How to create inverse 2D-LUT

From 2D - LUTs:

X = LUT(a; b)

Y = LUT(a; b)

How could I get 2 inverse LUTs, also:

a = LUT(X; Y)

b = LUT(X; Y)

Thank you!

Best Regards

Tony

Answer 1

[Namnendra]

Hi,

Creating inverse lookup tables (LUTs) from existing 2D LUTs can be challenging, especially if the original LUTs are not bijective (i.e., they don't have a one-to-one mapping). However, if you know that the relationship between inputs and outputs is invertible, you can proceed with the following steps:

Assumptions

1. Invertibility: The original LUTs are invertible, meaning each pair of outputs `(X, Y)` corresponds to a unique pair of inputs `(a, b)`.

2. Data Coverage: The original LUTs cover the range of interest for both input and output spaces.

Steps to Create Inverse LUTs

1. Data Preparation

- Collect Data: Ensure you have a complete dataset of inputs `(a, b)` and their corresponding outputs `(X, Y)` from the original LUTs.

- Grid the Data: If the original LUTs are defined over a grid, ensure your data covers this grid adequately.

2. Construct the Inverse Mapping

- Unique Mapping: Ensure that for each `(X, Y)`, there is a unique `(a, b)`. If not, you may need to refine your dataset or use an interpolation method that can handle non-unique mappings.

3. Interpolation Method

- Scattered Interpolation: Use scattered data interpolation methods (e.g., `scatteredInterpolant` in MATLAB) to create the inverse LUTs:

- MATLAB Example:

% Assume you have vectors a, b, X, and Y from the original LUTs
    F_a = scatteredInterpolant(X, Y, a, 'linear', 'none');
    F_b = scatteredInterpolant(X, Y, b, 'linear', 'none');

    % To find a and b for given (X, Y)
    a_estimated = F_a(X_query, Y_query);
    b_estimated = F_b(X_query, Y_query);

4. Validation

- Check Accuracy: Validate the inverse LUTs by checking if they correctly map `(X, Y)` back to `(a, b)` over a range of test points.

- Error Analysis: Analyze the interpolation error and adjust the interpolation method or dataset if necessary.

Additional Considerations

- Non-Bijective Cases: If the original LUTs are not bijective, consider using optimization techniques or additional constraints to resolve ambiguities.

- Smoothing and Regularization: Apply smoothing or regularization techniques if the inverse mapping is sensitive to noise or if the data is sparse.

- Dimensionality Reduction: If the LUTs are large, consider dimensionality reduction techniques to simplify the inverse mapping process.

By following these steps, you can construct inverse LUTs to estimate the original inputs `(a, b)` from the outputs `(X, Y)`.