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)`.
