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Bi-Linear Interpolation

Brett Towler, USFWS, 1/11/2015

Bi-linear interpolation is an extension of linear interpolation for interpolating functions of two variables (e.g., x and y) on a regular 2D grid. The procedure, below, performs linear interpolation first in one direction, and then again in the other direction. Note that although each step is linear, the interpolation as a whole is not linear but instead quadratic. The code below would work for any size matrix.

x y M bilinear A M 112 M cols submatrix : B M 2 M rows 11 submatrix : C M 2 M rows 2 M cols submatrix : j 1 A length 2 range CC j el B C 1 C rows j j submatrix y linterp : for A transpose CC x linterp 51 line :

1. Given the system [X, Y, XYZ]

X, Y are in ascending order of, any values representative of your system.

X 012345678910111 mat :

Y 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1111 mat :

XYZ 131814371440142614391437143314311422143214354781 - 305 - 490 - 727 - 199 - 864 - 259 - 140 - 2185 - 142 - 117 - 9976 - 3460 - 4357 - 5703 - 3064 - 5997 - 3463 - 2859 - 8049 - 2945 - 2803 - 14964 - 9005 - 10596 - 11972 - 8004 - 12181 - 8986 - 7414 - 13644 - 7632 - 7207 - 19953 - 16129 - 17258 - 18127 - 14822 - 18264 - 16120 - 14003 - 19128 - 14343 - 13777 - 24941 - 23501 - 23915 - 24239 - 23005 - 24278 - 23509 - 22369 - 24611 - 22715 - 22019 - 29933 - 30862 - 30649 - 30448 - 31111 - 30401 - 30858 - 31430 - 30162 - 31256 - 31698 - 35204 - 38293 - 37420 - 36747 - 39289 - 36636 - 38284 - 40228 - 35904 - 39831 - 40493 - 40403 - 45629 - 44141 - 42983 - 46735 - 42819 - 45625 - 47304 - 41594 - 47070 - 47462 - 45556 - 51682 - 50676 - 49176 - 52254 - 48957 - 51688 - 52543 - 47238 - 52429 - 52620 - 50676 - 55619 - 55266 - 54704 - 55830 - 54577 - 55631 - 55957 - 52850 - 55916 - 55983 - 1111 mat :

2. Construct the Bi-linear matrix

New x/y X stack Y transpose XYZ 0.001 * stack augment :

Observe: Y direction ==> ... X direction [up/down] ... "x/y" any alpha numeric character New x/y 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 1.318 1.437 1.44 1.426 1.439 1.437 1.433 1.431 1.422 1.432 1.435 1 4.781 - 0.305 - 0.49 - 0.727 - 0.199 - 0.864 - 0.259 - 0.14 - 2.185 - 0.142 - 0.117 - 2 9.976 - 3.46 - 4.357 - 5.703 - 3.064 - 5.997 - 3.463 - 2.859 - 8.049 - 2.945 - 2.803 - 3 14.964 - 9.005 - 10.596 - 11.972 - 8.004 - 12.181 - 8.986 - 7.414 - 13.644 - 7.632 - 7.207 - 4 19.953 - 16.129 - 17.258 - 18.127 - 14.822 - 18.264 - 16.12 - 14.003 - 19.128 - 14.343 - 13.777 - 5 24.941 - 23.501 - 23.915 - 24.239 - 23.005 - 24.278 - 23.509 - 22.369 - 24.611 - 22.715 - 22.019 - 6 29.933 - 30.862 - 30.649 - 30.448 - 31.111 - 30.401 - 30.858 - 31.43 - 30.162 - 31.256 - 31.698 - 7 35.204 - 38.293 - 37.42 - 36.747 - 39.289 - 36.636 - 38.284 - 40.228 - 35.904 - 39.831 - 40.493 - 8 40.403 - 45.629 - 44.141 - 42.983 - 46.735 - 42.819 - 45.625 - 47.304 - 41.594 - 47.07 - 47.462 - 9 45.556 - 51.682 - 50.676 - 49.176 - 52.254 - 48.957 - 51.688 - 52.543 - 47.238 - 52.429 - 52.62 - 10 50.676 - 55.619 - 55.266 - 54.704 - 55.83 - 54.577 - 55.631 - 55.957 - 52.85 - 55.916 - 55.983 - 1212 mat

... Sanity check ...
interpolation is 0ver
   (row_1,col_1,New) 0.25 2 New bilinear 0.3 2 New bilinear 0.3 2.5 New bilinear 31 mat 5.03 - 5.703 - 8.838 - 31 mat

u v f u v u v New bilinear 31 mat :

M u v f 010102020 CreateMesh :

User/orient. γ 1101 - 101 - 0033 mat :

45°<= 11 - 011000033 mat

45°=> 1101 - 1000033 mat

Vertical 01010000033 mat

Horizontal 10001000033 mat

1101 - 101 - 0033 mat

plot M γ * 1 col M γ * 2 col augment :

plot

x y surface x y New bilinear :

Azimuth: 54.25 Elevation: 72.5