Biomedical Imaging

This can be written in more compact form by using vector notation

$\begin{array}{l}\mathcal{F}\left[f\left(\underset{\_}{x}\right)\right]\left(\underset{\_}{k}\right)=\frac{1}{{\sqrt{2\pi }}^d}{\displaystyle \underset{{ℝ}^d}{\int }{d}^{d}xf\left(\underset{\_}{x}\right)e^{-i\underset{\_}{k}\cdot \underset{\_}{x}}}\left(2.108\right)\\ {\mathcal{F}}^{-1}\left[\tilde{f}\left(\underset{\_}{x}\right)\right]\left(\underset{\_}{k}\right)=\frac{1}{{\sqrt{2\pi }}^d}{\displaystyle \underset{{ℝ}^d}{\int }{d}^{d}x\tilde{f}\left(\underset{\_}{k}\right)e^{+i\underset{\_}{k}\cdot \underset{\_}{x}}}\left(2.109\right)\end{array}$

All properties of the Fourier transform derived for d = 1 can be readily extended to d > 1.

Fourier theory defined for suitable vector spaces of (nonperiodic) square integrable functions can be linked to the analysis of Fourier series applicable to periodic functions. In fact, the definition of the Fourier transform is often motivated by the taking the limit L → ∞ of a periodic function, i.e., fL: x ∈ ℝ → f(x) ∈ ℂ with period L, i.e., fL(x + nL) = fL(x) ∀ n ∈ ℤ. This function can be represented by its Fourier series (provided ...