Research On The Properties And General Properties Of Random Probability Frames

The frames are similar to a base system that span a vector space.They are overcomplete and allowing for linear dependency,which can be used to reduce noise,obtain other desirable feature unviable with orthonormal bases,or find sparse representations.They are useful in some field such as in compression sensing,sphere coding,wavelet analysis and signal processing.In a finite set,a frame is spanning set,yet many applications require possess additional properties to improve these application.Therefore,building particular frames is particularly hot.For example,a special kind of frame called a finite unit normal tight frame(FUNTF),which is similar to an orthonormal basis and can facilitate the representation of the Parseval frame for compressing sensing patterns.If the elements in the finite unit normal tight frame(FUNTF)are selected randomly,then another special frame will be called the probability unit normal tight frame(PUNTF).The main work of this paper is to generalize the frame theory to the probability frame theory,and to further extend the probability frame theory to the probabilistic p-frame.First,the frame is extended to the random probabilistic frame.The main difference is that the latter mainly focuses on the fact that the elements in the finite framework are randomly selected,that is,the form of the series of frames becomes the integral form.Random probabilistic frame is the case of finite frame whose elements are chosen for randomly selecting.The finite tight frame extends the orthogonal basis by creating redundant,linearly dependency,uniform spherical distributions to approximate the finite unit standard tight frames(FUNTFs).From the point of view random probability,we study the tight frame and the finite unit standard frames(FUNTFs)so as to weaken the point selection conditions to obtain the approximate finite frame.In other words,the points can be chosen from any probability tight frame,and they do not necessarily have to be uniformly distributed or have standard unit points.With the definition of random probability framework,some of the properties of the frame theory can be extended to the probabilistic framework,such as frame bounds in frame theory,a large class operators of frame,frame identities and inequalities can be extended to the probability frame boundary,large class operators of the probability framework,probability framework identities and inequalities.These probabilistic frame bounds and probabilistic frame inequalities provide theoretical support for ensuring the validity and stability of data in second random sampling in compressive sensing.This is the main work of the dissertation.Probabilistic frame theory is extended to probabilistic p-framework.The main difference is that we can distinguish different types of frame potentials by studying the optimal configuration of points on the unit sphere.Further,we use the random probability distribution on the sphere to approximate the optimal configuration.In the probability set,we characterize these optimal distributions by using the special classes of probabilistic frameworks.In this paper,we extend the frame potential to the probabilistic p-frame.According to the weak density property of probabilistic p-frame,we can narrow the scope of its probabilistic p-frame potential.At the end of the paper,the relationship between the framework theory and the statistical morphological analysis is also shown.