Abstract
The most typical applications of multidimensional signals concern images, of both the still (i.e., photography, fax) and dynamic (i.e., movie, television) types. This chapter introduces the fundamentals of images, using multidimensional signals as their mathematical model. In the most general case, 3D images with motion, the source signal may be expressed as i(x,y,z,t), (x,y,z,t)∈ℝ4, where x,y,z are space coordinates, t is time, and i represents information about (x,y,z) at time t. This information may refer to luminous intensity (luminance) or to color (chrominance), in which case the information signal i is vector valued. Both space and time coordinates are continuous parameters, and therefore the domain is ℝ4. Often the z coordinate is disregarded, and, for the purpose of the chapter, chrominance is neglected. In this way, instead of a 4D signal, we consider a 3D signal ℓ(x,y,t),(x,y,t)∈ℝ3, where ℓ is the luminance. At first, still images, constant in time, are considered, where the corresponding source signal is ℓ(x,y), (x,y)∈ℝ2.
In the second part of the chapter images will be considered in a less standard framework, investigating the possibility of an image reconstruction from its projections. This problem, born in astronomy, finds its most important application in medicine (computer-aided tomography) and nowadays is investigated and applied in several other disciplines.
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Notes
- 1.
Since framing is actually a source-image truncation, also in the vertical direction, the band cannot be limited in strict sense. Therefore, the parameter B y can only correspond to a conventional band.
- 2.
The frame limitation is usually neglected in literature. Its introduction, however, is essential for a correct formulation of the scanning process.
- 3.
The operator \(\mathcal{C}\) of this transform is related to the Hankel operator by
$$\mathcal{C}=\mathcal{F}_r^{-1} \mathcal{H}=\mathcal{F}_a \mathcal{R}_p^{-1} \mathcal{F}_a^{-1}.$$ - 4.
With the constraints discussed below.
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Cariolaro, G. (2011). Study of Images. In: Unified Signal Theory. Springer, London. https://doi.org/10.1007/978-0-85729-464-7_17
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