Example 8 - Spline Tension

Apply spline tension to interpolaton, gradient, derivative, or smoothing routines to reduce the occurance of undershoot / overshoot inconsistencies in the solution.

The values to determine the degree of tension is stored in sigma. Using the routine get_spline_tension_factors will determine the smallest tension factor such that the spline preserves the local shape properties (monotonicity and convexity) of the data. If sigma is zero everywhere, then no tension is active.

We walk through a number of routines that we have explored in previous notebooks, but in this case demonstrating the use of tensioned splines.


import stripy as stripy
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
xmin = 0.0
xmax = 10.0
ymin = 0.0
ymax = 10.0
extent = [xmin, xmax, ymin, ymax]

spacingX = 0.5
spacingY = 0.5

mesh = stripy.cartesian_meshes.elliptical_mesh(extent, spacingX, spacingY, refinement_levels=3)
def analytic(lons, lats, k1, k2):
     return  np.cos(k1*lons) * np.sin(k2*lats) 

def analytic_noisy(lons, lats, k1, k2, noise, short):
     return  np.cos(k1*lons) * np.sin(k2*lats) + short * (np.cos(k1*5.0*lons) * np.sin(k2*5.0*lats)) +  noise * np.random.random(lons.shape)

data   = analytic(mesh.x, mesh.y, 1.0, 1.0)
data_n = analytic_noisy(mesh.x, mesh.y, 1.0, 1.0, 0.1, 0.0)
# get tension factors from the data
sigma   = mesh.get_spline_tension_factors(data, tol=1e-6)
sigma_n = mesh.get_spline_tension_factors(data_n, tol=1e-6)

assert sigma.any(), "if this raises an error, then no tension is active"
assert sigma_n.any(), "if this raises an error, then no tension is active"

Smoothing with tension

Tension is applied simply by supplying sigma. It’s effect is most noticible near the poles where there are edge artefacts in the solution.

stripy_smoothed,  dds, err = mesh.smoothing(data, np.ones_like(data_n), 10.0, 0.1, 0.01)
stripy_smoothed2, dds, err = mesh.smoothing(data, np.ones_like(data_n), 10.0, 0.1, 0.01, sigma=sigma)
def axes_mesh_field(fig, ax, mesh, field, label):
    im = ax.tripcolor(mesh.x, mesh.y, mesh.simplices, field, cmap="PiYG", label=label)
    fig.colorbar(im, ax=ax)

fig, axes = plt.subplots(1,3, figsize=(16,4))

axes_mesh_field(fig, axes[0], mesh, stripy_smoothed, "no tension")
axes_mesh_field(fig, axes[1], mesh, stripy_smoothed2, "spline tension")
axes_mesh_field(fig, axes[2], mesh, stripy_smoothed - stripy_smoothed2, "difference")

Interpolation with tension

Only applies to cubic interpolation. The effect of supplying a non-negative sigma is to produce a more linear interpolation. For regions that lie outside the hull, cubic extrapolation produces wild oscillations in the solution which can be mitigated with spline tension.

# set up a discontinuous mesh and offset

mask_points = mesh.x < 5.0
cmesh = stripy.Triangulation(mesh.x[mask_points]+5, mesh.y[mask_points]+5)

cdata   = analytic(cmesh.x, cmesh.y, 1.0, 1.0)
csigma  = cmesh.get_spline_tension_factors(cdata, tol=1e-6)
# interpolate back to original mesh

grid_z1, ierr = cmesh.interpolate_cubic(mesh.x, mesh.y, cdata) # no tension
grid_z2, ierr = cmesh.interpolate_cubic(mesh.x, mesh.y, cdata, sigma=csigma) # tension
fig, axes = plt.subplots(1,3, figsize=(16,4))

axes_mesh_field(fig, axes[0], mesh, grid_z1, "no tension")
axes_mesh_field(fig, axes[1], mesh, grid_z2, "spline tension")
axes_mesh_field(fig, axes[2], mesh, grid_z1 - grid_z2, "difference")

Gradients with tension

Pass sigma to the following routines that involve derivatives:

  • gradient_lonlat

  • gradient_xyz

  • derivatives_lonlat

Again, the largest difference is visible at the poles.

dx1, dy1 = mesh.gradient(data, nit=5, tol=1e-6) # no tension
dx2, dy2 = mesh.gradient(data, nit=5, tol=1e-6, sigma=sigma) # tension
fig, axes = plt.subplots(2,3, figsize=(16,9))

axes_mesh_field(fig, axes[0,0], mesh, dx1, "x derivative no tension")
axes_mesh_field(fig, axes[0,1], mesh, dx2, "x derivative spline tension")
axes_mesh_field(fig, axes[0,2], mesh, dx1 - dx2, "x derivative difference")

axes_mesh_field(fig, axes[1,0], mesh, dy1, "y derivative no tension")
axes_mesh_field(fig, axes[1,1], mesh, dy2, "y derivative spline tension")
axes_mesh_field(fig, axes[1,2], mesh, dy1 - dy2, "x derivative difference")

The next notebook is Ex9-Voronoi-Diagram