1.1 Breast cancer . . . . 9

Contents

Introduction 5

1 Digital mammography 9

1.1 Breast cancer . . . . 9

1.2 X-ray imaging . . . . 13

1.2.1 The image formation . . . . 13

1.2.2 Contrast and Signal to Noise Ratio . . . . 14

1.3 Digital mammographic systems . . . . 16

1.3.1 Indirect detection Systems . . . . 18

1.3.2 Direct detection Systems . . . . 19

1.4 Some basics on Semiconductors pixel Detectors . . . . 22

1.4.1 Principal Detectors characteristics . . . . 24

1.4.2 Read-out electronics: Photon Counting Mode versus Integration Mode . . . . 26

2 Image Quality Metrics for digital systems 31 2.1 Assessment of the image quality . . . . 32

2.2 Transfer Functions analysis . . . . 33

2.3 Modulation Transfer Function . . . . 35

2.3.1 The MTF for digital imaging systems . . . . 37

2.4 The Noise Power Spectrum . . . . 39

2.4.1 NPS for digital imaging systems . . . . 43

2.5 The Detective Quantum Efficiency . . . . 47

2.6 Transfer function analysis of an ideal SPC system . . . . . 48

2.6.1 The Presampling MTF of the ideal SPC detector . 49 2.6.2 NPS and DQE of the ideal SPC detector . . . . 51

2.7 Contrast-Threshold Analysis . . . . 53

3 Development and Calibration of the IMI Demonstrator 57 3.1 The GaAs detectors . . . . 58

3.2 The Medipix1 chip . . . . 60

3.3 The IMI Demonstrator . . . . 63

3.4 System Calibration . . . . 66

3.4.1 Threshold Calibration . . . . 67

3.4.2 Other preliminary measurements . . . . 70

3.4.3 Image acquisition set-up . . . . 73

4 Monte Carlo simulation of the radiation transport in GaAs pixel detectors 75 4.1 Signal sharing in GaAs detectors . . . . 76

4.2 Monte Carlo simulations of the radiation transport . . . . 81

4.3 Simulation of the Charge Sharing: an analytical model . . 82

4.4 Simulation of the IMI detector response . . . . 87

5 Assessment of IMI prototype image quality: Experimental results 95 5.1 Experimental Set-up . . . . 96

5.2 Presampling MTF of the IMI prototype . . . . 98

5.2.1 Materials and Methods . . . . 98

5.2.2 Results and Discussions . . . . 101

5.3 NPS of the IMI prototype . . . . 107

5.4 DQE of the IMI prototype . . . . 109

Contents 3

5.4.1 Measurement of the detection efficiency . . . . 111 5.4.2 Effect on the DQE of the flat-field equalization. . . 112 5.5 Measurement of the Thickness-Threshold curves . . . . 115

Conclusions 121

Bibliography 125

Introduction

Breast cancer is the tumor disease more diffused in the female population of developed countries and the second cause of death. Early diagnosis has been proofed to be effective in reducing the mortality due to breast cancer.

Mammography is up to date the most effective examination to perform an early diagnosis of the tumor.

A mammographic imaging system to be efficient in the breast cancer detection must be able to produce mammograms with high Contrast and high Signal to Noise Ratio (SNR). This allows the radiologist to visualize tumor masses (low-contrast rather large objects, of the order of few mm) as well as microcalcifications clusters (high-contrast but small dimensions, of the order of hundreds of µm). In order to cope with these require- ments a mammographic system must have high Quantum Efficiency (QE), low noise and high spatial resolution. Because of these tight requirements, mammography is one of the most technically demanding radiological imag- ing techniques.

Nowadays, although screen-film mammography is still the most widespread imaging system for mammography, digital systems are rapidly replacing conventional ones due to several significant advantages, the most impor- tant being the linear response over a wide range of exposures.

Most commercial digital systems use an intermediate layer of scintillat-

ing material (like CsI) to convert x rays into visible light pulses, coupled to an array of light-sensitive elements used to collect the light emitted by the phosphor (as in the CCD or TFT based detectors). This approach, on one hand presents unquestionable advantages like a quite simple read-out electronics; on the other hand it shows considerable drawbacks like lim- ited spatial resolution (due to the light spread in the conversion layer) and losses in the signal to noise ratio due to the simultaneous integration of the noise and the signal. With the aim to overcome these limitations, new detection and read-out technologies have been developed by the scientific community.

The present work is about an innovative digital mammographic system developed in the framework of the research project “Integrated Mammo- graphic Imaging” (IMI), funded by the Italian Ministry for University and research (MIUR) and carried out by a consortium of Universities and five high tech companies. This system, although at a prototypal stage, is the first full-field mammographic imaging device based on GaAs pixel detec- tors and on Single Photon Counting (SPC) read-out electronics.

The GaAs pixel sensors represent a technological advance respect to the scintillators based detectors, since the problems related to the light scatter are suppressed. The GaAs sensors developed for the IMI prototype are composed of a 200µm thick Semi-Insulating GaAs crystal with a matrix of 64×64 square electrodes (pixels 150µm in side and with 20µm of inter-pixel distance) deposited on the surface. The electrodes are connected through bump-bonding to an equally segmented read-out chip, named Medipix1.

The Medipix1 chip works in SPC mode, for which the noise can be totally rejected by setting a discrimination threshold above the noise level and by counting only the photons with energy over threshold.

Each assembly, constituted by a GaAs sensor coupled to Medipix1,

is the basic detection unit of about 1.2cm

of active area. In the IMI

prototype, 18 of these detection units are mounted onto a chip-board,

which is scanned to cover the standard (18 × 24)cm

exposure area.

7

The aim of this thesis is to perform a complete characterization of the IMI prototype and to assess its imaging capabilities. Chapter 3 reports the system calibration and optimization for the mammographic imaging use.

