Using fish-covering model, this paper intuitively explains how to extend Hartley's information formula to the generalized information formula step by step for measuring subjective information: metrical information (such as conveyed by thermometers), sensory information (such as conveyed by color vision), and semantic information (such as conveyed by weather forecasts). The pivotal step is to differentiate condition probability and logical condition probability of a message. The paper illustrates the rationality of the formula, discusses the coherence of the generalized information formula and Popper's knowledge evolution theory. For optimizing data compression, the paper discusses rate-of-limiting-errors and its similarity to complexity-distortion based on Kolmogorov's complexity theory, and improves the rate-distortion theory into the rate-fidelity theory by replacing Shannon's distortion with subjective mutual information. It is proved that both the rate-distortion function and the rate-fidelity function are equivalent to a rate-of-limiting-errors function with a group of fuzzy sets as limiting condition, and can be expressed by a formula of generalized mutual information for lossy coding, or by a formula of generalized entropy for lossless coding. By analyzing the rate-fidelity function related to visual discrimination and digitized bits of pixels of images, the paper concludes that subjective information is less than or equal to objective (Shannon's) information; there is an optimal matching point at which two kinds of information are equal; the matching information increases with visual discrimination (defined by confusing probability) rising; for given visual discrimination, too high resolution of images or too much objective information is wasteful.