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Adaptive systems are useful when the signals or images are changing with time. For example, with adaptive wavelets, different filters are used for different parts of the signal: the signal itself should indicate whether a high or low order filter should be used. With adaptive optics, rapidly varying atmospheric wavefront distortions in a medium changing with time is reduced using optics: i.e. in astronomical adaptive optical systems, a system of control-driven deformable mirrors eliminates distortion produced by a medium changing with time. Adaptive wavelets has the potential for achieving the same objective while reducing cost. Adaptive optics provides real-time compensation for aberrations produced by atmospheric turbulence, jitter, and the optics. The adaptive optics subsystem consists of a Wavefront Sensor, Real-Time Reconstructor and Server Compensator, Deformable Mirror, Tilt Correction, Optical Assembly, and Adaptive Optics Control. The Wavefront Sensor senses phase difference and wavefront tilts. The Real-Time Reconstructor and Servo Compensation system computes the Deformable Mirror actuator. The Tilt Correction system corrects wavefront tilt errors and angle of arrival jitter caused by atmospheric turbulence, mount vibration, wobble dynamics lag and system vibration. In summation, adaptive optics systems are highly complex and both assembly and maintenance very expensive. Adaptive wavelets offers the potential of simplifying the system and reducing the cost. The ultimate goal is higher image resolution. Adaptive systems are important when the signals or environments are changing in time. With adaptive lifting, the prediction/update filters or wavelet/scaling functions are chosen in a fixed fashion. They can be chosen in such a way that a signal is approximated with very high accuracy using only a limited number of coefficients. Different prediction filters can be used for different parts of the signal. A high or low order prediction filter is chosen based on the signal itself. For example, the space-adaptive approach, the prediction filter depends on local information of the image pixels of one of two complementary groups. For applying adaptive wavelet lifting to optical images modulated by atmospheric turbulence, certain assumptions can be made: (1) the image is radially symmetric, and (2) the atmospheric turbulence is to some degree periodic. After that, choice of the prediction filter will take into account the characteristics of the optical image being investigated. Phase, for example, is never an easy problem.
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