# Hyperspectral Super-Resolution Technique Using Histogram Matching and Endmember Optimization
## 3 Proposed Solution  
### Overall Scheme  
In this paper, we propose a solution to the problem formulated in Section 2, using histogram matching and the endmember abundance optimization process proposed in [PERSON] et al. [28]. As briefly explained in Section 1, [28] defined a high-resolution HSI as a factorization of spectral endmembers and its per-pixel abundance. The authors of [28] find a high-resolution HSI by optimizing the spectral endmembers to a low-resolution HSI, and the per-pixel endmember abundances to a high-resolution RGB, image. The optimization step for endmember abundance is adopted in this method to find the per-pixel endmember abundance. The overall framework of the proposed solution is displayed in Figure 1. The HSR procedure proposed in this paper consists of three steps:  
Step 1. Creating an RGB image from a low-resolution HSI, using a spectral response function of a commercial RGB camera. Step 2. Matching the histogram of a high-resolution RGB image with that of a low-resolution RGB image, obtained in Step 1. Step 3. Optimizing endmember abundance of a high-resolution HSI toward the histogram-matched high-resolution RGB image obtained in Step 2. In Step 1, a low-resolution RGB image is reconstructed from a low-resolution HSI by factorizing a spectral response function of a commercial RGB camera. The reconstructed low-resolution RGB image is used as a color distribution standard point for both the low-resolution HSI and high-resolution RGB image. In Step 2, the histogram of the high-resolution RGB image is matched with the low-resolution RGB image obtained in Step 1. Since the high-resolution RGB image has the same color distribution as the low-resolution RGB image, it is less necessary to match the color distribution using an optimization process. In Step 3, the per-pixel endmember abundance optimization process of [28] is adopted, to match the spatial distribution of endmember abundance between the high-resolution HSI and the high-resolution RGB image. The per-pixel endmember abundance optimization is performed by reducing the subtraction between the target high-resolution RGB image and the high-resolution RGB image obtained from the high-resolution HSI. Because this optimization process disregards the objective function, representing the difference between low- and high-resolution HSIs, the endmember abundance freely converges to the pixel location of the high-resolution RGB image. Each part of the proposed method is formulated as explained below. Step 1 factorizes the spectral response function of a RGB camera with a low-resolution HSI to reconstruct a low-resolution RGB image. An RGB image is reconstructed the equation formulated in [29]:  
\[I_{ch}=\int_{400 mm}^{720 nm}S_{ch}(\lambda)L(\lambda)R_{x}(\lambda)d\lambda \tag{5}\]  
where \(S_{ch}(\lambda)\) (\(ch=\text{R},\text{G},\text{B}\)) is the spectral response function of an camera for each channel of RGB, \(L(\lambda)\) is spectral power distribution of an illuminant and \(R_{x}(\lambda)\) is a spectral reflectance of a point in an image. Equation (5) can also be written in a matrix form, for a discrete summation along the spectral range of a hyperspectral camera with a given bandwidth. Step 2 matches the histogram of an RGB image to that of a low-resolution RGB image **L**, obtained from an HSI. The histogram-matching process is a minimization of grayscale transformation \(T\) in the following equation:  
\[\underset{T}{\text{min}}\left|c_{1}(T(g))-c_{0}(k)\right| \tag{6}\]  
where \(g\) is a specific index on a gray scale, \(c_{0}\) is the cumulative distribution function of **L**'s histogram and \(c_{1}\) is the cumulative distribution function of **I**'s histogram for all intensities \(k\) on a gray scale. Since the histogram equalization is defined on a gray scale, it has to be iteratively performed on each channel of a RGB image. \(T\) is a function that finds the index on \(c_{0}\) that has the value most similar to the value of \(c_{1}\) at a particular index \(g\) on grayscale.