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CRC Press eBooks

Image Restoration and Information

Francis T.S. Yu, Francis T.S. Yu

Published Oct 8, 2018• 0 citations
Publisher

Abstract

In Chap. 6 we presented in detail the relationship between observation and information, arriving at a very important result, namely, that for every observation there is a compensatory increase in entropy of other sources. We have shown that every physical observation, as well as every physical device, is ultimately limited by Heisenberg's uncertainty principle from quantum mechanics. In observing the most minute object or a small distance, there is a practical limit at which great accuracy cannot be achieved. Also, the smaller the distances or objects to be observed are, the more costly the energy (or entropy) required. In some instances, the amount of energy required may even approach infinity.

DOI
10.1201/9781315214498-13

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In Chap. 6 we presented in detail the relationship between observation and information, arriving at a very important result, namely, that for every observation there is a compensatory increase in entropy of other sources. We have shown that every physical observation, as well as every physical device, is ultimately limited by Heisenberg's uncertainty principle from quantum mechanics. In observing the most minute object or a small distance, there is a practical limit at which great accuracy cannot be achieved. Also, the smaller the distances or objects to be observed are, the more costly the energy (or entropy) required. In some instances, the amount of energy required may even approach infinity.
Light is not just the major source of energy that supports life and it is also a very viable information carrier. In this article, we showed that there is an intimate relationship between information and light as a carrier. We discussed the Uncertainty Principle of Heisenberg as related to information, in which we showed that every bit of information (or quanta of light) is limited by an unit cell of information and it is associated with a cost of entropy. Examples are given within the limit and beyond the Uncertainty Principle. We have shown that time and frequency resolutions cannot be observed simultaneously and yet the imaging can also be obtained within the limit of Certainty! We have further shown a crucial connection between the uncertainty principle and the relativity theory of Einstein, in which we see that very high speed object is conceivable to be detected at the expensed of the observation speed.
Light is not just the major source of energy that supports life and it is also a very viable information carrier. In this article, we showed that there is an intimate relationship between information and light as a carrier. We discussed the Uncertainty Principle of Heisenberg as related to information, in which we showed that every bit of information (or quanta of light) is limited by an unit cell of information and it is associated with a cost of entropy. Examples are given within the limit and beyond the Uncertainty Principle. We have shown that time and frequency resolutions cannot be observed simultaneously and yet the imaging can also be obtained within the limit of Certainty! We have further shown a crucial connection between the uncertainty principle and the relativity theory of Einstein, in which we see that very high speed object is conceivable to be detected at the expensed of the observation speed.
In the classical theory of light, an observation can be made as small as we please. However, for some very small objects this assumption does not hold true. For example, let us take a Mach-Zehnder interferometer [1], as shown in Fig. 6.1. It can be seen that only a very small fraction of light is required to illuminate the object transparency s(x, y). This small fraction of light, however, carries all the information that we intended. However, the other path of light carries almost all the energy of the light source. The corresponding irradiance distributed on the photographic emulsion is [2] /(/?, q) = R2 + \S(p, q)\2 + 2RS(p, q)cos[oc0p + >(/?, q)] (6.1) where R is the background (reference) beam, S(p, q) is the corresponding Fourier spectrum of the object transparency, (/?, q) are the corresponding spatial frequency coordinates, ao is an arbitrary constant, and S(p, q) = \S(p, q)\ exp [/>(/>, q)] (6.2) In our example, it is assumed that R » \S(p, q)\. Thus the interference term (the last term) is considered very small in comparison with the background irradiance. In principle, it is possible to enhance the weak interference term by means of a coherent optical processor [2], so it is simply a matter of making s(x, y) observable. In the classical theory, it is possible to observe with as weak an object beam as we wish. However, we know instinctively that this is not practical, since an increase in background light would at the same time increase background fluctuation. Thus the informational term (the interference term), could be completely buried in the background fluctuation. So in practice there exists a practical lower limit. Beyond this limit it is not possible to retrieve the information (the object).
One of the most fascinating principles in quantum mechanics must be Heisenberg’s uncertainty principle, which can be briefly stated as follows: every physical observation cannot be precisely determined without some degree of error or uncertainty. And it is by no means can one use the principle within the limit of certainty region, as will be shown in this Letter. Two of the most important pillars in modern physics must be Einstein’s relativity theory and Schr dinger’s contribution to quantum mechanics. Yet, there is a profound connection between these discoveries by means of the uncertainty relationship, in which we shown that the observation of a high-speed object is conceivable if the speed of the observer keeps up with object’s speed.
Since every substance has a price tag of information that includes all the building blocks in our universe, one cannot simply ignore information when dealing with science. We have shown that there is a profound connection between information and entropy, a quantity that has been well accepted in science. Without this connection, information would be more difficult to apply in science. Two of the most important pillars in modern physics must be Einstein’s relativity theory and Schrödinger’s quantum mechanics. However, there exists a profound connection between them, by means of Heisenberg’s uncertainty principle. Due to the uncertainty relation, we see that every bit of information takes time and energy to transfer, create, and observe. Since one cannot create something from nothing, anything to be created needs a huge amount of energy and a great deal of entropy to make it happen.
Modern science is eliminating the old, unrealistic approach to experimental errors. The Heisenberg principle of uncertainty has effected a fundamental change in the scientific attitude toward experimental errors and toward the degree of exactness in stating physical laws. Quantum physics has brought about the realization that a measurement or a mere observation at the quantum level perturbs the observed system in such a way that corrections cannot be made for the ``loading'' effect of the measuring instrument. There is something more basic and fundamental to our limitations in measurements than the lack of perfection in experimental planning and equipment. This article examines the types of uncertainty in measurements or observations. The interest then focuses on the fundamental importance of the relation between experimental uncertainty in measuring, observing, or in any way identifying a physical quantity, and the energy required to accomplish this in a certain allotted interval of time. This last relation has a direct bearing on the speed limitations of measurements.