1. An electronic endoscopic apparatus comprising: an endoscope; and a processor for processing an image obtained by the electronic endoscope, wherein the processor includes a light source, an illumination light amount control means, a brightness image production means, a high brightness pixel group detection means, a light amount adjustment means and a conversion means, and wherein the illumination light amount control means controls the light amount of illumination light, which is emitted from the light source and guided to a leading edge of the endoscope, and wherein the brightness image production means produces a brightness image representing the brightness distribution of the image, and wherein the high brightness pixel group detection means detects a high brightness pixel group in the brightness image, and wherein the high brightness pixel group is a group of pixels next to one another, of which the brightness values exceed a predetermined value, and of which the number is greater than or equal to a predetermined number, and wherein when the high brightness pixel group detection means detects a high brightness pixel group, the light amount adjustment means instructs the illumination light amount control means to reduce the light amount until the high brightness pixel group detection means does not detect any high brightness pixel group, and wherein the conversion means converts the brightness of the image so that the brightness value of a pixel that forms the brightness image, and of which the brightness value is less than or equal to a predetermined value, becomes greater than or equal to the predetermined value.