## 3 Proposed Methodology
### Fast Local Laplacian Filter  
The Fast Local Laplacian Filter (FLLF) technique is an advanced approach for enhancing the contrast of images while preserving their fine details and textures. Unlike traditional global contrast enhancement methods that can lead to oversaturation or loss of subtle details, the FLLF technique operates locally, adapting to the unique characteristics of different image regions [10], as shown in Figure 1.  
At the core of the FLLF technique lies the Laplacian pyramid. This pyramid is constructed by generating a series of Gaussian pyramids of varying scales and then computing the differences between consecutive levels. Mathematically, for a Gaussian pyramid level \(i\) denoted as \(Gi\), the corresponding Laplacian pyramid level \(Li\) is calculated as:  
\[Li=Gi-resize(Gi+1) \tag{1}\]  
where \(resize(Gi+1)\) represents the Gaussian pyramid level \(Gi+1\) resized to the dimensions of \(Gi\). This process generates a pyramid that captures local image variations and details at different scales. The FLLF technique introduces the concept of the modified Laplacian (\(ML\)), obtained through a bilateral filter applied to each level of the Laplacian pyramid [4]. The bilateral filter is governed by two main factors: the spatial distance, \(d\), and the range similarity, \(r\). It preserves edges and reduces noise while enhancing the contrast shown in Figure 1. Mathematically, the modified Laplacian \(ML\) at level \(i\) can be expressed as:  
\[MLi=bilateralFilter(Li,d,r) \tag{2}\]  
where \(bilateralFilter(Li,d,r)\) applies the bilateral filter to the Laplacian pyramid level \(Li\) with parameters \(d\) and \(r\). The final step involves reconstructing the enhanced image by combining the modified Laplacian \(ML\) with the base image \(G_{0}\). The reconstructed image \(E\) is calculated as:  
\[E=G_{0}+resize(ML_{0})+\sum i=n\;resize(MLi) \tag{3}\]  
where \(n\) is the number of pyramid levels. In this equation, \(G_{0}\) is the base image, \(resize(ML_{0})\) is the resized version of the modified Laplacian at the top level of the pyramid, and \(\sum i=n\;resize(MLi)\) represents the sum of resized modified Laplacian levels across all pyramid levels. This reconstruction process adapts the contrast enhancement locally, allowing the technique to enhance different image regions individually based on their unique characteristics. Equation (3) can be broken down into more detail:  
Figure 1: The schematic framework of the proposed technique. * **E**: This is the enhanced image to be created, which will have improved contrast and details. * **G\({}_{0}\)**: This represents the base image, which is typically the original input image to be enhanced. * _resize_(\(\mathbf{ML}_{0}\)): The modified Laplacian at the top level of the pyramid \((MLi)\) is resized to match the dimensions of the base image (\(G_{0}\)). This means ensuring that the \((MLi)\) and \(G_{0}\) have the same width and height. * \(\sum i=n\)_Resize_(\(MLi\)): The sum (\(\sum\)) is taken over all levels (\(i\)) of the Laplacian pyramid from 1 to 'n'. For each level, the corresponding modified Laplacian \((MLi)\) is resized to match the dimensions of the base image (\(G_{0}\)) and then added to the result. So, this equation shows that the enhanced image '\(E\)' is formed by adding three components:
* The original base image \(G_{0}\). * The modified Laplacian at the top level of the Laplacian pyramid (_resize_(\(ML_{0}\))), ensuring that it matches the dimensions of \(G_{0}\). * The sum of the modified Laplacian levels from all levels of the pyramid (\(\sum i=n\)_Resize_(\(MLi\))), where each level has been resized to match \(G_{0}\). This process combines the base image with information from the Laplacian pyramid, including the local image variations and details captured at different scales, to create an enhanced image that adapts contrast locally.