1. Introduction

The two-dimensional Fourier transform is widely applied in many fields [1–9]. In this chapter, the ill-posedness of the problem for computing two-dimensional Fourier transform is analyzed on a pair of spaces by the theory and examples in detail. A two-dimensional regularized Fourier series is presented with the proof of the convergence property and some experimental results.

First, we describe the band-limited signals.

Definition. For two positive Ω1, Ω<sup>2</sup> ∈ R, a function f ∈L<sup>2</sup> R<sup>2</sup> � � is said to be bandlimited if

$$\hat{f}\left(\mathbf{o}\_1,\mathbf{o}\_2\right) = \mathbf{0}, \forall (\mathbf{o}\_1,\mathbf{o}\_2) \in \mathbb{R}^2 \\ \langle -\mathcal{Q}\_1,\mathcal{Q}\_1 \rangle \times [-\mathcal{Q}\_2,\mathcal{Q}\_2].$$

Here ^f is the Fourier transform of:

$$F(f)(\boldsymbol{\alpha}\_1, \boldsymbol{\alpha}\_2) = \hat{f}(\boldsymbol{\alpha}\_1, \boldsymbol{\alpha}\_2) = \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} f(t\_1, t\_2) e^{it\_1 \boldsymbol{\alpha}\_1 + it\_2 \boldsymbol{\alpha}\_2} dt\_1 dt\_2 \ (\boldsymbol{\alpha}\_1, \boldsymbol{\alpha}\_2) \in \mathbb{R}^2. \tag{1}$$

We will consider the problem of computing ^fð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> from f tð Þ <sup>1</sup>; <sup>t</sup><sup>2</sup> .

For band-limited signals, we have the following sampling theorem [4, 10, 11]. For the two-dimensional band-limited function above, we have

$$f(t\_1, t\_2) = \sum\_{n\_1 = -\infty}^{\infty} \sum\_{n\_2 = -\infty}^{\infty} f(n\_1 H\_1, n\_2 H\_2) \frac{\sin \mathcal{Q}\_1(t\_1 - n\_1 H\_1)}{\mathcal{Q}\_1(t\_1 - n\_1 H\_1)} \frac{\sin \mathcal{Q}\_2(t\_2 - n\_2 H\_2)}{\mathcal{Q}\_2(t\_2 - n\_2 H\_2)},\tag{2}$$

where H<sup>1</sup> ≔ π=Ω<sup>1</sup> and H<sup>2</sup> ≔ π=Ω2.

Calculating the Fourier transform of f tð Þ <sup>1</sup>; t<sup>2</sup> by the formula (2), we have the formula which is same as the Fourier series

$$\hat{f}(\mathfrak{w}\_{1},\mathfrak{w}\_{2}) = H\_{1}H\_{2} \sum\_{n\_{1} = -\infty}^{\infty} \sum\_{n\_{2} = -\infty}^{\infty} f(n\_{1}H\_{1}, n\_{2}H\_{2}) e^{in\_{1}H\_{1}\mathfrak{w}\_{1} + in\_{2}H\_{2}\mathfrak{w}\_{2}} P\_{\mathfrak{A}}(\mathfrak{w}\_{1}, \mathfrak{w}\_{2}),\tag{3}$$

where PΩð Þ¼ ω1; ω<sup>2</sup> 1½ �� � �Ω1;Ω<sup>1</sup> ½ � <sup>Ω</sup>2;Ω<sup>2</sup> (ω1, ω2) is the characteristic function of ½ �� � �Ω1; Ω<sup>1</sup> ½ � Ω2; Ω<sup>2</sup> .

In many practical problems, the samples f g f nð Þ <sup>1</sup>H1; n2H<sup>2</sup> are noisy:

$$f(n\_1H\_1, n\_2H\_2) = f\_T(n\_1H\_1, n\_2H\_2) + \eta(n\_1H\_1, n\_2H\_2),\tag{4}$$

where f g ηð Þ n1H1; n2H<sup>2</sup> is the noise

$$|\eta(n\_1H\_1, n\_2H\_2)| \le \delta,\tag{5}$$

Definition 2.1 Assume A: D ! U is an operator in which D and U are metric

of determining a solution z in the space D from the "initial data" u in the space U

i. For every element u∈ U, there exists a solution z in the space D; in other

ρUð Þ u1; u<sup>2</sup> < δ ) ρDð Þ z1; z<sup>2</sup> < ∈: In other words, the inverse mapping A�<sup>1</sup> is uniformly continuous. Problems that

