Communications in Mathematical Sciences

Volume 11 (2013)

Number 2

On the continuity of images by transmission imaging

Pages: 573 – 595



Chunlin Wu (Department of Mathematics, National University of Singapore)


Transmission imaging is an important imaging technique which is widely used in astronomy, medical diagnosis, and biology science, whose imaging principle is quite different from that of reflection imaging used in our everyday life. Images by reflection imaging are usually modeled as discontinuous functions and even piecewise constant functions in most cases. This discontinuity property is the basis for many successful image processing techniques such as the popular total variation (TV) regularization. In this paper we prove that almost all images by transmission imaging are continuous functions. For the convenience of description, we will consider transmission imaging with parallel line geometry of wave beam, which is a fundamental geometry in transmission imaging and has been extensively applied in microscopes. In this kind of imaging, people take images of the physical scene from many different projection directions. We will prove that for almost every projection direction, the generated image is a continuous function, even if the density function of the physical scene is discontinuous. If the density functions of the objects to be imaged are radial regardless of some coordinate shifts, then all the projection directions generate continuous images. This continuity property has not been published yet in the literature. As a straightforward application, we finally present a simple yet effective improvement of TV regularization for Poisson noise (which is the most significant noise in transmission imaging) removal. Numerical examples and comparisons verify our analysis and demonstrate the effectiveness of the improved model.


transmission imaging, reflection imaging, Radon transform, parallel line geometry, continuity, measure zero, radial function, Poisson noise, variational method

2010 Mathematics Subject Classification

68U10, 90C90, 92C55

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