# Weighted Similarity-Confidence Laplacian Synthesis for High-Resolution Art Painting Completion
## 3 Methodology
### Weighted Similarity-Confidence Laplacian Synthesis  
Addressing a missing region characterized by complicated structures and textures, as commonly encountered in damaged art paintings, presents a formidable challenge in the realm of image completion. Numerous recent research endeavors have endeavored to tackle this complicated issue [30; 31]. However, the persistent issues of blurriness and color discrepancies have proven to be substantial impediments. Moreover, when the missing region lies arbitrarily within the boundary of objects, it intensifies the challenge, leading to the generation of unsatisfactory results marked by significant color divergences, especially in the context of complex structures and textures at high resolutions. In response to these challenges, our proposed approach advocates a proficient problem-solving strategy, involving a consistent collaboration on multi-regions of Weighted Laplacian Synthesis and patch-based completion. The detailed explanation of our approach is depicted in Figure 1. Commencing the process, we utilize a high-resolution painting with a size of around 1600 \(\times\) 2136 pixels, denoted as \(I_{in}\) in Figure 1a, featuring damaged holes. Subsequently, we perform segmentation on \(I_{in}\) and its corresponding mask images, \(M_{in}\), dividing them into 16 multi-regions through the separation of pixels into distinct patches (Figure 1b). Each region encompasses approximately 400 \(\times\) 400 pixels, resembling the standard size of compressed images but exhibiting more homogeneity in pixels. This approach facilitates a more specific and precise completion of the missing regions within the local area. Initially, the exploitation of a Laplacian pyramid completion technique proves effective for high-resolution paintings, utilizing the abundance of pixels to make consistent completion decisions across various resolutions. Inspired by [PERSON] et al. [32], we integrate Laplacian \((Lp)\) and upsampled Gaussian \((Up)\) pyramids to progressively enhance completion quality from the coarsest to the finest layer, as exemplified in Figure 1c. Furthermore, to address the challenge of structure diffusion involving the segmentation of a non-homogeneous missing region along the object boundary into smaller, homogeneous regions, we employ texture synthesis. This process aims to generate a high-quality surface with similar intensities through the estimation of the Laplacian of a Gaussian (LoG). According to [32], the computation of LoG is costly due to its complicated function. However, the Laplacian of a Gaussian operates similarly to the Difference of Gaussian (DoG), which is integral for considering edge structures in the convolution process. Consequently, our choice is to apply the Laplacian of a Gaussian pyramid at different levels, ensuring consistent performance in edge awareness and base textures with more manageable computations. In the texture synthesis process, the initial step entails constructing a Gaussian pyramid \((Gs)\) to extract pixel intensities at each level for analyzing the base structure. Following this, the construction establishes the Laplacian pyramid to enhance edge awareness:  
\[ \begin{split} Gs_{i-1}&=downsample(Gs_{i})\\ Up_{i-1}&=upsample(Gs_{i-2})\\ Lp_{i}&=Gs_{i}-Up_{i}\end{split}  \tag{1}\]  
where \(i\) is the current level of the pyramid. \(Gs_{i-1}\) represents a downsampled Gaussian of \(Gs_{i}\), while \(Up_{i-1}\) denotes an upsampled Gaussian of \(Gs_{i-2}\). \(Lp_{i}\) is then transformed into a Laplacian image using \(Gs_{i}\) and \(Up_{i}\) as a basis, following the approach outlined in [32]. Following the construction of Laplacian and Gaussian pyramids, the next step involves identifying matching patches at each level. This is accomplished by improving a nearest neighbor search algorithm that approximates the most similar areas between the source \(S\) and target \(T\), denoted as \(E_{i}(T,S)\). The basis for this approximation relies on the minimum normalized distance.