A general comparative study of the multiplicative homomorphic, log-ratio and logarithmic image processing approaches

Abstract This article presents a comparative study of the multiplicative homomorphic image processing (MHIP), the log-ratio image processing (LRIP) and the logarithmic image processing (LIP). These three image processing approaches are based on abstract linear mathematics and provide specific operations and structures that have opened up new pathways to the development of image processing techniques. The MHIP approach was designed for the processing of multiplied images, the LRIP approach was introduced to overcome the out-of-range problem associated with many image processing techniques, while the LIP approach was developed for the processing of images valued in a bounded intensity range. First, it is claimed that an image processing framework must be physically relevant, mathematically consistent, computationally tractable and practically fruitful. It is also pointed out that the classical linear image processing (CLIP) is not adapted to non-linear and/or bounded range images or imaging systems, such as transmitted light images, practical digital images or the human brightness perception system. Then, the importance and usefulness of several mathematical fields, such as abstract linear algebra and abstract analysis, for image representation and processing within such image settings are discussed. Third, the MHIP, LRIP and LIP approaches are presented, focusing on their distinctive ideas, structures and properties for image representation and processing, rather than an in-depth review. Next, a study of the relationships and differences between their image representations and basic algebraic operations is detailed. Finally, a general comparative discussion is developed, showing the main physical, mathematical, computational and practical characteristics of these three abstract-linear-mathematics-based image processing approaches, and summarizing their respective advantages and disadvantages. It is concluded and highlighted through real application examples in both image enhancement and edge detection areas that the LIP approach surpasses the two other approaches, although, from a strictly practical point of view, a detailed quantitative comparative study on real applications is now necessary.

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