Color images can be described mathematically by $3$-dimensional vector-valued image functions. The three components of a vector-valued image function represent respectively the compositions/intensity of three primary colors, namely, red (R), green (G), and blue (B). An interesting and intensely studied question is how to extend the successful PDE and variational methods for gray image processing to color image processing. Apparently, there is no unique way for achieving the goal. Moreover, as the trivial adaptation of the methods for gray image processing to color image processing by treating the RGB channels independently would give unsatisfactory results in general, more advanced and sophisticated approaches and methods must be developed for color image processing.
  In this talk I shall first introduce a variational and PDE method for color image denoising based on the chromaticity and brightness decomposition. The method consists of the standard TV (total variation) model for denoising the brightness and a cTV (color TV) model for denoising chromaticity. The cTV models then lead to very interesting variational and PDE problems of p-harmonic maps, linear growth maps, and their corresponding heat flows. I shall then give an overview of recent developments in heat flows for linear growth maps into the unit sphere, in particular, I shall discuss a newly developed BV (bounded variation) weak solution theory for the 1-harmonic map heat flow. Finally, I shall come beck to applications of these geometric flows to color image denoising based on the chromaticity and brightness decomposition. I shall also present some numerical methods and computer simulations for the proposed color image denoising models.