Iterative Refinement

This section of the manual describes an approximation method for an arbitrary two-dimensional convex set $S$ and a given error bound $ε$ using support vectors. The basic idea is to add new supporting directions whenever the approximation error is still bigger than $ε$.

Local approximations

The polygonal approximation of an arbitrary lazy convex set S is represented by a list of local approximations or refinements. More precisely, a local approximation is a triple $(p_1, p_2, q)$, where:

  • $p_1$ and $p_2$ belong to $S$
  • the segments $(p_1 q)$ and $(p_2 q)$ belong to support lines of $S$

Since $S$ is assumed to be convex, the segment $(p_1 p_2)$ is inside $S$. Taking each support line $(p_1 q)$ of a given list of local approximations of $S$, we can build a polygon in constraint representation that overapproximates S.

The type LocalApproximation{N} implements a local approximation; it is parametric in the numeric type N, and also contains additional information regarding the quality of the approximation: The refinable field is a boolean that is true whenever the approximation can be improved, and err is an upper bound on the exact Hausdorff distance of the approximation with respect to the exact set S.

Given the unit ball in the 2-norm, below we plot the local approximation along the East and North directions.

using Plots, LazySets, LazySets.Approximations

b = Ball2(zeros(2), 1.)

plot(b, 1e-3, aspectratio=1, alpha=0.3, legend=false)

plot!(Singleton([1.0, 0.0]), annotations=(1.1, 0.1, text("p1")), color="green")
plot!(Singleton([0.0, 1.0]), annotations=(0.1, 1.1, text("p2")), color="green")
plot!(Singleton([1.0, 1.0]), annotations=(1.09, 1.1, text("q")))
plot!(Singleton([0.0, 0.0]), annotations=(0.1, 0.0, text("0")), color="green")
plot!(annotations=(1.4, 0.1, text("d1")))
plot!(annotations=(0.1, 1.4, text("d2")))
plot!(annotations=(0.75, 0.8, text("ndir")))

plot!(x->x, x->1., -0.8, 1.3, line=1, color="black", linestyle=:dash)
plot!(x->1., x->x, -0.8, 1.3, line=1, color="black", linestyle=:dash)
plot!(x->x+1, x->0., 0.0, 0.4, line=1, color="red", linestyle=:solid, arrow=true)
plot!(x->0., x->x+1, 0.0, 0.4, line=1, color="red", linestyle=:solid, arrow=true)
plot!(x->-x, x->x+1, -1.2, .2, line=1., color="black", linestyle=:dashdot)
plot!(x->x+.6, x->x+.6, -.1, .08, line=1, color="red", linestyle=:solid, arrow=true)

We can instantiate and append this approximation to a fresh PolygonalOverapproximation object, which is a type that wraps a set and a list of LocalApproximations. The approximation is refinable, since it can be "split" along ndir, where ndir is the direction normal to the line $(p_1 p_2)$ (shown dash-dotted in the figure), providing two approximations which are closer to the given set in Hausdorff distance.

import LazySets.Approximations:PolygonalOverapproximation, addapproximation!

Ω = PolygonalOverapproximation(b)
p1, d1, p2, d2 = [1.0, 0.0], [1.0, 0.0], [0.0, 1.0], [0.0, 1.0]
approx_EAST_NORTH = addapproximation!(Ω, p1, d1, p2, d2)

approx_EAST_NORTH.refinable
true

The associated error is $\sqrt{2}-1≈0.414213$, which is the distance between the point $q$ and the intersection between the line $(0 q)$ and the circle. Actually this point corresponds to the support vector of the set b along ndir.

approx_EAST_NORTH.err
0.414213562373095

The refined approximation is computed next.

Refinement

Basically, the refinement step consists of splitting the local approximation $(p_1, p_2, q)$ into two local approximations $(p_1, s, q')$ and $(s, p_2, q'')$, where s is the support vector of $S$ along ndir.