In this section, we discuss the ill-posedness of <sup>A</sup>^<sup>f</sup> <sup>¼</sup> <sup>f</sup> on the pair of Banach

<sup>∞</sup> is the space a nð Þ: <sup>n</sup>∊Z<sup>2</sup> � � of bounded sequences. The norm of <sup>l</sup>

j j a nð Þ ,

Az ¼ u: (6)

)), where ^fð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> is given by the Fourier

<sup>A</sup>^<sup>f</sup> <sup>¼</sup> f, (7)

<sup>2</sup> h i <sup>P</sup>Ωð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> ,

<sup>∞</sup> is

(8)

spaces with distances ρDð Þ ∗; ∗ and ρUð Þ ∗; ∗ , respectively. The problem

Computation of Two-Dimensional Fourier Transforms for Noisy Band-Limited Signals

is said to be well-posed on the pair of metric spaces ð Þ D; U in the sense of

ii. The solution is unique; in other words, the mapping A is injective.

iii. The problem is stable in the spaces ð Þ D; U : ∀ ∈>0, ∃δ>0, such that

Hadamard if the following three conditions are satisfied:

violate any of the three conditions are said to be ill-posed.

The operator A in Eq. (6) is defined by the following formula:

k ka <sup>l</sup>

<sup>∞</sup> ¼ sup n∈Z<sup>2</sup>

iii. The stability condition is not satisfied. The proof is similar to the proof

Based on the one-dimensional regularized Fourier series in [12], we construct

f nð Þ <sup>1</sup>H1; <sup>n</sup>2H<sup>2</sup> ein1H1ω1þin2H2ω<sup>2</sup>

<sup>2</sup> h i <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

words, the mapping A is surjective.

DOI: http://dx.doi.org/10.5772/intechopen.81542

½ �� � �Ω1; <sup>Ω</sup><sup>1</sup> ½ � <sup>Ω</sup>2; <sup>Ω</sup><sup>2</sup> , l<sup>∞</sup>(Z<sup>2</sup>

where ¼ f g f nð Þ <sup>1</sup>H1; n2H<sup>2</sup> : n1∊Z; n2∊Z .

i. The existence condition is not satisfied.

ii. The uniqueness condition is satisfied.

the two-dimensional regularized Fourier series:

1 þ 2πα þ 2παð Þ n1H<sup>1</sup>

3. The regularized Fourier series

∑ ∞ n2¼�∞

spaces (L<sup>2</sup>

series in Eq. (3).

As usual, l

defined by

where

in [10].

^<sup>f</sup> <sup>α</sup>ð Þ¼ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup>

47

H1H<sup>2</sup> ∑ ∞ n1¼�∞

and f <sup>T</sup> ∈L<sup>2</sup> is the exact band-limited signal.

The noise in the two-dimensional case is discussed in [5, 6], and the Tikhonov regularization method is used. However, there is too much computation in the Tikhonov regularization method since the solution of an Euler equation is required.

The ill-posedness in the one-dimensional case is considered in [12, 13]. The regularized Fourier series

$$\hat{f}\_a(\mathbf{o}) = H \sum\_{n = -\infty}^{\infty} \frac{f(nH)e^{inH\alpha}}{1 + 2\pi\alpha + 2\pi\alpha \left(n\_1 H\_1\right)^2} P\_\Omega(\mathbf{o})$$

in [12] is given based on the regularized Fourier transform

$$F\_a[f] = \int\_{-\infty}^{\infty} \frac{f(t)e^{i\alpha t}dt}{1 + 2\pi\alpha + 2\pi\alpha t^2}$$

in [14]. The regularized Fourier transform was found by finding the minimizer of the Tikhonov's smoothing functional.

In this chapter, we will find a reliable algorithm for this ill-posed problem using a two-dimensional regularized Fourier series. In Section 2, the ill-posedness is discussed in the two-dimensional case. In Section 3, the regularized Fourier series and the proof of the convergence property are given. The bias and variance of regularized Fourier series are given in Section 4. The algorithm and the experimental results of numerical examples are given in Section 5. Finally, the conclusion is given in Section 6.