In particular we have calibrated the discrimination thresholds in terms of the absolute energy scale, evaluated the linearity and the stability of the system and developed an equalization procedure to make the pixels response uniform.

With the aim to investigate the advantages and the limits of GaAs de- tectors and SPC electronics, we have simulated with a Monte Carlo Code (MCNP 5), the radiation transport into the GaAs crystals. In Chapter 4 this simulation shows that when high Z materials are used in pixel detec- tors, the signal sharing due to the high energy k-fluorescent photons (9.25 keV and 10.54 keV for the GaAs) may degrade the spectral and the spa- tial resolution of the detector. This effect is due to the fluorescent photons escaping, which are re-absorbed far away the pixel in which the primary photon interacted. Nevertheless we show that the SPC electronics is use- ful in rejecting these spurious events and then in preserving the spatial resolution of the detector.

In order to quantitatively assess the imaging capabilities of the IMI system, two methodologies have been followed. The first one is based on the experimental measurement of the transfer functions of the system.

The Modulation Transfer Function (MTF) , the Noise Power Spectrum (NPS) and the Detective Quantum Efficiency (DQE) are figures of merit that describe the spatial resolution, noise and SNR transfer capability of the system. The theoretical background of the transfer functions analysis for digital imaging is reported in Chapter 2, and in the Chapter 5 the experimental measurements of these function for the IMI prototype are reported and discussed.

This analysis allows an objective evaluation of the performance of the

X-ray detector, but it does not take into account other aspects of the image

visualization. For this reason another approach, based on the Contrast

Threshold (CT) analysis, has been also used. This analysis takes into account also the visual processes of the human observer.

For the CT measurements several images of a contrast detail phan-

tom (CDMAM) have been acquired and presented to three independent

observers for the scoring. The CDMAM phantom consist of a matrix of

square cells with dots of varying size and contrast. The observer’s task is

the detection of the dots. The responses have been evaluated to determine

the minimum detectable contrast as a function of the dot size.

CHAPTER 1

Digital mammography

In this chapter we introduce the topic of the x-ray mammography, that is the most effective tool for the diagnosis of the breast cancer. We describe the radiation detection and the image formation stages for both screen-film and digital systems. Moreover we illustrate the state of the art of digital mammographic imaging systems and discuss how new technologies, such as pixel semiconductor detectors and Single Photon Counting electronics, are effective in the improvement of the image quality and of the diagnosis capabilities of a mammographic system.

1.1 Breast cancer

Breast cancer is the most common type of cancer and the second most common cause of death for women in developed countries. According to the American Cancer Society, about 1.3 million women are annually diagnosed with breast cancer worldwide, and about 465.000 die from this disease [1]. Although breast cancer rates are rising in many western countries, mortality has decreased in some of these countries as a result of extended screening programs and improved early detection.

Because there is not certainty of the cause of most breast cancers, the

most effective means of prevent death from breast cancer is detection when the cancer is in situ or minimally invasive. The most effective means cur- rently available for accomplishing this is mammography; the X radiography of the breast. Mammography is used both for examining symptomatic patients (diagnostic mammography) and for screening of asymptomatic women in selected age groups.

In particular systematic early detection through screening has a sub- stantial effect on breast cancer mortality reduction. In fact it has been shown that a breast cancer screening programme can reduce mortality in the age group of 40 − 74 years by up to 40% [2]. On the other hand there is an increase in the risk of cancer related to radiation dose absorbed by the breast during the examination. Even though the risk related to the radiation dose in one individual examination is very low, the hazard re- lated to the population as a whole should not be neglected. For this reason in screening exams the dose delivered to the patient must be kept to the minimal value.

The ultimate purpose of mammography, performed on both symp- tomatic and asymptomatic patients, is the diagnosis of the breast can- cer. This can be accomplished on the basis of four types of signs on the mammogram [3]:

• peculiar characteristic morphology of tumor masses

• presence of mineral deposits as specks called microcalcifications

• architectural distortion of normal tissue patterns caused by the dis- ease

• asymmetry between corresponding regions of images of the left and right breast.

Tumor masses and microcalcifications are visualized in the mammo-

gram due to differential x-ray attenuation between these structures and

normal breast tissues. To allow the visualization of these structures, the

1.1. Breast cancer 11

imaging system must precisely measure the transmitted x-ray intensity through all regions of the breast and must detect the small contrast asso- ciated to these structures to produce visible signals that exceed the per- ceptual threshold of the viewer.

In the breast two kind of non-pathologyc tissues are present: fibroglan- dular breast tissue and adipose tissue. Fig.1.1 shows the linear attenuation coefficient of these normal tissues and of the breast carcinoma as a func- tion of the x-ray energy. In order to visualize the tumor masses in the mammogram, the contrast due to the difference in attenuation coefficient between the pathologic tissue and the surround must be visible to the human viewer. In fig. 1.2 typical values of contrast between a tumoral mass and the normal tissues are plotted as a function of the x-ray energy.

Unfortunately this contrast is very small, in particular when the tumor is surrounded by fibroglandular tissue as shown in fig. 1.3, where two mam- mograms show a tumoral mass surrounded by adipose tissue (fig. a) and by fibroglandular tissue (fig. b).

Figure 1.1: X-ray linear attenuation coefficient of the fat, fibroglandular tissue and ductal carcinoma as a function of the x-ray energy.

Since the contrast decreases rapidly with increasing x-ray energy, mam-

mography is conventionally carried out with low energy spectra, typically

Figure 1.2: Intrinsic contrast of the ductal carcinoma and of the microcalcifica- tions respect to the normal breast tissue as a function of the x-ray energy (monoenergetic case).

Figure 1.3: Two mammograms show a ductal carcinoma surrounded by fat (in

a) or fibroglandular tissue (in b). In the latter case the contrast

between the carcinoma and the surrounding tissue is subtle.

