1. A video correction apparatus, comprising: a Gaussian filter coefficient calculator configured to extract a filter coefficient of at least one Gaussian filter for performing a filtering process for extracting an illumination component from one color component image; a resolution adjuster configured to extract an optimized resolution value for the filtering process based on the filter coefficient, and adjust a resolution of the color component image to the optimized resolution value; an illumination component extractor configured to extract the illumination component of the color component image from the color component image with the adjusted resolution; a reflection component extractor configured to extract a reflection component of the color component image by removing the illumination component from pixel values of the color component image; a color adjuster configured to generate a sum image by adding the reflection component to the color component image; and an image synthesizer configured to output a corrected image generated by readjusting a resolution of the sum image to the resolution of the color component image.