Experiments with maximum likelihood method for image motion deblurring

Relative motion between the camera and the object results in the recording of a motion-blurred image. Under certain idealized conditions, such blurring can be mathematically corrected. We refer to this as (motion deblurring‘. We start with some idealized assumptions under which the motion deblurring problem is a linear inverse problem with certain positivity constraints; LININPOS problems, for short. Such problems, even in the case of no statistical noise, can be solved using the maximum likelihood/EM approach in the following sense. If they have a solution, the ML/EM iterative method will converge to it; otherwise, it will converge to the nearest approximation of a solution, where ‘nearest‘is interpreted in a likelihood sense or, equivalently, in a Kullback-Leibler information divergence sense. We apply the ML/EM algorithm to such problems and discuss certain special cases, such as motion along linear or circular paths with or without acceleration. The idealized assumptions under which the method is developed are hardly ever satisfied in real applications, so we experiment with the method under conditions that violate these assumptions. Specifically, we experimented with an image created through a computer-simulated digital motion blurring corrupted with noise, and with an image of a moving toy cart recorded with a 35 mm camera while in motion. The gross violations of the idealized assumptions, especially in the toy cart example, led to a host of very difficult problems which always occur under real-life conditions and need to be addressed. We discuss these problems in detail and propose some ‘engineering solutions’ that, when put together, appear to lead to a good methodology for certain motion deblurring problems. Some of the issues we discuss, in various degrees of detail, include estimating the speed of motion which is referred to as ‘blur identification’ non-zero-background artefacts and pre- and postprocessing of the images to remove such artefacts; the need to ‘stabilize’ the solution because of the inherent ill-posedness of the problem; and computer implemetation.