1.2. X-ray imaging 13

using an x-ray tube with molybdenum anode operated at a potential of 28kV p, with additional molybdenum beam filtration. Nevertheless the breast attenuates x rays very strongly at these energies and, therefore, to obtain adequate signal from the image receptor, a relatively high dose compared to general radiography is received by the breast. On the other hand the dose should be kept to a minimum to reduce the risk from the radiation exposure.

As regards the detection of microcalcifications, it is equally complicated because they have dimensions of the order of hundreds of µm, therefore, the spatial resolution of the imaging system must be very high. Because of these requirements, mammography is one of the most technically demand- ing radiological imaging techniques.

1.2 X-ray imaging

The radiographic image is formed by the interaction of x-ray photons with the patient tissues and after that with a photon detector. It is therefore a spatial distribution of those photons which are transmitted through the patient and are recorded by the detector. This distribution gives a measure of the probability that a photon will pass through the patient without in- teracting, and this probability will itself depend upon the sum of the x-ray attenuating proprieties of all the tissues that the photon crosses. The im- age is therefore a two-dimensional projection along the photon path of the three-dimensional distribution of the tissues x-ray attenuating proprieties.

1.2.1 The image formation

In what is to follow we have a simple mathematical model of the radio-

graphic imaging process. In Fig. 1.4 we consider a monochromatic x-ray

source that emits photons of energy E parallel to the z direction. In this

figure the patient is replaced by an uniform block of tissue of thickness t

and linear attenuation coefficient µ

, containing an embedded block of ‘tar-

get’ tissue of thickness x and linear attenuation coefficient µ

. When the beam passes through the patient, the r rays spatial distribution changes in relation to the tissue attenuation proprieties. This photon spatial distri- bution is called quantic image. Now we can record the quantic image with a photon detector placed in the xy plane. We assume that each photon interacting in the receptor is locally absorbed and that the response of the receptor is linear with the detected energy, so that the image may be con- sidered as a distribution of absorbed energy. If there are N photons per unit area incident on the patient and we neglect the scattered radiation, than the energy absorbed, I(x, y)dxdy, in area dxdy of the detector can be expressed as

I(x, y) = N ε(E)Eexp



−

Z

µ(x, y, z)dz



(1.1) where ε(E) is the detector efficiency as a function of the energy, µ(x, y, z) is the linear attenuation coefficient and the line integral is over all tissues along the path of the photons reaching the point (x, y). This equation represents the recorded image.

1.2.2 Contrast and Signal to Noise Ratio

The ability to distinguish structures of different tissues in a radiological image depends upon both the tissue attenuation proprieties and the perfor- mance of the imaging system in transferring this information content into the image. This capability represents the Image Quality of the radiogram and it can be quantified, as first approximation, in terms of two physical parameters, namely Contrast and Signal to Noise Ratio (SNR).

Referring to Fig. 1.4, the contrast C of the target is defined in terms of the image signal I

(target signal) and I

(background signal), defined in the 1.1 and related respectively to detector area outside and inside the

“shadow” image of the target. C is defined as C = I

− I

I

(1.2)

1.2. X-ray imaging 15

Figure 1.4: The uniform block of tissue of thickness t and linear attenuation coefficient µ

represents the patient. It contains an embedded block of ‘target’ tissue of thickness x and linear attenuation coefficient µ

. I

and I

represent the image signals relative respectively to the area below the target and the background

Substituting I

and I

with the expressions given in the 1.1, we obtain C = 1 − exp [−(µ

− µ

)x] (1.3) The factors that affect the contrast are therefore the thickness of the tar- get and the difference in linear attenuation coefficients, which depend in general on the photon energy.

The SNR is defined as:

SN R = I

− I

q

σ

+ σ

(1.4)

where σ

and σ

represent respectively the image variance relative to the target and the background.

In real systems the image contrast and the SNR are functions of the

spatial frequency, and they depend upon several physical characteristics of

the imaging system such as the spatial resolution, the noise of the detector

and the electronics, the detector efficiency and the beam geometry. In the

following section we discuss in detail these factors and how they affect the image quality.

1.3 Digital mammographic systems

A good detector for radiography, and for mammography in particular, must show a good spatial resolution to maximize the contrast of the small targets, a low noise to maximize the SNR and a high detection efficiency to minimize the dose delivered to the patient.

Nowadays screen-film mammography is still the most common tech- nique used in mammography. It is and will continue to be a valuable tool for detection and diagnosis of breast cancer. In fact, screen-film mammog- raphy has several important characteristics that become evident when one attempts to develop an alternative, improved technology. Among others the very high spatial resolution (in excess of 20 line-pairs/mm) and the cost is relatively low. Nevertheless a problem of this system is the low efficiency in the diagnostic energy range. It may be improved by using in- tensifying screens, composed of a 70−350µm thick layer of active phosphor crystals. The intensifying screen absorbs the x rays and re-emits fluores- cent photons (at a wavelength at which the film is more sensitive) that are detected in the film. While the intensifying screens improve the efficiency, nevertheless the spatial resolution is affected by the lateral spread of the fluorescent photons, consequently a compromise must be found between efficiency and spatial resolution. Another disadvantage of screen-film sys- tems is related to their limited exposure range, due to the non-linearity of its characteristic response curve [4].

To overcome these limitations digital mammography is rapidly replac-

ing traditional method. Digital systems in fact have in general a linear

response over a wide range of exposure. Moreover using high Z materials

as sensors it could be possible to increase the efficiency or alternatively

reduce the dose to the patient. These systems also have other advantages

related to the possibility of software image optimization, archiving and

1.3. Digital mammographic systems 17

transmission of the digital images [5].

Digital mammography replaces the screen-film image receptor with a detector which provides an electronic signal proportional to the energy in- tensity (or to the photons number) of x rays transmitted by the breast.

The digitized signal, stored in a computer, represents the final digital mam- mogram that may be processed and displayed on a soft copy or hard copy device.

