Integrated representation for discrete Fourier and wavelet transforms using vector notation

Many mathematical operations are implemented easily through transform domain operations. Multiple transform domain operations are used independently in large and complex applications. There is a need to develop integrated representations for multiple transform domain operations. This paper presents an integrated mathematical representation for the discrete Fourier transformation and the discrete wavelet transformation. The proposed combined representation utilizes the powerful vector notation. A mathematical operator, called the star operator, is formulated that merges coefficients from different transform domains. The star operator implements both convolution and correlation processes in a weighted fashion to compute the aggregated representation. The application of the proposed mathematical formulation is demonstrated successfully through merging transform domain representations of time-domain and image-domain representations. Heart sound signals and magnetic resonance images are used to describe transform-domain data merging applications. The significance of the proposed technique is demonstrated through merging time-domain and image-domain representations in a single-stage that may be implemented as the primary processing engine inside a typical digital image processing and analysis system.