Double Interpolation using Lookup Tables in MATLAB

Double Interpolation is a mathematical method used to calculate values within a set of data points. It is basically an advanced version of interpolation, as it is used to interpolate a function that depends on two variables. In other words, the double interpolation is a method of estimating value of a function within a set of a data points by combining two distinct interpolation processes.

In this tutorial, I will explain how to calculate double interpolation using lookup table in MATLAB. But before let's get a basic overview of terms "interpolation", "double interpolation", and "lookup table".

What is Interpolation?

Interpolation is a mathematical method of determining values within a set of data points. It helps us to define a mathematical function that can relate with the given set of data points. Then, using this function, we can estimate or calculate the values at those points which are not given in the data set.

Overall, interpolation is a mathematical tool which allows us to fill the gaps between the data points. Based on parameters like nature of data points, accuracy, and use cases, there are several different types of interpolation techniques, such as linear interpolation, polynomial interpolation, double interpolation, spline interpolation, etc.

This tutorial is primarily meant for double interpolation. So, let's get an overview of double interpolation.

What is Double Interpolation?

Double interpolation is an advanced version of interpolation. It is also known as bivariate interpolation. This type of interpolation is utilized in cases when the interpolation function depends on two variables to estimate the values between data points.

The double interpolation is an interpolation method which combines two distinct interpolation processes to calculate the values withing a set of data points.

For example, consider a set of data point arranged in an array. In this set of data points, let one variable is varying along the columns while the other variable is varying along the rows of the array. Now, the double interpolation first estimates the values between data points withing the array along one dimension and then along the other dimension.

Double interpolation is widely used in digital image processing, engineering analysis, image transformation, etc.

What is a Lookup Table?

A lookup table is a kind of data structure used to store precomputed data or values for specific input data. It allows us to retrieve data more efficiently and faster without performing any complex computations.

In MATLAB programming, a typical lookup table consists of two main elements namely, input values and output values. A lookup table stores specific input values that can be scalar values, vectors, multidimensional arrays, etc. The output values are precalculated values for input values.

Lookup tables are widely used in different applications of engineering and technology such as digital image processing, optimization, sensor calibration, control systems, etc.

The use of lookup tables significantly reduces the time required for processing by speeding up calculations.

How to Calculate Double Interpolation using MATLAB?

In MATLAB, there is a built-in function 'interp2' which is used to perform double interpolation. The syntax of this function is given as follows:

value = interp2(X, Y, V, Xq, Yq);

Where, X and Y are the vectors containing the coordinates of the data points, V is a matrix containing the function values at each data point, and Xq¬ and Yq are the coordinates of the query points.

Now, let us discuss the step-by-step process to perform the double interpolation using lookup table in MATLAB.

Process of Double Interpolation using Lookup Table in MATLAB:

Here are the steps to perform the double interpolation using a lookup table in MATLAB-

Step (1) – Create a lookup table.

Step (2) – Use the lookup table to perform double interpolation for a set of input values by using the 'interp2' function.

Now, let us consider an example program in MATLAB to understand how to implement the code to calculate the double interpolation using lookup table.

Example

Consider a mathematical function f(a, b) = a² + b². Let we have to perform doble interpolation for a given set of input values by using a lookup table. The following is the MATLAB example to perform this interpolation.

% MATLAB code to perform double interpolation using lookup table
% Define the input values for a and b
a = [0, 1, 2, 3, 4, 5, 6];
b = [0, 1, 2, 3, 4, 5, 6];

% Create a meshgrid of values of a and b
[A, B] = meshgrid(a, b);

% Calculate the values of function for each combination of a and b
V = A.^2 + B.^2;

% Create the lookup table
LT = {a, b, V};

% Define the input values to perform double interpolation
a_in = 2.5;
b_in = 3.5;

% Perform double interpolation using the lookup table
value = interp2(A, B, V, a_in, b_in);

% Display the meshgrid of A and B values
disp('Grid of A and B values:');
[A, B]

% Display the output of double interpolation
disp(['Interpolated value at (2.5, 3.5) is: ', num2str(value)]);

Output

Grid of A and B values:
ans =

   0   1   2   3   4   5   6   0   0   0   0   0   0   0
   0   1   2   3   4   5   6   1   1   1   1   1   1   1
   0   1   2   3   4   5   6   2   2   2   2   2   2   2
   0   1   2   3   4   5   6   3   3   3   3   3   3   3
   0   1   2   3   4   5   6   4   4   4   4   4   4   4
   0   1   2   3   4   5   6   5   5   5   5   5   5   5
   0   1   2   3   4   5   6   6   6   6   6   6   6   6

Interpolated value at (2.5, 3.5) is: 19

Conclusion

In this tutorial, I explained what are interpolation, double interpolation, and lookup table in detail, and also how to calculate the double interpolation using lookup table with the help of example program.

In conclusion, MATLAB provides a built-in function 'interp2' that allows us to calculate the double interpolation to estimate values between data points. It is widely used in fields like image processing, engineering analysis, control systems, etc.