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Discrete orthogonal moments such as Tchebichef moments and Krawtchouk moments are more powerful in image representation than traditional continuous orthogonal moments. However, less work has been done for the summarisation of these discrete orthogonal moments. This study proposes two general forms which will simplify and group the discrete orthogonal Tchebichef and Krawtchouk polynomials and their corresponding moments, and discusses their importance in theories and applications. Besides, the proposed general form can be used to obtain other three discrete orthogonal moments: Hahn moments, Charlier moments and Meixner moments. Computations of these discrete orthogonal polynomials are also discussed in this task, including the recurrence relation with respect to variable x and order n. Some properties of these discrete orthogonal moments, which are of particular value to image processing applications, such as energy compact capability and signal decorrelation, are also presented. Finally, the study evaluates these discrete orthogonal moments in terms of the capacity of image reconstruction and image compression, and discusses the importance of the proposed general form in theories and engineering.