The detection of an x-ray image may conceptually be divided into three separate stages.

• The first stage is the interaction of the transmitted x rays within a suitable detection medium to generate a measurable response.

• The second stage is the measurement and the storage of this response by means of a dedicated read-out electronics.

• In the third stage the data are transmitted to a computer for the manipulation and the visualization.

Several different technologies have been developed for digital x-ray imaging, and consequently today several different types of digital systems for mammography are available and commonly used in clinical exams [3].

An important distinction among these can be made at the first stage of the image formation: the photon detection stage, which can be indirect or direct. Indirect detectors use an intermediate layer of scintillating material (like CsI) to convert x rays beforehand into visible light pulses, which after are collected by photons sensitive elements; while direct-detection systems convert x rays into electrical charges inside the detector itself. Indirect detection shows an important drawback respect to the direct detection re- lated to the light spread in the conversion layer, than may limit the spatial resolution of the detector (refer to Fig. 1.5).

Another important distinction can be made in the second stage: the

signal formation. In general the detector gives a response proportional to

Figure 1.5: The response to a narrow line beam of x ray photons for screen- film and digital detectors. Screen-film and indirect detectors have broader spatial response then direct detectors.

the energy collected during the acquisition time, this approach is called Charge Integration Mode. Lately the improvement in the microelectronics has allowed the development of new integrated read-out electronics based on Single Photon Counting (SPC) architectures. These systems gives a response proportional to the number of detected photons, moreover a dis- crimination threshold allowed to count only those photons with energy above a threshold.

In the following we give a brief description of the principal imaging systems developed for mammography.

1.3.1 Indirect detection Systems

Computed Radiography Systems These systems rely on phosphor

screens that exploit photostimulated luminescence. Energy from x-ray

absorption causes electrons in the phosphor layer to be temporarily freed

and then captured and stored in traps within the crystal lattice. The

number of filled traps is proportional to the absorbed x-ray signal. The

image is then read out by placing the screen in a separate reader, where it

is scanned with a laser beam. The detector-element size is usually equal

to 100µm, but it can be improved to 50µm. The potential advantages of

1.3. Digital mammographic systems 19

this technology are the small detector-element size and the relatively lower cost. Potential disadvantages are loss of spatial resolution due to scattering of the laser light during readout, and the addition of noise associated with the low collection efficiency of emitted light.

Flat-Panel Phosphor Systems In these systems, x rays are absorbed in a CsI(Tl) phosphor layer. The phosphor is deposited on a flat plate composed of amorphous silicon on which a rectangular array of photodi- odes is used to record the light emitted by the phosphor. Each detector element on the array contains the photodiode and a thin-film transistor (TFT) switch. Also for these systems the pixel size is in general equal to 100µm. The most common flat-panel system is the Senographe 2000D (GE Medical Systems). Potential limitations of this technology are the difficul- ties in further reducing detector element size, and the possible presence of

“memory” in the detector that causes the signal to depend on its previous exposure history (“ghosting” effect).

Scanning Phosphor-CCD System In these systems the detector again uses a CsI(Tl) phosphor. Here however, the phosphor is deposited on a coupling plate that consists of millions of optical fibers. The fiberoptics conduct light from the phosphor to a CCD array with minimal spread.

The CCD is a chip containing rows and columns of light-sensitive elements, which converts the light into an electronic signal that is digitized. To ac- quire the image, the detector scan across the breast. There are however disadvantages related to the scanning such as the total image acquisition time.

1.3.2 Direct detection Systems

The direct-detection systems directly convert x rays into electrical charges

inside the detector itself, generally constituted by a solid-state material

as a semiconductor. The most common system based on direct detection

of the x rays are the Selenium flat-panels and the semiconductor pixel detectors.

Differently by the previous described systems, in these systems the elec- tronics read-out is physically separated by the x-ray sensor. The important advantage of this feature is the possibility to combine the read-out with sensors of different materials (Silicon, Selenium, GaAs, CdTe, etc.).

Selenium Flat-Panels The Selenium based flat-panels use as radia- tion absorber a slab of amorphous selenium that is a suitable material for mammographic detectors because of a high x-ray absorption efficiency (close to 100% in the mammographic energy range). Moreover it shows an extremely high intrinsic resolution, a low noise and a well-established manufacturing process.

The charge, generated into the a-Se detector, is collected by a pixel elec- trode and accumulated onto its capacitance. On each pixel, the electrode and the storage capacitor are connected to a TFT switch, similar to that used by Csl(Tl) Flat-Panel systems, where, in this case, the photodiode elements are replaced by electrodes.

Systems based on Si pixel detectors and SPC electronics Pixel detectors are formed by a semiconductor crystal which is segmented to form a matrix of diodes (pixels) depleted by a reverse bias voltage applied on its electrodes. The ionization particle incoming into depleted volume of the diode creates a charge which is collected by the electric field and brought to contacts where it is taken by read-out electronics for further processing.

The read-out electronics is generally constituted by an integrated circuit which contains a read-out chain for each detector pixel. Some examples of read-out chips for pixel detectors are Medipix [6], Pilatus [7] or XPAD[8], which are based on SPC architectures.

Lately novel systems, based on Silicon pixel detectors and on SPC read-

out begin to be used in medical applications. Two direct detection SPC

1.3. Digital mammographic systems 21

systems optimized for mammographic imaging have recently been proposed to the scientific community: the Dear-Mama [9], and the Sectra MicroDose Mammography (MDM) system [10]. The Dear-Mama is a prototypal de- vice based on 700µm thick Si detectors divided into 55µm square pixels and coupled to the Medipix2 chip [11]. The MDM is a commercial system based on silicon strip detectors in edge-on geometry with an effective pixel pitch of 50µm. Both the systems use Si detector, but given the low atomic number of silicon (Z = 14), several millimeter thick crystals are required to completely absorb the x-rays. This conflicts with the general rule that the detector thickness should to be kept of the order of the pixel size to reduce the charge spread effect and improve the spatial resolution.

