In the square root velocity framework and similar approaches, the computation of shape space distances and the registration of curves requires the solution of a nonconvex variational problem. In this paper, we present a new PDE-based method for solving this problem numerically. The method is constructed from numerical approximation of the Hamilton--Jacobi--Bellman equation for the variational problem and has quadratic complexity and global convergence for the distance estimate. In conjunction, we propose a backtracking scheme for approximating solutions of the registration problem, which additionally can be used to compute shape space geodesics. The methods have linear numerical convergence and improved efficiency compared previous global solvers.


  1. shape registration
  2. curve matching
  3. square root velocity transform
  4. dynamic programming
  5. Hamilton--Jacobi--Bellman equation

MSC codes

  1. 65D19
  2. 49L20
  3. 65K10
  4. 49Q10

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Information & Authors


Published In

cover image SIAM Journal on Imaging Sciences
SIAM Journal on Imaging Sciences
Pages: 762 - 796
ISSN (online): 1936-4954


Submitted: 31 March 2021
Accepted: 17 December 2021
Published online: 7 June 2022


  1. shape registration
  2. curve matching
  3. square root velocity transform
  4. dynamic programming
  5. Hamilton--Jacobi--Bellman equation

MSC codes

  1. 65D19
  2. 49L20
  3. 65K10
  4. 49Q10



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