Method and system for interpolating data
A method and apparatus for interpolating data on a data grid having a plurality of data grid points to provide an interpolated data value at an interpolated point offset from a data grid point by offsets x, y, the method comprising the steps of obtaining a data value and a plurality of derivative values for each of a set of data grid points defining a sub-grid, defining a 4×4 data value matrix from the data values and derivative values, defining a plurality of 4×4 transformation matrices, calculating a 4×4 coefficient matrix from the transformation matrices and the data value matrix, defining a x vector based on the offset x and a y vector based on the offset y, and calculating the interpolated data value from the x vector, y vector and coefficient matrix.
1. A method of estimating a physical characteristic at a geographic location adjacent to at least one geographic location at which a measurement of the physical characteristic was made, measurements of the physical characteristic being represented as data on a data grid having a plurality of data grid points, each data grid point corresponding to a geographic location and each data grid point having a data value representing the physical characteristic at the corresponding geographic location, to provide an interpolated data value at an interpolated point offset from a data grid point by offsets x, y, the method comprising the steps of:
obtaining the data value and a plurality of derivative values for each of a set of data grid points defining a sub-grid,
defining a 4×4 data value matrix from the data values and derivative values,
defining a plurality of 4×4 transformation matrices,
calculating a 4×4 coefficient matrix from the transformation matrices and the data value matrix,
defining a x vector based on the offset x and a y vector based on the offset y, and
calculating the interpolated data value representing the physical characteristic at the corresponding geographic location from the x vector, y vector and coefficient matrix;
wherein the data value matrix is defined by:
data value matrix
IV
=
(
P
i
,
j
P
i
,
j
x
′
P
i
,
j
y
′
P
i
,
j
xy
′
P
i
+
1
,
j
P
i
+
1
,
j
x
′
P
i
+
1
,
j
y
′
P
i
+
1
,
j
xy
′
P
i
,
j
+
1
P
i
,
j
+
1
x
′
P
i
,
j
+
1
y
′
P
i
,
j
+
1
xy
′
P
i
+
1
,
j
+
1
P
i
+
1
,
j
+
1
x
′
P
i
+
1
,
j
+
1
y
′
P
i
+
1
,
j
+
1
xy
′
)
,
where P i,j is a data value with x,y coordinates in the data grid of i,j, P i,j x′ is the first order derivative in the x direction of P i,j , P i,j y′ is the first order derivative in the y direction of P i,j , and P i,j xy′ is the second-order derivative in the x-v direction of P i,j .
2. A method of estimating a physical characteristic at a geographic location adjacent to at least one geographic location at which a measurement of the physical characteristic was made, measurements of the physical characteristic being represented as data on a data grid having a plurality of data grid points, each data grid point corresponding to a geographic location and each data grid point having a data value representing the physical characteristic at the corresponding geographic location, to provide an interpolated data value at an interpolated point offset from a data grid point by offsets x, y, the method comprising the steps of:
obtaining the data value and a plurality of derivative values for each of a set of data grid points defining a sub-grid,
defining a 4×4 data value matrix from the data values and derivative values,
defining a plurality of 4×4 transformation matrices,
calculating a 4×4 coefficient matrix from the transformation matrices and the data value matrix,
defining a x vector based on the offset x and a y vector based on the offset y, and
calculating the interpolated data value representing the physical characteristic at the corresponding geographic location from the x vector, y vector and coefficient matrix;
wherein the plurality of transformation matrices are defined by:
transformation
matrix
A
11
=
(
1
0
0
0
0
0
0
0
-
3
3
0
0
2
-
2
0
0
)
transformation
matrix
A
12
=
(
0
0
0
0
1
0
0
0
-
2
-
1
0
0
1
1
0
0
)
transformation
matrix
A
31
=
(
-
3
0
3
0
0
0
0
0
9
-
9
-
9
9
-
6
6
6
-
6
)
transformation
matrix
A
32
=
(
0
0
0
0
-
3
0
3
0
6
3
-
6
-
3
-
3
-
3
3
3
)
transformation
matrix
A
33
=
(
-
2
0
-
1
0
0
0
0
0
6
-
6
3
-
3
-
4
4
-
2
2
)
transformation
matrix
A
34
=
(
0
0
0
0
-
2
0
-
1
0
4
2
2
1
-
2
-
2
-
1
-
1
)
transformation
matrix
A
41
=
(
2
0
-
2
0
0
0
0
0
-
6
6
6
-
6
4
-
4
-
4
4
)
transformation
matrix
A
42
=
(
0
0
0
0
2
0
-
2
0
-
4
-
2
4
2
2
2
-
2
-
2
)
transformation
matrix
A
43
=
(
1
0
1
0
0
0
0
0
-
3
3
-
3
3
2
-
2
2
-
2
)
transformation
matrix
A
44
=
(
0
0
0
0
1
0
1
0
-
2
-
1
-
2
-
1
1
1
1
1
)
.
3. A method according to claim 1 wherein the x vector is defined by
x vector {right arrow over ( x )}=(1, x,x 2 ,x 3 )
and the y vector is defined by
v vector {right arrow over ( y )}=(1, y,y 2 ,y 3 ).
4. A method according to claim 1 wherein the coefficient matrix is calculated from:
coefficient matrix C [1] =A 11 IV [1] +A 12 IV [2]
coefficient matrix C [2] =A 11 IV [3] +A 12 IV [4]
coefficient matrix C [3] =A 31 IV [1] +A 32 IV [2] +A 33 IV [3] +A 34 IV [4]
coefficient matrix C [4] =A 41 IV [1] +A 42 IV [2] +A 43 IV [3] +A 44 IV [4].
5. A method according to claim 1 wherein the interpolated data value V is calculated by:
V={right arrow over (y)} ·( C·{right arrow over (x)} T )
where {right arrow over (y)} is the y vector, C is a coefficient matrix, and {right arrow over (x)} T is a transpose of the x vector.
6. A method according to claim 1 wherein at least some of the method steps are performed using GPU.
7. A method according to claim 1 comprising performing the method for a plurality of interpolated data points.
8. A method according to claim 1 wherein the data values comprise meteorological data.
9. A system operable to perform a method according to claim 1 , the system comprising a GPU programmed to run a plurality of shaders to perform at least some of the method steps for each of a plurality of interpolated points.
10. A method according to claim 1 , the data values comprising GIS data and the data grid being displayed on an electronic display, wherein the interpolated data value is provided in response to an instruction to zoom in on the display and the interpolated data value is used to display the data grid at a higher zoom and corresponding higher resolution.
11. A method according to claim 1 , wherein the physical characteristic comprises a meteorological characteristic, further comprising forecasting weather at a geographic location in between geographic locations where measured meteorological data is available using the interpolated data value.