The IMI prototype The IMI prototype [12] is a mammographic imag- ing system based on pixel detectors in GaAs and on SPC read-out electron- ics (Medipix1 chip [6]). The X-ray sensing element is a slab of crystalline GaAs, 200µm thick, 1.2cm

of area, segmented in a matrix of 64×64 pixels at a pitch of 170µm. The use of an highly absorbing material like GaAs overcome the problem related to the use of Silicon, and allow the almost total interaction of x rays (in the mammographic energy range) in only 200µm thick crystal.

The IMI prototype has been developed in the framework of the Inte- grated Mammographic Imaging Project (IMI Project) [13], a collaboration among Italian high-technology industries and public research institutions.

This project has been supported by the Italian Ministry of Education, University and Research (MIUR) and by five Italian High Tech compa- nies, AMS, CAEN, Gilardoni, LABEN and Poly.Hi.Tech., in collaboration with the Universities and the Istituto Nazionale di Fisica Nucleare (INFN) sections of Ferrara, Pisa and Roma “La Sapienza”.

The development and the characterization of the IMI prototype form

the main subjects of this thesis. The system is described in depth in

Chapter 3 and the characterization of the system in terms of image quality

is reported in Chapters 4 and 5.

1.4 Some basics on Semiconductors pixel De- tectors

Semiconductors pixels detectors were originally developed for applications in high energy physics as particles detectors with high spatial resolution.

As we will demonstrate they represent a technological advance in medical imaging, eliminating problems associated with light scatter inherent in indirect conversion systems. Further, since the sensor and the electronics are physically independent, it is possible to separately optimize them. The sensor and readout electronics in fact, can be made of different materials and manufactured with different technologies and processes.

Referring to Fig. 1.6 when a semiconductor absorbs the x rays, an electric charge is released in the material in the form of electron-hole pairs, which drift towards the respective electrodes. The charges are collected by the pixellated electrode and the signal is processed by the read-out electronics relative to the pixel in which the photon has been absorbed.

The read-out circuit generates an electric signal proportional to the charge collected by each pixel and transfers the digitalized signal to the acquisition system. The digital signal finally constitutes the image.

The potential advantages of such a system are the high spatial resolu- tion and the high efficiency that can be achieved, thanks to sensor material with high density and high atomic number.

Si and Ge have been widely used as x-ray detectors, but Ge requires a

low temperature to reduce the leakage current and Si is not efficient enough

due to its low atomic number. So the research is directed toward the inves-

tigation of compound semiconductors, as GaAs, CdTe and, more recently,

CdZnTe and InP, realized with materials with higher atomic number so as

to have a higher detection efficiency at room temperature. In Tab. 1.1 the

main physical characteristics of these semiconductors are compared.

1.4. Some basics on Semiconductors pixel Detectors 23

Figure 1.6: A pixel detector scheme. A photon is absorbed in the semiconductor crystal, where an electric field is applied. The electron-hole pairs created by the photons drift towards the respective electrodes. The charges are collected by the pixellated electrode and the signal is processed by the read-out electronics relative to the pixel in which the photon has been absorbed.

Table 1.1: The physical characteristics of the main compound semiconductors

Material Density (g/cm

)

Atomic number

Band gap (eV )

Ionization Energy (eV )

Hole mobility (cm

/V s )

Electron mobility (cm

/V s )

Si 2.33 14 1.12 3.6 450 1450

Ge 5.33 32 0.67 2.9 1900 3900

GaAs 5.32 31/33 1.42 4.3 400 8000

CdTe 6.06 48/52 1.52 4.43 80 1000

CdZnTe 6 48/30/52 1.6 5 120 1350

HgI

6.4 80/53 2.13 4.3 4 100

InP 4.79 49/15 1.35 4.2 150 4600

1.4.1 Principal Detectors characteristics

A good choice of the detector material is a nodal point for the develop- ment of a mammography system, since it is fundamental in determining the intrinsic quality of the produced images, such as the SNR and the Contrast. To accomplish these general requirements a good detector for digital mammography should have a high efficiency, a low level of noise and a high spatial resolution.

Efficiency The most important characteristic of an x-ray detector is the capability to absorb x rays. In the mammographic energy range the pho- tons interact in the matter via the photoelectric absorption and the co- herent or inchoerent scattering. The photon probability to interact in the detector medium is determined by the linear attenuation coefficient, which is characteristic of the medium and it is energy dependent. The interaction probability is expressed as:

η(E) = 1 − e

where x is the thickness of the detector and µ(E) is the linear attenuation coefficient of the medium as a function of the incident photons energy.

The detector efficiency in detecting the radiation quanta however is determined also by the ability in collecting the charge released into the detector and by the read-out electronics. Therefore the efficiency of a de- tector system may be expressed by means the quantum efficiency, defined as the ratio between the number of recorded pulses and the number of incident quanta, except the spurious events. In terms of energy it can be expressed as the collected share of the incident x rays energy. The recorded events do not involve the spurious ones due to the noise of the detector, since their presence would artificially augment the efficiency instead of reducing it.

Since the detection efficiency is strongly dependent on the linear attenu-

ation coefficient of the detector medium, high-Z semiconductor compounds

1.4. Some basics on Semiconductors pixel Detectors 25

such as CdTe, CdZnTe and GaAs are more desirable than semiconductor with lower Z as Silicon. For example, for x rays at 44keV , the detected quanta in 300µm of GaAs are 10 times more abundant than in 700µm Si, as reported in [14].

Detector noise In a semiconductor detector the noise is principally due to statistical fluctuation in the number of absorbed x rays, but there is an additional component that results from variation in the amount of energy released by x rays and in the number of electron-hole pairs produced.

