to finding a constrained minimum, or at least a local minimum, of a real-valued function \(\varphi\) defined on a product \({\mathbf SO}(3)^N\) of special orthogonal groups. The function in question will turn out to be quadratic and the constraint affine.

In order to take advantage of the Lie-group structure of \({\mathbf SO}(3)\), we preferred to treat this constrained optimization problem as one of finding the zeros of a gradient vector field on a sub-manifold of \({\mathbf SO} (3)^N\) rather than use the method of Lagrange multipliers. There is some evidence that the numerics of our approach may be slightly better conditioned, but more analysis is needed in order to justify this claim.

In Section 2 we introduce and motivate our model for the shape of the spine. We compare spines predicted by our model with measured ones for five patients.

Section 3 we describe Newton’s method for the problem: Find a zero for the map \(F:{\mathbf M}\to{\mathbf V}\) where \({\mathbf M}\) is a differentiable manifold, \({\mathbf V}\) is a Euclidean vector space and a zero is a point \(x\in{\mathbf M}\) such that \(F(x)=0\). In this section we introduce retractions and give some examples.

In Section 4 we study these questions on manifolds and even extend what is known on vector spaces. An alternative to solving least-square problems by the Newton-Gauss method is to use Newton’s method to find zeros of the gradient vector field grad \(f(x)\) where \(f(x)=\|F(x)\|^2/2\). In Section 5 we study Newton’s method for finding zeros of vector fields on manifolds. The Rayleigh Quotient Iteration for the eigenvalue problem arises naturally in this context.

In Section 6 we derive formulae for the Newton iteration for the gradient vector field associated with our model of the spine.

In order to take advantage of the Lie-group structure of \({\mathbf SO}(3)\), we preferred to treat this constrained optimization problem as one of finding the zeros of a gradient vector field on a sub-manifold of \({\mathbf SO} (3)^N\) rather than use the method of Lagrange multipliers. There is some evidence that the numerics of our approach may be slightly better conditioned, but more analysis is needed in order to justify this claim.

In Section 2 we introduce and motivate our model for the shape of the spine. We compare spines predicted by our model with measured ones for five patients.

Section 3 we describe Newton’s method for the problem: Find a zero for the map \(F:{\mathbf M}\to{\mathbf V}\) where \({\mathbf M}\) is a differentiable manifold, \({\mathbf V}\) is a Euclidean vector space and a zero is a point \(x\in{\mathbf M}\) such that \(F(x)=0\). In this section we introduce retractions and give some examples.

In Section 4 we study these questions on manifolds and even extend what is known on vector spaces. An alternative to solving least-square problems by the Newton-Gauss method is to use Newton’s method to find zeros of the gradient vector field grad \(f(x)\) where \(f(x)=\|F(x)\|^2/2\). In Section 5 we study Newton’s method for finding zeros of vector fields on manifolds. The Rayleigh Quotient Iteration for the eigenvalue problem arises naturally in this context.

In Section 6 we derive formulae for the Newton iteration for the gradient vector field associated with our model of the spine.