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322

COMPUTERIZED IMAGING TECHNIQUES I: SYNTHETIC APERTURE RADAR

Figure 17.6. Block diagram of SAR reconstruction with Fresnel approximation [Courtesy of Soumekh, 1999].

The block diagram for the algorithm is shown in Figure 17.6. When the approximations discussed above are not valid, more sophisticated image reconstruction algorithms are used, as discussed in the next section.

17.10ALGORITHMS FOR DIGITAL IMAGE RECONSTRUCTION

The goal of SAR imaging is to obtain the reﬂectivity function uðx; yÞ of the target area from the measured signal sðt; yÞ. There have been developed several digital reconstruction algorithms for this purpose, namely, the spatial frequency interpolation algorithm, the range stacking algorithm, the time domain correlation algorithm, and the backprojection algorithm [Soumekh]. The spatial frequency interpolation algorithm is brieﬂy discussed below.

17.10.1Spatial Frequency Interpolation

The 2-D Fourier transform of sðt; yÞ yields Sðf ; fyÞ, which can be written as Sðk; kyÞ where k ¼ 2pf =c and ky ¼ 2pfy.

After matched ﬁltering, it is necessary to map k to kx so that the reconstructed image can be obtained after 2-D inverse Fourier transform. The relationship between

kx, k, and ky is given by
kx2 ¼ 4k2 ky2	ð17:10-1Þ

When k and ky are sampled in equal intervals in order to use the DFT, kx is sampled in nonequal intervals. This is shown in Figure 17.7. Then, it is necessary to use interpolation in order to sample both kx and ky in a rectangle lattice. Interpolation

ALGORITHMS FOR DIGITAL IMAGE RECONSTRUCTION

323

Figure 17.7. SAR spatial frequency sampling for discrete data [Courtesy of Soumekh, 1999].

is not necessary only when kx ’ 2k, as discussed in the case of narrow bandwidth and narrow beamwidth approximation in Section 17.9.

Interpolation should be done with a lowpass signal. Note that the radar range swath is x 2 ½xc x0; xc þ x0&, y 2 ½yc y0; yc þ y0& where ðxc; ycÞ is the center of the target region. Since ðxc; ycÞ is not (0, 0), Sðkx; kyÞ is a bandpass signal with fast variations. In order to convert it to a lowpass signal, the following computation is performed:

S0ðkx; kyÞ ¼ Sðkx; kyÞe jðkxxcþkyycÞ

ð17:10-2Þ

Next S0ðkx; kyÞ is interpolated so that the sampled values of kx lie in regular intervals. The interpolated S0ðkx; kyÞ is inverse Fourier transformed in order to obtain the reconstructed image. The block diagram of the algorithm is shown in Figure 17.8.

324

COMPUTERIZED IMAGING TECHNIQUES I: SYNTHETIC APERTURE RADAR

Figure 17.8. Block diagram of SAR digital reconstruction algorithm with spatial frequency domain interpolation [Courtesy of Soumekh, 1999].

Interpolation is to be done with unevenly spaced data in the kx and k be sampled as

ky ¼ m ky ¼ kym

k ¼ n k ¼ kn

Then, the sampled values of kx are given by

k ¼ ½4n2ð kÞ2 m2ð k Þ2&1

xmn y

direction. Let ky

ð17:10-3Þ

ð17:10-4Þ

The function S0ðkx; kyÞ is to be interpolated for regularly sampled values of kx. This is done as follows [Soumekh, 1988]:

S0ðkx; kymÞ ¼ JmnS0ðkxmn; kymÞhðkx kxmnÞjkx kxmnj N kx ð17:10-5Þ n

where N is the number of sampled points along both positive and negative kx directions, kx is the desired sampling interval in the kx direction, hð Þ is the interpolating function to be discussed below, and Jm is the Jacobian given by

	d	½4k2 kym2	1		4kn	ð17:10-6Þ
Jmn ¼			&2	¼
	dw				c½4kn2 kym2 &
where k ¼ w=c. kx is chosen as
		kx ¼ i kx ¼ kxi				ð17:10-7Þ

ALGORITHMS FOR DIGITAL IMAGE RECONSTRUCTION

325

The choice of the function hð Þ is dictated by the sampling theorem in digital signal processing and communications. If a signal Vðkx; kyÞ is sampled as Vðn kx; m kyÞ it can be interpolated as

Vðkx; kyÞ ¼	X	Vðn kx; n kyÞ sin c		kx	n	ð17:10-8Þ

	n			kx
where
		sin cðvÞ ¼	sin pv			ð17:10-9Þ
			pv

The sin c function has inﬁnitely long tails. In order to avoid this problem in practise, it is replaced by its truncated version by using a window function wðkxÞ in the form

kx	wðkxÞ	ð17:10-10Þ
hðkxÞ ¼ sin c kx

wðkxÞ can be chosen in a number of ways as discussed in Section 14.2. For example, the Hamming window is given by

	8		kx
		0:54 þ 0:46 cos	p	jkxj N kx
w kx			N kx			17:10-11
ð	:	0		otherwise	ð		Þ
	Þ ¼ <

Computerized Imaging II: Image

Reconstruction from Projections

18.1INTRODUCTION

In the second part of computerized imaging involving Fourier-related transforms, image reconstruction from projections including tomography is discussed. The fundamental transform for this purpose is the Radon transform. In this chapter, the Radon transform and its inverse are ﬁrst described in detail, followed by imaging algorithms used in tomography and other related areas.

Computed tomography (CT) is mostly used as a medical imaging technique in which an area of the subject’s body that is not externally visible is investigated. A 3-D image of the object is obtained from a large series of 2-D x-ray measurements. An example of CT image is shown in Figure 18.1. CT is also used in other ﬁelds such as nondestructive materials testing.

This chapter consists of eight sections. The Radon transform is introduced in Section 18.2. The projection slice theorem that shows how the 1-D Fourier transforms of projections are equivalent to the slices in the 2-D Fourier transform plane of the image is discussed in Section 18.3. The inverse Radon transform (IRT) is covered in Section 18.4. The properties of the Radon transform are described in Section 18.5.

The remaining sections are on the reconstruction algorithms. Section 18.6 illustrates the sampling issues involved for the reconstruction of a 2-D signal from its projections. Section 18.7 covers the Fourier reconstruction algorithm, and Section 18.8 describes the ﬁltered backprojection algorithm.

18.2THE RADON TRANSFORM

Consider Figure 18.2. The x y axis are rotated by y to give the axis u v. The relationship between ðx; yÞ and ðu; vÞ is given by a plane rotation of

Diffraction, Fourier Optics and Imaging, by Okan K. Ersoy

326

THE RADON TRANSFORM

327

Figure 18.1. The CT image of the head showing cerebellum, temporal lobe and sinuses [Courtesy of Wikipedia].

y degrees:

y	¼	sin y cos y	v	ð		Þ
x		cos y sin y	u		18:2-1

The Radon transform pðu; yÞ of a signal gðx; yÞ shown in Figure 18.1 is the line integral of gðx; yÞ parallel to the v-axis at the distance u on the u-axis which makes the angle 0 y < p with the x-axis:

1ð

pðu; yÞ ¼

gðx; yÞdv

1ð

ð18:2-2Þ

¼gðu cos y v sin y; u sin y þ v cos yÞdv

Figure 18.2. Rotation of Cartesian coordinates for line integration along the v-axis.