We update the method of describing and assessing the process of the study of an abstract environment by a system, proposed earlier. We do not model any biological cognition mechanisms and consider the system as an agent equipped with an information processor (or a group of such agents), which makes a move in the environment, consumes information supplied by the environment, and gives out the next move (hence, the process is considered as a game). The system moves in an unknown environment and should recognize new objects located in it. In this case, the system should build comprehensive images of visible things and memorize them if necessary (and it should also choose the current goal set). The main problems here are object recognition, and the informational reward rating in the game. Thus, the main novelty of the paper is a new method of evaluating the amount of visual information about the object as the reward. In such a system, we suggest using a minimally pre-trained neural network to be responsible for the recognition: at first, we train the network only for Biederman geons (geometrical primitives). The geons are generated programmatically and we demonstrate that such a trained network recognizes geons in real objects quite well. We also offer to generate, procedurally, new objects from geon schemes (geon combinations in images) obtained from the environment and to store them in a database. In this case, we do not obtain new information about an object (i.e., our reward is maximal, thus the game and the object cognition process stop) when we stop getting new schemes of this kind. These schemes are generated from geons connected with the object. In the case of a possibly known item, the informational reward is maximal when we have no more detection uncertainty for any of the objects.