add function to compute an adapted grid of a function on an interval

src/PlotUtils.jl

@@ -15,11 +15,13 @@ export
     invisible,
     get_color_palette,
     isdark,
-    plot_color
+    plot_color,
+    adapted_grid
 
 include("color_utils.jl")
 include("color_gradients.jl")
 include("colors.jl")
+include("adapted_grid.jl")
 
 export
     optimize_ticks

src/adapted_grid.jl

@@ -0,0 +1,137 @@
+
+"""
+    adapted_grid(f, minmax::Tuple{Number, Number}; max_recursions = 7)
+
+Computes a grid `x` on the interval [minmax[1], minmax[2]] so that
+`plot(f, x)` gives a smooth "nice" plot.
+The method used is to create an initial uniform grid (21 points) and refine intervals
+where the second derivative is approximated to be large. When an interval
+becomes "straight enough" it is no longer divided. Functions are never evaluated
+exactly at the end points of the intervals.
+
+The parameter `max_recusions` computes how many times each interval is allowed to
+be refined.
+"""
+function adapted_grid(f, minmax::Tuple{Real, Real}; max_recursions = 7)
+    if minmax[1] >= minmax[2]
+        throw(ArgumentError("interval must be given as (min, max)"))
+    end
+
+    # When an interval has curvature smaller than this, stop refining it.
+    max_curvature = 0.05
+
+    # Initial number of points
+    n_points = 21
+    n_intervals = n_points ÷ 2
+    @assert isodd(n_points)
+
+    xs = collect(linspace(minmax[1], minmax[2], n_points))
+    # Move the first and last interior points a bit closer to the end points
+    xs[2] = xs[1] + (xs[2] - xs[1]) * 0.25
+    xs[end-1] = xs[end] - (xs[end] - xs[end-1]) * 0.25
+
+    # Wiggle interior points a bit to prevent aliasing and other degenerate cases
+    rng = MersenneTwister(1337)
+    rand_factor = 0.05
+    for i in 2:length(xs)-1
+        xs[i] += rand_factor * 2 * (rand(rng) - 0.5) * (xs[i+1] - xs[i-1])
+    end
+
+    n_tot_refinements = zeros(Int, n_intervals)
+
+    # We only evaluate the function on interior points
+    fs = [NaN; [f(x) for x in xs[2:end-1]]; NaN]
+    while true
+        curvatures = Vector{Float64}(n_intervals)
+        active = Vector{Bool}(n_intervals)
+        max_f = maximum(abs, fs)
+        # Guard against division by zero later
+        if max_f == 0
+            max_f = one(max_f)
+        end
+        # Skip first and last interval
+        for interval in 2:n_intervals-1
+            p = 2 * interval
+            tot_w = 0.0
+            # Do a small convolution
+            for (q,w) in ((-1, 0.25), (0, 0.5), (1, 0.25))
+                interval == 1 && q == -1 && continue
+                interval == n_intervals && q == 1 && continue
+                tot_w += w
+                i = p + q
+                # Estimate integral of second derivative over interval, use that as a refinement indicator
+                # https://mathformeremortals.wordpress.com/2013/01/12/a-numerical-second-derivative-from-three-points/
+                curvatures[interval] += abs(2 * ((fs[i+1] - fs[i]) / ((xs[i+1]-xs[i]) * (xs[i+1]-xs[i-1]))
+                                                -(fs[i] - fs[i-1]) / ((xs[i]-xs[i-1]) * (xs[i+1]-xs[i-1])))
+                                                * (xs[i+1] - xs[i-1])^2) / max_f * w
+            end
+            curvatures[interval] /= tot_w
+            # Only consider intervals that have not been refined too much and have a high enough curvature
+            active[interval] = n_tot_refinements[interval] < max_recursions && curvatures[interval] > max_curvature
+        end
+        # Approximate end intervals as being the same curvature as those next to it.
+        # This avoids computing the function in the end points
+        curvatures[1] = curvatures[2]
+        active[1] = active[2]
+        curvatures[end] = curvatures[end-1]
+        active[end] = active[end-1]
+
+        if all(x -> x >= max_recursions, n_tot_refinements[active])
+            break
+        end
+
+        n_target_refinements = n_intervals ÷ 2
+        interval_candidates = collect(1:n_intervals)[active]
+        n_refinements = min(n_target_refinements, length(interval_candidates))
+        perm = sortperm(curvatures[active])
+        intervals_to_refine = sort(interval_candidates[perm[length(perm) - n_refinements + 1:end]])
+        n_intervals_to_refine = length(intervals_to_refine)
+        n_new_points = 2*length(intervals_to_refine)
+
+        # Do division of the intervals
+        new_xs = similar(xs, n_points + n_new_points)
+        new_fs = similar(fs, n_points + n_new_points)
+        new_tot_refinements = similar(n_tot_refinements, n_intervals + n_intervals_to_refine)
+        k = 0
+        kk = 0
+        for i in 1:n_points
+            if iseven(i) # This is a point in an interval
+                interval = i ÷ 2
+                if interval in intervals_to_refine
+                    kk += 1
+                    new_tot_refinements[interval - 1 + kk] = n_tot_refinements[interval] + 1
+                    new_tot_refinements[interval + kk] = n_tot_refinements[interval] + 1
+
+                    k += 1
+                    new_xs[i - 1 + k] = (xs[i] + xs[i-1]) / 2
+                    new_fs[i - 1 + k] = f(new_xs[i-1 + k])
+
+                    new_xs[i + k] = xs[i]
+                    new_fs[i + k] = fs[i]
+
+                    new_xs[i + 1 + k] = (xs[i+1] + xs[i]) / 2
+                    new_fs[i + 1 + k] = f(new_xs[i + 1 + k])
+                    k += 1
+                else
+                    new_tot_refinements[interval + kk] = n_tot_refinements[interval]
+                    new_xs[i + k] = xs[i]
+                    new_fs[i + k] = fs[i]
+                end
+            else
+                new_xs[i + k] = xs[i]
+                # Don't evaluate function at end points
+                if !(i == 1 || i == n_points)
+                    new_fs[i + k] = fs[i]
+                end
+            end
+        end
+
+        xs = new_xs
+        fs = new_fs
+        n_tot_refinements = new_tot_refinements
+        n_points = n_points + n_new_points
+        n_intervals = n_points ÷ 2
+    end
+
+    return xs[2:end-1]
+end