To illustrate this, first let's add the remaining three approximations to Ω along the canonical directions, to build a box overapproximation of b.

import LazySets.Approximations: refine, tohrep

plot(b, 1e-3, aspectratio=1, alpha=0.3)

# initialize box directions
DIR_EAST, DIR_NORTH, DIR_WEST, DIR_SOUTH = [1., 0.], [0., 1.], [-1., 0.], [0., -1.]
pe, pn, pw, ps = σ(DIR_EAST, b), σ(DIR_NORTH, b), σ(DIR_WEST, b), σ(DIR_SOUTH, b)

Ω = PolygonalOverapproximation(b)
addapproximation!(Ω, ps, DIR_SOUTH, pe, DIR_EAST)
addapproximation!(Ω, pw, DIR_WEST, ps, DIR_SOUTH)
addapproximation!(Ω, pn, DIR_NORTH, pw, DIR_WEST)
addapproximation!(Ω, pe, DIR_EAST, pn, DIR_NORTH)

plot!(tohrep(Ω), alpha=0.2, color="orange")

Next we refine the first approximation of the list.

approx = pop!(Ω.approx_stack)
(r1, r2) = refine(approx, Ω.S)
push!(Ω.approx_stack, r2)
push!(Ω.approx_stack, r1)

plot(b, 1e-3, aspectratio=1, alpha=0.3)
plot!(tohrep(Ω), alpha=0.2, color="orange")

We call r1 and r2 the right and left approximations respectively, since they are saved in counter-clockwise order. We can check that the first two approximations are still refinable.

Ω.approx_stack[end].refinable,  Ω.approx_stack[end-1].refinable
(true, true)

Hence, we can make again a refinement of that approximation.

approx = pop!(Ω.approx_stack)
(r1, r2) = refine(approx, Ω.S)
push!(Ω.approx_stack, r2)
push!(Ω.approx_stack, r1)

plot(b, 1e-3, aspectratio=1, alpha=0.3)
plot!(tohrep(Ω), alpha=0.2, color="orange")

The criterion for an approximation being refinable is that we can properly define a normal direction ndir. This boils down to checking for the following "degenerate" cases:

  1. $p_1$ and $p_2$ overlap.
  2. $p_1$ and $q$ overlap.
  3. $p_2$ and $q$ overlap.

Moreover, we include the condition approx_error > TOL where TOL is the floating point epsilon in the given numerical precision.

Algorithm

Having presented the individual steps, we give the pseudocode of the iterative refinement algorithm, see _approximate(S, ε).

The algorithm consists of the following steps:

  1. Initialization. The approximation is initialized with box directions, i.e. it starts with four LocalApproximation objects. Let i=1.
  2. Refinement loop. If the local approximation at index i has an error greater than the threshold ε, then refine. Otherwise, increment i <- i+1.
  3. Redundancy check. Insert the refined right approximation at position i, and check whether the left approximation is redundant or not with respect to the one at position i+1. Checking for redundancy amounts to checking for overlap of both p1 and q. Then, either substitute at i+1 or insert (keeping the approximation at i+1) depending on the redundancy check.
  4. Stopping criterion. Terminate if the index i exceeds the current length of the approximations list; otherwise continue with step 2.

Observe that the algorithm finishes when all approximations are such that their associated error is smaller than ε, hence the Hausdorff distance between S and its polygonal overapproximation is no greater than ε.

Example

As a final example consider the iterative refinement of the ball b for different values of the approximation threshold ε.

import LazySets.Approximations:overapproximate, _approximate

p0 = plot(b, 1e-6, aspectratio=1)
p1 = plot!(p0, overapproximate(b, 1.), alpha=0.4, aspectratio=1)

p0 = plot(b, 1e-6, aspectratio=1)
p2 = plot!(p0, overapproximate(b, 0.1), alpha=0.4, aspectratio=1)

p0 = plot(b, 1e-6, aspectratio=1)
p3 = plot!(p0, overapproximate(b, 0.01), alpha=0.4, aspectratio=1)

plot(p1, p2, p3, layout=(1, 3))

Meanwhile, the number of constraints of the polygonal overapproximation increases, in this example by a power of 2 when the error is divided by a factor 10.

h = ε ->  length(_approximate(b, ε).constraints)
h(1.), h(0.1), h(0.01)
(4, 8, 32)
Note

Actually, the plotting function for an arbitrary ConvexSet plot(...), called recipe in the context of Plots.jl, is such that it receives a numeric argument ε and the routine itself calls overapproximate. However, some sets such as abstract polygons have their own plotting recipe and hence do not require the error threshold, since they are plotted exactly as the convex hull of their vertices.