Whats the Point Interpolation

1
Whats the Point?Interpolation Extrapolation
with a Regular Grid DEM

• David Kidner, Mark Dorey Derek Smith
• University of Glamorgan
• School of Computing
• Pontypridd
• WALES, U.K. CF37 1DL
• e-mail dbkidner_at_glam.ac.uk

2
Whats the Point?

• Digital Terrain Modelling and Grids
• Whats the Point ?
• Interpolation Algorithms
• Tests and Results
• Extrapolation Algorithms
• Data Compression
• Tests and Results
• Conclusions

3
A Digital Elevation Model (DEM)

• Regular grid of elevations represents the heights
  at discrete samples of a continuous surface
• vertices are sampled or interpolated
  independently
• represented in a 2D matrix
• No direct topological relationship between points
• 2D Grid imposes an implicit representation of
  surface form
• usually as a linear relationship between vertices
• Simple and convenient

4
Which Ones the DEM?
(a) Discrete Elevation Samples (b) Implicit
(Linear) Continuous Surface
5
Interpolation

• DEM resolution should be dependent upon the
  variability of each terrain surface
• but rarely is
• The requirement of the DEM is to represent the
  terrain surface such that elevations can be
  retrieved or inferred for any given location
• i.e. usually by interpolation
• The method of interpolation is often ignored
• Required for most, if not all applications

6
Whats the Point?

• Does Interpolation matter?
• Whats the height at D ?
• Wheres the 60m Contour(s) ?

7
Interpolation for Visibility Analysis

• Whats the profile through the cell ?

8
Interpolation for Visibility Analysis
(a) Linear with Diagonal (b) Linear without
Diagonal (c) Bilinear
20m Object (a) Completely Obscured (b)
Completely Visible (c) Partially Visible
9
Interpolation Algorithms

• Very small interpolation errors may lead to very
  large application errors
• visibility analysis, hydrological modelling,
  contouring, etc.
• Interpolation is flawed if we only consider the
  grid cell of the point to be estimated
• Most GIS only consider the 4 vertices of the grid
  cell !
• bilinear interpolation

10
Interpolation Alternatives

• For the most part, we can use polynomial
  interpolation of the form
• hi a00 a10x a01y a20x2 a11xy
  a02y2 a30x3
• a21x2y a12xy2 a03y3 a31x3y a22x2y2
  a13xy3
• a32x3y2 a23x2y3 a33x3y3 amnxmyn
• solved from the set of simultaneous equations
  which are set up, one for each point.

11
Interpolation Alternatives

• Level Plane (1 coefficient)
• Linear Plane (3)
• Double Linear and Bilinear (4)
• Biquadratic (8 or 9)
• Bicubic (12 or 16)
• Biquintic (36)
• Jancaitis Biquadratic, Piecewise Cubics, etc.

12
Linear 1 Linear 2 Double Linear
Bilinear
Biquadratic Bicubic Jancaitis
Biquintic (9 term) (16 term)
(36 term)
13
Results (1) Test Surface Functions
(Franke, 1979 Akima, 1996)
14
Results (1) Test Surface Functions
15
Results (1) Test Surface Functions
16
Results (1) Test Surface Functions

• Higher-order interpolation algorithms will always
  out-perform linear techniques

17
Results (2) Actual Terrain

• Based on Ordnance Survey data for S. Wales
• 150,000 Scale (50 m) DEMs and 110,000 Scale (10
  m) DEMs
• Higher-order interpolation algorithms will always
  out-perform linear techniques
• By 3 to 10 (of the RMSE)
• Less correlation as to which algorithm performs
  best

18
Extrapolation

• Interpolation outside the spatial extent
• Extrapolation can be considered to be at the
  heart of the best techniques for spatial data
  compression
• i.e. what is the next symbol in the series
• or standing on the surface and given my field of
  view, what is the elevation if I take one step
  backwards?

19
Why do we need DEM compression?

• Seen as yesterdays problem ?
• expensive hardware small capacity disks, etc.
• File/Internet Transfer
• Higher Resolutions

2m LiDAR DEM
20
DEM Extrapolation Prediction
21
DEM Transformation for Compression
22
TerrainExtrapolators
23
(No Transcript)
24
(No Transcript)
25
Frequency Distribution of 15x15 Corrections
26
O.S. South Wales 1201x801 DEM
27
Prediction (extrapolation) Corrections
28
(No Transcript)
29
Data Compression Results

• GZIPped DEM requires a storage capacity of 261
  of the best extrapolator and Arithmetic Coding
  method

30
Summary

• Mathematical modelling has now largely been
  forgotten by todays GIS developers
• Many GIS techniques are of limited value and may
  propagate through to application error (e.g.
  visibility analysis)
• For DEM Interpolation
• dont use linear algorithms
• Mathematical modelling offers significant savings
  for spatial data compression