# Per-Pixel Manifold-Based Color Calibration Technique
## 1 Introduction
Another way of performing the color correction is to solve a manifold optimization problem defined globally for the entire image. Such an approach can be found in [8; 22; 23]. The authors of [8; 23] define color correction as an optimization problem with priors both from the source image and from the target image. From the source image, the priors are color characteristics and color distribution, while from the target image, the priors tend to preserve the spatial and temporal structure of the image, after color correction. Similarly, the paper [22] introduces a method for global optimization of pixel values for large baseline multiview sequences, preserving the structure and spatio-temporal consistency. The paper [24] introduces a parametric linear method and a nonparametric nonlinear method to deal with different color changes between pairs of images. The method proposed is a modified Laplacian Eigenmaps, which is a nonlinear manifold learning approach. The used goal function for color transfer is a quadratic cost function with a quadratic regularizer. The described method only searches for image sample values of a target image, matching the color characteristics of a source image. In many applications, only a fraction of the pixels from one image directly correspond to any other image. For example, in image stitching, images only partially overlap. Similarly, in view synthesis, synthesized images from a virtual camera contain holes caused by disocclusions, where some parts of the scene are not visible due to perspective changes between the source view and the virtual view created with the source view data. Thus, only part of the real image has corresponding pixels in the virtual image. Yet we can obtain color transformation for all of the pixels in the images in order to seamlessly combine them together. Color characteristics can vary across images due to effects like ambient occlusion, nonuniform lighting, shadows, and non-Lambertian reflections. Most color calibration techniques do not work reliably for these cases, only averaging color distribution and color characteristics across entire images. But in many applications, we can retain that non-uniformity during the color transformation from one image to the other, since they are inherent for some objects and materials. We can preserve these features for a color-corrected view. To obtain a proper transformation that preserves the aforementioned effects, one needs to estimate per-pixel color transformation from one image to the other. An important factor of novelty of the work presented in this paper is the fact that the optimization algorithm does not work on the pixel values themselves, but rather on the transformations applied to those pixels. In this way, our method can use optimization constraints that do not refer to pixel values, but rather to the transformations that are applied to the pixel values. In this way, it is possible to obtain a smooth transformation field that can better capture the specific properties of a given scene. Additionally, the presented method includes parameters that can be used to adjust the smoothness of the transformation manifold. We also provide values of the parameters that were found to provide good results for all the tested sequences. In this paper, we describe such a method and provide experimental results obtained with the use of the method. In this paper, we first present the formal description of the optimization problem in a simple abstract space and then provide an analytical solution for this space. From there, we move on to a more complex case, where the problem is defined in a 2D grid, normally used in image and video processing. We present the problem formulation in this 2D grid space and provide a solution for such a space. The final part of derivation considers the occlusions and non-overlapping parts of the scene, the phenomena usually encountered in practical applications.