OPTICAL IMAGING AND SPECTROSCOPY Phần 2 ppt

Figure 2.28 Base transmission pattern, tiled mask, and inversion deconvolution for p ¼ 5.

Figure 2.29 Base transmission pattern, tiled mask, and inversion deconvolution for p ¼ 11.

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implemented under cyclic boundary conditions rather than using zero padding. In

contrast with a pinhole system, the number of pixels in the reconstructed coded aper￾ture image is equal to the number of pixels in the base transmission pattern.

Figures 2.31–2.33 are simulations of coded aperture imaging with the 59 59-element

MURA code. As illustrated in the figure, the measured 59 59-element data are

strongly positive. For this image the maximum noise-free measurement value is

100, and the minimum value is 58, for a measurement dynamic range of ,2. We

will discuss noise and entropic measures of sensor system performance at various

points in this text, in our first encounter with a multiplex measurement system we

simply note that isomorphic measurement of the image would produce a much

higher measurement dynamic range for this image.

In practice, noise sensitivity is a primary concern in coded aperture and other mul￾tiplex sensor systems. For the MURA-based coded aperture system, Gottesman and

Fenimore [102] argue that the pixel signal-to-noise ratio is

SNRij ¼ Nfij ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Nfij þ N P

kl fkl þ P

kl Bkl p (2:47)

where N is the number of holes in the coded aperture and Bkl is the noise in the (kl)th

pixel. The form of the SNR in this case is determined by signal-dependent, or “shot,”

noise. We discuss the noise sources in electronic optical detectors in Chapter 5 and

Figure 2.30 Base transmission pattern, tiled mask, and inversion deconvolution for p ¼ 59.

2.5 PINHOLE AND CODED APERTURE IMAGING 39

Figure 2.31 Coded aperture imaging simulation with no noise for the 59 59-element code

of Fig. 2.30.

Figure 2.32 Coded aperture imaging simulation with shot noise for the 59 59-element

code of Fig. 2.30.

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derive the square-root characteristic form of shot noise in particular. For the 59 59

MURA aperture, N ¼ 1749. If we assume that the object consists of binary values 1

and 0, the maximum pixel SNR falls from 41 for a point object to 3 for an object with

200 points active. The smiley face object of Fig. 2.31 consists of 155 points.

Dependence of the SNR on object complexity is a unique feature of multiplex

sensor systems. The equivalent of Eqn. (2.47) for a focal imaging system is

SNRij ¼ Nfij ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Nfij þ Bij p (2:48)

This system produces an SNR of approximately ffiffiffiffi

N p independent of the number of

points in the object.

As with the canonical wave and correlation field multiplex systems presented

in Sections 10.2 and 6.4.2, coded aperture imaging provides a very high depth

field image but also suffers from the same SNR deterioration in proportion to

source complexity.

2.6 PROJECTION TOMOGRAPHY

To this point we have considered images as two-dimensional distributions, despite

the fact that target objects and the space in which they are embedded are typically

Figure 2.33 Coded aperture imaging simulation with additive noise for the 59 59-element

code of Fig. 2.30.

2.6 PROJECTION TOMOGRAPHY 41

three-dimensional. Historically, images were two-dimensional because focal imaging

is a plane-to-plane transformation and because photochemical and electronic detector

arrays are typically 2D films or focal planes. Using computational image synthesis,

however, it is now common to form 3D images from multiplex measurements. Of

course, visualization and display of 3D images then presents new and different

challenges.

A variety of methods have been applied to 3D imaging, including techniques

derived from analogy with biological stereo vision systems and actively illuminated

acoustic and optical ranging systems. Each approach has advantages specific to tar￾geted object classes and applications. Ranging and stereo vision are best adapted

to opaque objects where the goal is to estimate a surface topology embedded in

three dimensions.

The present section and the next briefly overview tomographic methods for multi￾dimensional imaging. These sections rely on analytical techniques and concepts, such

as linear transform theory, the Fourier transform and vector spaces, which are not for￾mally introduced until Chapter 3. The reader unfamiliar with these concepts may find

it useful to read the first few sections of that chapter before proceeding. Our survey of

computed tomography is necessarily brief; detailed surveys are presented by Kak and

Slaney [131] and Buzug [37].

Tomography relies on a simple 3D extension of the density-based object model

that we have applied in this chapter. The word tomography is derived from the

Greek tomos, meaning slice or section, and graphia, meaning describing. The

word predates computational methods and originally referred to an analog technique

for imaging a cross section of a moving object. While tomography is sometimes used

to refer to any method for measuring 3D distributions (i.e., optical coherence

tomography; Section 6.5), computed tomography (CT) generally refers to the projec￾tion methods described in this section.

Despite our focus on 3D imaging, we begin by considering tomography of 2D

objects using a one-dimensional detector array. 2D analysis is mathematically

simpler and is relevant to common X-ray illumination and measurement hardware.

2D slice tomography systems are illustrated in Fig. 2.34. In parallel beam systems,

a collimated beam of X rays illuminates the object. The object is rotated in front of

the X-ray source and one-dimensional detector opposite the source measures the inte￾grated absoption along a line through the object for each ray component.

As always, the object is described by a density function f(x, y). Defining, as

illustrated in Fig. 2.35, l to be the distance of a particular ray from the origin, u to

be the angle between a normal to the ray and the x axis, and a to be the distance

along the ray, measurements collected by a parallel beam tomography system take

the form

g lð Þ¼ , u

ð

f lð Þ cos u a sin u, lsin u þ a cos u da (2:49)

where g(l, u) is the Radon transform of f(x, y). The Radon transform is defined

for f [ L2

(Rn

) as the integral of f over all hyperplanes of dimension n 2 1. Each

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