On the Optimal Choice of Quality Metric in Image Compression

There exist many di erent lossy compression meth ods and most of these methods have several tunable parameters In di erent situations di erent methods lead to di erent quality reconstruction so it is impor tant to select in each situation the best compression method A natural idea is to select the compression method for which the average value of some metric d I e I is the smallest possible The question is then which quality metric should we choose In this paper we show that under certain reasonable symmetry con ditions L metrics d I e I R jI x e I x jp dx are the best and that the optimal value of p can be se lected depending on the expected relative size r of the informative part of the image Formulation of the Problem Image Compression Is Necessary Images tend to take up a lot of computer space so in many applications where we cannot store the orig inal images we must use image compression Ideally we would like to use a lossless compression but unfor tunately there are serious limitations on how much we can compress without losing information For a more radical compression we must therefore use lossy com pression schemes In these schemes some information about the image is lost as a result for every point x the intensity e I x of reconstructed image e I at this point may be slightly di erent from the intensity I x of the original image I at this point It Is Important to Select Optimal Or At Least Good Enough Compression Scheme There exist many di erent compression schemes from standard ones like gif jpg jpg etc to spe cially designed ones Most of these schemes comes with one or several parameters which we can select One of the reasons why so many di erent schemes co exist is that in di erent applications di erent schemes with di erent values of parameters work better It is vitally important to select an appropriate compression scheme i e a scheme which provides the best compression ratio within the same accuracy How can we do that The Notion of a Quality Metric Intuitively the quality of a compression scheme can be characterized by how close the decompressed image is to the original one In order words the quality of a compression scheme can be described by using an appropriate metric d I e I on the set of all images Such metrics describing the distance d I e I between the two images I and e I are called quality metrics It Is Important to Select Optimal Or At Least Good Enough Quality Met ric If we select a quality metric then we can choose the optimal compression scheme as the one for which the average value of the selected metric is the smallest possible So within this approach in order to select the optimal compression scheme we must rst select the appropriate quality metric In some cases it is clear how to select the quality metric For example in some practical applications we are interested in only one characteristic c I of the observed image I e g we may only need to know the total intensity c I within a certain zone which characterizes the tumor size In such cases our goal is to reconstruct the value c I as closely as possible so we can take the absolute value jc I c e I j of the di erence c I c e I as the desired metric d I e I jc I c e I j In such applications the choice of the best compression is straightforward there is no need to store the entire image I it is su cient to store only the single value c I as the compressed image This compression is in general extremely lossy but from the viewpoint of the problem of reconstructing the value c I this compression is lossless Similarly if we intend to use only a few character istics ci I i m of an image I it is natural to compress an image I by storing only the values of these characteristics c I cm I Thus we get a drastic compression ratio and a perfect reconstruction of all desired values ci I In many practical situations however we do not know a priori which characteristics we will be inter ested in depending on the situation we may use the stored image to evaluate many di erent characteris tics How can we determine the metric in this case The larger the di erence I x e I x I x be tween the two images the larger the distance d I e I should be Thus it is natural to de ne the desired dis tance in terms of the di erence I x The question is how exactly What We Are Planning to Do In this paper we propose a three step solution to this question First we use some reasonable arguments to de scribe a general class of quality metrics Second we use natural symmetry requirements to select a subclass of quality metric characterized by a single parameter p Finally we show how to select the best value of the parameter p depending on the image As a result we get a data driven technique for select ing the optimal quality metric and thus of the optimal compression scheme First Step Using Reasonable Argu ments to Select a General Class Of Quality Metrics How to Describe Preferences There Exists a General Formalism The necessity to describe preferences i e to de scribe the utility of di erent alternatives for di erent people is extremely important in decision making in cluding decision making under con ict also known under a somewhat misleading name of game theory To describe these preferences utilities a special util ity theory has been developed see e g The mathematical formalism of utility theory comes from the observation that sometimes when a person faces several alternatives A An instead of choosing one of these alternatives this person may choose a probabilistic combination of them i e A with probability p A with a probability p etc For example if two alternatives are of equal value to a per son that person will probably choose the rst one with probability and the second one with the same prob ability Such probabilistic combinations are called somewhat misleadingly lotteries In view of this re alistic possibility it is desirable to consider the pref erence relation not only for the original alternatives but also for arbitrary lotteries combining these alter natives Each original alternative Ai can be viewed as a degenerate lottery in which this alternative Ai ap pears with probability and every other alternative Aj Ai appear with probability The main theorem of utility theory states that if we have an ordering relation L L between such lot teries with the meaning L is preferable to L and if this relation satis es natural consistency conditions such as transitivity etc then there exists a function u from the set L of all possible lotteries into the set R of real numbers for which L L if and only if u L u L and for every lottery L in which each alternative Ai appears with probability pi we have