Moreover other sources of noise may arise from the fluctuations of the detector leakage current or by the read-out electronics.

Among semiconductors Silicon shows the best noise characteristics, be- cause it has high purity, low leakage current and high charge collection efficiency (c.c.e.). The Silicon is in fact the best developed material, from a technological point of view, with a well consolidated process of realization.

On the other hand in the last 15 years GaAs detectors have been optimized as x-ray image receptors, and devices with spectroscopic and imaging ca- pabilities adequate for medical applications have been produced [15]. To mention a recent example, G. Bertuccio [14] has characterized GaAs detec- tors with leakage current, noise and spectroscopic capabilities comparable to, or much better than, silicon devices.

Spatial resolution The spatial resolution of an x-ray imaging system is determined by multiple factors depending both by the detector and by the read-out electronics. Principally it is due to the sampling aperture of the detector (i.e., to the size of the electrode pixels). Moreover other effects related to the radiation and charge transport may degrade the resolution.

An important problem for the resolution of the pixel detectors is related

to the signal sharing. This can arise from x-ray scattering, fluorescent

photons or from charge diffusion, where some of the induced charge drifts

to neighbouring pixels and thus contribute in the wrong photon position

detection. The last effect is called charge-sharing. If the pixel size of a pixellated detector is small compared to the wafer thickness, signal sharing may reduce both spatial resolution and energy resolution of the detector, as shown in Fig. 1.7. In this figure the signal generated by an incident photon is shared in more than one pixel. This is due to the re-absorption of the fluorescent photon in a neighboring pixel and to the diffusion of the charge carriers while they drift toward the electrodes.

The signal sharing is an important problem that may affect significantly the imaging performance of a pixel detector. In the Chapter 4 we inves- tigate, by means of Monte Carlo simulations of the radiation and charge transport, how the signal sharing influences the image quality of the IMI prototype detector.

Figure 1.7: The signal sharing takes place when the charge generated by an incident photon is collected in more than one pixel. This may be do to the re-absorption of the fluorescent photons or to the diffusion of the charge carriers while they drift toward the electrodes.

1.4.2 Read-out electronics: Photon Counting Mode versus Integration Mode

The most common digital detectors (as the TFT or CCD based systems)

work via charge integrating principle. The sum of the charge accumulated

1.4. Some basics on Semiconductors pixel Detectors 27

in a pixel corresponds to the total x-ray energy absorbed by that pixel in the image. Pixel detectors instead are commonly read out by SPC electronics.

Semiconductor pixel detectors require an array of electronics pixels that collect charge signals. Each pixel of a detector must be connected to a front end circuit, which processes the signals and transfers them to a computer during a readout sequence.

Single Photon Counting read-out electronics has become possible thanks to advances in microelectronics, which allows the deigns and fabrication of chips with pixellated front-end electronics of small dimensions. The mea- sured charge pulse defines the energy of the absorbed photon, and there- fore it is possible to perform spectroscopic discrimination on each pixel by means an energy threshold. The final image represents for each pixel the number of photons which have released in the correspondent detector pixel an amount of energy higher than the energy threshold.

The SPC electronics show several advantages in respect to the charge- integration one. These are lower noise, higher spatial resolution and higher detected contrast. In the following we discuss these advantages point by point.

Noise The fundamental source of noise in a x-ray imaging detector is

due to statistical fluctuation in the number of absorbed x rays, but in the

charge integration systems there may be additional noise components that

result from variation in the amount of charge produced per x-ray and from

the electronic noise of the read-out. Both of these last sources of noise

can be eliminated if, instead of accumulating charge from multiple x rays,

the absorbed quanta are simply counted. Thus, each x-ray produces one

count, regardless of its energy or how much charge it actually deposits on

the detector (Fig. 1.8).

Figure 1.8: The energy threshold of the SPC electronics allow to discriminate the noise arising from the electronics and form the produced charge variability.

Image contrast In mammography policromathic x rays in the energy range of (10 − 30)keV are commonly used. As shown in Fig. 1.2, the contrast of the typical objects of interest in mammography is strongly decreasing with the photon energy. Consequently low energy photons carry greater contrast information than higher energy ones, due to their higher differential absorption. In charge integrating detectors, since high energy x rays have the capacity to produce more charge, then they will be weighted more then low energy ones contrary to their lower information content.

In SPC systems each photon produces one count regardless of its energy, therefore all photons are weighted equally, which is a more efficient use of the information.

Spatial Resolution As explained before, when a photon interacts in a

detector pixel, the signal sharing may degrade the spatial response, since

the photon energy may be collected not only by the pixel where the interac-

tion originally occurred, but also by the neighbours. If the SPC electronics

is used, then the energy threshold may be suitably set so that the signal

collected by the neighbouring pixels do not overstep the threshold and

one hit is assigned only to the pixel where the photon has originally been

1.4. Some basics on Semiconductors pixel Detectors 29

absorbed, as illustrated in Fig. 1.9.

Figure 1.9: In Charge-Integration systems the signal sharing affects the spatial

resolution, instead in SPC system, if the energy threshold discrim-

ination is suitably chosen, the event is assigned only to the pixel

mostly interested by the event.

CHAPTER 2

Image Quality Metrics for digital systems

In this chapter we define the concept of image quality for digital imaging systems and we report on the theoretical and mathematical background of two evaluation methods used to assess the imaging capabilities of the IMI prototype.

The first method uses an approach based on the linear-system theory to quantify the performances of the imaging system. In the following we de- fine the Modulation Transfer Function (MTF), the Noise Power Spectrum (NPS) and the Detective Quantum Efficiency (DQE), useful descriptors of the contrast, noise, and spatial resolution of an imaging system.

We have also developed a theoretical model that describes the image quality of an ideal photon counter detector. We have calculated the ex- pected MTF, NPS and DQE of such a system that will be compared, in Chapter 5, with the experimental measurements performed on the IMI Prototype.

