Color constancy algorithms can provide us with illuminant invariant descriptions for a scene, and it is often accomplished by illuminant estimation. Most statistics-based methods estimate the illuminant color from the information provided by all pixels of an image. However, this research reveals that, for most images, the color of many pixels is quite different from the illuminant, and these pixels may severely trim the performance of statistics-based methods. Based on this fact, we propose a color constancy algorithm that finds a subset of image pixels with r, g components similar to those of the illuminant through a shallow neural network, and this subset of pixels is called illuminant close pixels (ICPs). Then the illuminant color is estimated from these pixels by some statistics-based methods. The proposed method has been evaluated and investigated on two benchmark datasets. Compared to using all pixels in an image, these statistics-based methods have been efficiently improved using ICPs.