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
import cartopy.crs as ccrs
%matplotlib inline

mesh = stripy.spherical_meshes.icosahedral_mesh(refinement_levels=4)

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.lons, mesh.lats, 5.0, 2.0)
data_n = analytic_noisy(mesh.lons, mesh.lats, 5.0, 2.0, 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)

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)

import k3d

plot = k3d.plot(camera_auto_fit=False, grid_visible=False, 
                menu_visibility=True, axes_helper=False )

indices = mesh.simplices.astype(np.uint32)
points = np.column_stack(mesh.points.T).astype(np.float32)

mesh_viewer = k3d.mesh(points, indices, wireframe=False, attribute=data_n,
                   color_map=k3d.colormaps.paraview_color_maps.Cool_to_Warm, 
                   name="splines",
                   flat_shading=False, opacity=1.0  )


plot   += mesh_viewer
plot.display()

## ## ## 

from ipywidgets import interact, interactive
import ipywidgets as widgets

choices = { "original": data_n,
             "smoothed": stripy_smoothed, 
             "smooth with tension": stripy_smoothed2, 
             "delta": stripy_smoothed - stripy_smoothed2  }

@interact(choice=choices.keys())
def chooser(choice):
    mesh_viewer.attribute = choices[choice].astype(np.float32)
    range = choices[choice].max() * 0.2
    mesh_viewer.color_range = [0, range]
    return 

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

mask_points = mesh.lats < np.pi/3
cmesh = stripy.sTriangulation(mesh.lons[mask_points], mesh.lats[mask_points])

cdata   = analytic(cmesh.lons, cmesh.lats, 5.0, 2.0)
csigma  = cmesh.get_spline_tension_factors(cdata, tol=1e-6)

grid_z1, ierr = cmesh.interpolate_cubic(mesh.lons, mesh.lats, cdata) # no tension
grid_z2, ierr = cmesh.interpolate_cubic(mesh.lons, mesh.lats, cdata, sigma=csigma) # tension

import k3d

plot = k3d.plot(camera_auto_fit=False, grid_visible=False, 
                menu_visibility=True, axes_helper=False )

indices = mesh.simplices.astype(np.uint32)
points = np.column_stack(mesh.points.T).astype(np.float32)

mesh_viewer = k3d.mesh(points, indices, wireframe=False, attribute=data_n,
                   color_map=k3d.colormaps.paraview_color_maps.Cool_to_Warm, 
                   name="splines",
                   flat_shading=False, opacity=1.0  )


plot += mesh_viewer
#plot += background
plot.display()

## ## ## 

from ipywidgets import interact, interactive
import ipywidgets as widgets

choices = {  
             "cubic": grid_z1, 
             "cubic with tension": grid_z2, 
             "delta": grid_z1 - grid_z2  }

@interact(choice=choices.keys())
def chooser(choice):
    mesh_viewer.attribute = choices[choice].astype(np.float32)
    range = choices[choice].max() * 0.2
    mesh_viewer.color_range = [-range, range]
    return 

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.

dlon1, dlat1 = mesh.gradient_lonlat(data, nit=5, tol=1e-6) # no tension
dlon2, dlat2 = mesh.gradient_lonlat(data, nit=5, tol=1e-6, sigma=sigma) # tension

import k3d

plot = k3d.plot(camera_auto_fit=False, grid_visible=False, 
                menu_visibility=True, axes_helper=False )

indices = mesh.simplices.astype(np.uint32)
points = np.column_stack(mesh.points.T).astype(np.float32)

mesh_viewer = k3d.mesh(points, indices, wireframe=False, attribute=data_n,
                   color_map=k3d.colormaps.paraview_color_maps.Cool_to_Warm, 
                   name="splines",
                   flat_shading=False, opacity=1.0  )


plot += mesh_viewer
plot.display()

## ## ## 

from ipywidgets import interact, interactive
import ipywidgets as widgets

choices = { "dlon": dlon1,
            "dlat": dlat1,
            "dlon + tension": dlon2,
            "dlat + tension": dlat2,
            "dlon delta": dlon1 - dlon2,
            "dlat delta": dlat1 - dlat2 }


@interact(choice=choices.keys())
def chooser(choice):
    mesh_viewer.attribute = choices[choice].astype(np.float32)
    range = choices[choice].max() * 0.2
    mesh_viewer.color_range = [-range, range]
    return 

The next notebook is Ex9-Voronoi-Diagram

EMSC 4033 Computational Geoscience

Example 8 - Spline Tension

Contents

Example 8 - Spline Tension¶

Contents¶

Smoothing with tension¶

Interpolation with tension¶

Gradients with tension¶