# Illustration of the Gradient of a Second Order Difference

This example explains how to compute the gradient of the second order difference mid point model using `adjoint_Jacobi_field`

s.

This example also illustrates the `PowerManifold`

manifold as well as `ArmijoLinesearch`

.

We first initialize the manifold

`using Manopt, Manifolds`

and we define some colors from Paul Tol

```
using Colors
black = RGBA{Float64}(colorant"#000000")
TolVibrantBlue = RGBA{Float64}(colorant"#0077BB") # points
TolVibrantOrange = RGBA{Float64}(colorant"#EE7733") # results
TolVibrantCyan = RGBA{Float64}(colorant"#33BBEE") # vectors
TolVibrantTeal = RGBA{Float64}(colorant"#009988") # geo
```

Assume we have two points $x,y$ on the equator of the Sphere $\mathcal M = \mathbb S^2$ and a point $y$ near the north pole

```
M = Sphere(2)
p = [1., 0., 0.]
q = [0., 1., 0.]
c = mid_point(M,p,q)
r = shortest_geodesic(M, [0., 0., 1.], c, 0.1)
[c,r]
```

```
2-element Array{Array{Float64,1},1}:
[0.7071067811865475, 0.7071067811865475, 0.0]
[0.11061587104123713, 0.11061587104123713, 0.9876883405951378]
```

Now the second order absolute difference can be stated as (see [Bačák, Bergmann, Steidl, Weinmann, 2016])

where $\mathcal C_{x,z}$ is the set of all mid points $g(\frac{1}{2};x,z)$, where $g$ is a (not necessarily minimizing) geodesic connecting $x$ and $z$.

For illustration we further define the point opposite of

`c2 = -c`

```
3-element Array{Float64,1}:
-0.7071067811865475
-0.7071067811865475
-0.0
```

and draw the geodesic connecting $y$ and the nearest mid point $c$, namely

```
T = [0:0.1:1.0...]
geoPts_yc = shortest_geodesic(M,r,c,T)
```

looks as follows using the `asymptote_export_S2_signals`

export

```
asymptote_export_S2_signals("secondOrderData.asy";
render = asyResolution,
curves = [ geoPts_yc ],
points = [ [x,y,z], [c,c2] ],
colors=Dict(:curves => [TolVibrantTeal], :points => [black, TolVibrantBlue]),
dotSize = 3.5, lineWidth = 0.75, cameraPosition = (1.2,1.,.5)
)
render_asymptote("SecondOrderData.asy"; render=2)
```

Since we moved $r$ 10% along the geodesic from the north pole to $c$, the distance to $c$ is $\frac{9\pi}{20}\approx 1.4137$, and this is also what

`costTV2(M, (p,r,q) )`

`1.413716694115407`

returns, see `costTV2`

for reference. But also its gradient can be easily computed since it is just a distance with respect to $y$ and a concatenation of a geodesic, where the start or end point is the argument, respectively, with a distance. Hence the adjoint differentials `adjoint_differential_geodesic_startpoint`

and `adjoint_differential_geodesic_endpoint`

can be employed, see `∇TV2`

for details. we obtain

`(Xp, Xr, Xq) = ∇TV2(M, (p,r,q) )`

`([-0.0, -4.9676995583751974e-18, -0.7071067811865475], [-0.6984011233337104, -0.6984011233337102, 0.15643446504023084], [4.9676995583751974e-18, 0.0, -0.7071067811865475])`

When we aim to minimize this, we look at the negative gradient, i.e. we can draw this as

```
asymptote_export_S2_signals("SecondOrderGradient.asy";
points = [ [x,y,z], [c,c2] ],
colors=Dict(:tvectors => [TolVibrantCyan], :points => [black, TolVibrantBlue]),
dotSize = 3.5, lineWidth = 0.75, cameraPosition = (1.2,1.,.5)
)
render_asymptote("SecondOrderGradient.asy"; render=2)
```

If we now perform a gradient step, we obtain the three points

`pn, rn, qn = exp.(Ref(M), [p,r,q], [-Xp,-Xr,-Xq])`

```
3-element Array{Array{Float64,1},1}:
[0.7602445970756302, 4.563951614149274e-18, 0.6496369390800624]
[0.6474502912317517, 0.6474502912317516, 0.4020152245473301]
[-4.563951614149274e-18, 0.7602445970756302, 0.6496369390800624]
```

as well we the new mid point

```
cn = mid_point(M,pn,qn)
geoPts_yncn = shortest_geodesic(M,rn,cn,T)
```

and obtain the new situation

```
asymptote_export_S2_signals("SecondOrderMin1.asy";
points = [ [x,y,z], [c,c2,cn], [xn,yn,zn] ],
curves = [ geoPts_yncn ] ,
tVectors = [Tuple.([ [p, -Xp], [r, Xr], [q, Xq] ])],
colors=Dict(:tvectors => [TolVibrantCyan],
:points => [black, TolVibrantBlue, TolVibrantOrange],
:curves => [TolVibrantTeal]
),
dotSize = 3.5, lineWidth = 0.75, cameraPosition = (1.2,1.,.5)
)
render_asymptote("SecondOrderMin1.asy"; render=2)
```

`#md`

One can see, that this step slightly “overshoots”, i.e. $r$ is now even below $c$. and the cost function is still at

`costTV2(M, (pn, rn, qn) )`

`0.46579428818288593`

But we can also search for the best step size using `ArmijoLinesearch`

on the `PowerManifold`

manifold $\mathcal N = \mathcal M^3 = (\mathbb S^2)^3$

```
x = [p,r,q]
N = PowerManifold(M, NestedPowerRepresentation(),3)
s = ArmijoLinesearch(1.0,ExponentialRetraction(),0.999,0.96)(N, x,
x -> costTV2(M, Tuple(x)),
[ ∇TV2(M, (p,r,q))... ] # transform from tuple to PowTVector
)
```

`0.7348020742679654`

and for the new points

```
pm, rm, qm = exp.(Ref(M), [p,r,q], s*[-Xp,-Xr,-Xq])
cm = mid_point(M,pm,qm)
geoPts_xmzm = shortest_geodesic(M,pm,qm,T)
```

we obtain again with

```
asymptote_export_S2_signals("SecondOrderMin2.asy";
points = [ [x,y,z], [c,c2,cm], [xm,ym,zm] ],
curves = [ geoPts_xmzm ] ,
tVectors = [Tuple.( [-ξx, -ξy, -ξz], [x, y, z] )],
colors=Dict(:tvectors => [TolVibrantCyan],
:points => [black, TolVibrantBlue, TolVibrantOrange],
:curves => [TolVibrantTeal]
),
dotSize = 3.5, lineWidth = 0.75, cameraPosition = (1.2,1.,.5)
)
```

Here, the cost function yields

`costTV2( M, (pm, rm, qm) )`

`0.0012555303255725315`

which is nearly zero, as one can also see, since the new center $c$ and $r$ are quite close.

## Literature

- [Bačák, Bergmann, Steidl, Weinmann, 2016]
Bačák, M; Bergmann, R.; Steidl, G; Weinmann, A.:
A second order nonsmooth variational model for restoring manifold-valued images. , SIAM Journal on Scientific Computations, Volume 38, Number 1, pp. A567–597, doi: 10.1137/15M101988X