The second method used to assess the image quality of the imaging

devices is based on the Contrast-Threshold (CT) analysis, and it describes

the ability of the human observer to detect objects of given contrasts and

sizes into the images acquired by the system. The theoretical foundation for the CT analysis is in the work of Rose [16], which will be briefly de- scribed at the end of the chapter.

2.1 Assessment of the image quality

The rapid diffusion of new radiographic imaging technologies required the definition of new standard methods to evaluate and compare them.

The image-quality assessment of a radiographic system is necessary for both scientists and radiologists. The radiologists need to verify the suit- ability of an imaging system for their clinical needs; to compare different systems or to perform the periodical quality tests that are required in clinical activities.

On the other hand the scientists need a set of physical image quality parameters, objective and simple to measure, that aims to quantitatively characterize the imaging device and to suggest a guideline for the devel- oping of new technologies. Unfortunately an exhaustive method for the image quality assessment that meets all these requirements is not easy to find.

The ultimate definition of the image quality for a radiographic device may be expressed as its ability to improve the diagnostic accuracy for a specific diagnostic task. However define and quantify the diagnostic accuracy in a number of measurable parameters is not straightforward.

The physicists have conducted a number of studies to find a set of physical quantities somehow connected to the diagnostic image quality, such as the resolution, noise, and signal to noise transfer performance of the devices.

All these quantities can be easily assessed by measuring in the spatial

frequency domain the Modulation Transfer Function (MTF), the Noise

Power Spectrum (NPS), and the Detective Quantum Efficiency (DQE),

which describe the spatial resolution, the noise, and the SNR transfer

properties of the system.

2.2. Transfer Functions analysis 33

Although the evaluation of these physical parameters provides useful information about the system, it does not give a complete picture of the system imaging quality in terms of diagnostic accuracy, because improved physical performance may not lead to improved performance in the diag- nosis or, counterintuitively, a degradation of physical performance, such as slightly decreased resolution or smoothing, may sometimes improve the performance in the diagnosis.

To give a description of the imaging quality in terms of diagnosis ca- pabilities, the visual process of the human observer must be taken into account. Since the observer receives the final presentation of radiographic images and converts that into a diagnostic finding, it is important to in- clude the observer’s performance contribution in the overall scheme of as- sessing the image quality. In order to accomplish this task, a well known technique is the Contrast-Threshold analysis [17]. Through this analysis it is possible to quantitatively assess the imaging performance of the sys- tem by means of the observer evaluation of the radiographs of a specific test-phantom.

2.2 Transfer Functions analysis

One way to characterize an imaging system is to describe the input-output relationship of the parameters useful in the description of the image signal and the noise. Figure 2.1 shows input and output images of a hypothetical imaging system in which the degradation of image resolution (fig. a), noise (fig. b), and both of them (fig. c) occur. This figure shows four objects of different sizes, that are surrounded by an uniform background. To evaluate the system capabilities in transferring the signals related to an object, the contrast transfer factor may be used. It is defined as

T

= C

C

where C

is the image contrast defined in the Eq. 1.2, and C

is the

same quantity calculated on the distribution of the x rays incident on the

detector (quantum image, see section 1.2.2).

The problem is that T

depends on the size of the object. This is illustrated in Fig. 2.1, where the system transfers large-area contrast fairly well, but small-area contrast poorly. Transfer theory must therefore be tied to the object size. Consequently it is customary to describe the signal and noise transfer capabilities in the spatial-frequency domain. MTF, NPS and DQE are the main descriptors of an imaging system response in the spatial-frequency domain.

Figure 2.1: Input-Output images of an hypothetical system that introduce a

degradation in the spatial resolution (a), noise(b) and in both of

them (c). There are four detail sizes in the input image. It can be

seen how the contrast of the output depends on the detail size.

2.3. Modulation Transfer Function 35

2.3 Modulation Transfer Function

The M T F is a useful descriptor of the spatial resolution and of the contrast transfer capabilities of an imaging device.

The spatial resolution of a system is the minimum distance at which two objects can be placed and still be perceived as distinct objects. The present definition is not very practical, because it depends on the shape of the object used.

To overcome this problem, an alternative descriptor of the spatial res- olution may be found by considering the system point spread function (PSF ). Given a Dirac delta function (δ(x − x

)) as input signal, the cor- responding system output S{δ(x − x

)} is called the system PSF.

Linear-Systems theory assures that the input-output relations of a lin- ear shift-invariant (LSI) system are completely specified by the P SF [18].

In fact the response d(x), to an input function h(x) may be expressed as the convolution product:

d(x) = S{h(x)} = h(x) ∗ P SF (x) (2.1) Since the convolution product in the spatial domain is equivalent to a simple multiplication in the spatial frequency domain [19], the Eq. 2.1 may be expressed in terms of Fourier transforms as

D(u) = H(u) · T (u) (2.2)

where D(u) and H(u) represent the Fourier transforms of d(x) and h(x), and T (u) is the Fourier transform of the P SF as a function of the spatial frequency u:

T (u) = F {P SF (x)} (2.3)

This is called the system characteristic function. Fig. 2.2 represents the input-output relationships in the spatial and frequency domains.

From equation 2.2 we can express T (u) as:

T (u) = D(u)

H(u) (2.4)

Figure 2.2: Signal-transfer characteristics can be represented either as convo- lution with the P SF in the spatial domain (left column) then as multiplication with the characteristic function T (u) in the spatial- frequency domain (right column).

that represents the ratio of the frequency content output vs the frequency content input. If we consider a sinusoidal input of frequency u, from equa- tion 2.2 we can see that the output is identical to the input scaled by the frequency-dependent factor T (u), as illustrated in Fig. 2.3. Therefore the characteristic function, contrary to T

defined previously, is able to represent the system signal transfer capability as a function of the object size.

