Algorithm 1.

LDDMM-based DWI registration

Input: I0, I1, {μj}, {qi}, λ1, λ2, b, n, {σl} and {α} (step size)

Initialization {υtp ∣ υtp ← 0, ∀p}

1: estimate kernel weights {al}

2: repeat

3:  for each p do

4:    ${\varphi }_{t_p,1}\leftarrow (\mathrm{id}+\frac{1}{n}{\mathit{\upsilon }}_{t_{n-1-p}})\circ {\varphi }_{t_{p+1},1}$

5:    ${\varphi }_{t_p,0}\leftarrow {\varphi }_{t_{p-1},0}\circ (\mathrm{id}-\frac{1}{n}{\mathit{\upsilon }}_{t_{p-1}})$

6:  end for

7:  compute current energy E using (4)

8:  for each p do

9:   compute gradient (∇υ E)tp using (5)

10:  end for

11:  (E′, α) ← linesearch (E, {υtp}, {α}, {(∇υ E)tp})

12:  if E′ < E then

13:   for each p do

14:    υtp ← υtp − α(∇υ E)tp

15:   End For

16:  end if

17: until E′ ≥ E ∨ α = inf {α} ∨ maximum number of iterations

18: function linesearch (E, {υtp}, {α}, {(∇υ E)tp})

19:  for each α do

20:   for each p do

21:     ${\mathit{\upsilon }}_{t_p}^{\prime }\leftarrow {\mathit{\upsilon }}_{t_p}-\alpha {({\nabla }_{\mathit{\upsilon }}E)}_{t_p}$

22:   end for

23:   for each p do

24:     ${\varphi }_{t_p,1}\leftarrow (\mathrm{id}+\frac{1}{n}{\mathit{\upsilon }}_{t_{n-1-p}}^{\prime })\circ {\varphi }_{t_{p+1},1}$

25:     ${\varphi }_{t_p,0}\leftarrow {\varphi }_{t_{p-1},0}\circ (\mathrm{id}-\frac{1}{n}{\mathit{\upsilon }}_{t_{p-1}}^{\prime })$

26:   end for

27:   compute current energy E′ using (4)

28:   if E′ < E then

29:    return E′, α

30:   end if

31:  end for

32:  return E, inf {α}

33: end function