To characterize the spatial response of an imaging system a different descriptor, named the Modulation Transfer Function (MTF) is commonly used. It is defined as:

M T F (u) = |T (u)|

T (0) (2.5)

where M T F (u) has a value of unity at u = 0 by definition. It can be noted

that the M T F is not a complete descriptor of a system as the characteristic

function T (u) because the phase information and the scaling constant have

been discarded.

2.3. Modulation Transfer Function 37

Figure 2.3: A sinusoidal signal at the input of an LSI system will produce a sinusoidal output with the same frequency, scaled by the frequency- dependent factor T (u).

For bi-dimensional systems (as in the case of imaging devices) the M T F is a function of the spatial frequencies u and v, M T F (u, v). However, similarly to the one-dimensional case, we can define the line spread function (LSF ) as the the system response to a delta line function. Also in this case the M T F (u) is defined as the Fourier transform of the LSF normalized to the zero value:

M T F (u) = |L(u)|

L(0) where L(u) = F {LSF (x)} (2.6) The MTF defined here represents the system response only in the orthog- onal direction in respect to the input delta line. However for systems with a rotationally symmetric P SF , the M T F (u, v) is also symmetric and it is totally specified by the M T F (u), calculated in any one direction.

Generally for bi-dimensional systems the one dimensional MTF is con- sidered, and it is obtained by measuring the system response to a slit or an edge.

2.3.1 The MTF for digital imaging systems

In digital imaging systems the detected signal is sampled by the electronics

read-out with a given sampling distance correspondent to the detector

inter-pixel distance. The inverse of the sampling distance is the sampling rate. The Nyquist-Shannon sampling theorem states that an analog signal, that has been sampled, can be perfectly reconstructed from the samples if the highest frequency in the original signal is below the Nyquist frequency (F

), defined as half of the sampling rate. If the analog signal has some components at frequencies higher than the Nyquist one, the signal will be undersampled, and any frequency components of the original signal higher than the Nyquist frequency will appear as a lower frequency in the sampled image. This effect is called aliasing [19].

Almost all digital imaging systems are undersampled to some degree.

An undersampled digital system is no longer adequately described by the LSI model, because it is not space-invariant. The sampled point spread function of the digital system (P SF

) will then depend on the phase of the sampling comb with respect the impulse function and the system response to a generic input may not be calculated as the convolution product of the input signal with the P SF

(i.e. the equations 2.1 and 2.2 are no more valid if the P SF

is undersampled).

An way to overcome this problem in characterizing an undersampled digital system is by means of the presampled MTF (P M T F ) that describe the system response up to, but not including, the stage of sampling. The P M T F is mathematically defined as the absolute value of the Fourier transform, normalized to the zero-frequency value, of the presampled P SF (P SF

).

P M T F (u) = F {P SF

(x)}

F {P SF

(x)}|

(2.7) The (P SF

) and the P SF

coincide only for systems that are not un- dersampled. The P M T F includes the response of the system to the blur from the x-ray sensor and the aperture function due to the shape and size of active area of the pixel, but it does not include the effect of sampling.

Misinterpretations of the P M T F may occur when it is used for eval-

uating and comparing the spatial resolution of an undersampled digital

imaging system. The P M T F in fact accurately represents the response

2.4. The Noise Power Spectrum 39

of a digital system to a single sinusoid input only for frequencies below the Nyquist one. If the P M T F were found to extend beyond the Nyquist frequency imposed by the sampling rate, some frequency components of the input signals would be aliased in the final sampled image. Hence the P M T F cannot be used to derive the system response to a delta input, and in general to any input signal with spectral components over the Nyquist frequency.

As an example in Fig. 2.4 we present two hypothetical systems that have different P M T F s but the same sampled response to a delta input (same P SF

). In the right column are shown the P M T F s of these sys- tems, and in the right column the fourier transforms of the P SF

. The system 1 in a) is an undersampled system that has a typical P M T F that extend over the Nyquist frequency (on the left). The Fourier transform of the P SF

(on the right) is obtained with infinite copies of the P M T F , shifted by multiples of the sampling rate and combined by addition. The second system in b) has a nicely band-limited response. This system is not aliased, then the P M T F coincides with the Fourier transform of the P SF

. One should note that these two systems have vastly different presampling M T F s, but, they respond equivalently to the image of a slit.

2.4 The Noise Power Spectrum

The M T F is useful in characterizing the deterministic component of the system response, but in real systems the output images are also affected by a random fluctuation due to the stochastic nature of the signal (image noise) [18]. Although the output for a system with random fluctuations cannot be predicted, it may be possible to determine its statistical prop- erties.

In the image-intensity domain, the statistical property of the noise may

be represented by the sampling variance of the image matrix (σ

), defined

Figure 2.4: The P M T F and the Fourier transform of the P SF

for two hy- pothetical digital imaging systems are shown. These two functions are equal only for not undersampled systems. The system 1 is un- dersampled whereas the system 2 is nicely band-limited. Note that the P M T F curves are quite different, but the sampled responses to an impulsive input are identical.

as

σ

=

N

X

i=1

(I

− ¯ I)

(2.8)

where ¯ I is the expectation image pixel value, and N is the number of pixels in the image. Nevertheless this noise characterization is not exhaustive.

The Fig. 2.5 for example represents two signals with identical variance, but with very different profiles. In a) the noise is correlated only over a very short distance, while in b) it is correlated over a longer distance. To take into account also the noise correlation, the second order statics must be used.

If d(x, y) is a real random variable expressed as a function of x and y, the autocorrelation function R

may be defined as

R

(x

, x

+ x, y

, y

+ y) = E{d(x

, y

)d(x

+ x, y

+ y)} (2.9)

where E is the expectation value operator. The autocorrelation describes

the correlation of d(x

, y

) with itself at a location displaced by (x, y).

2.4. The Noise Power Spectrum 41

Figure 2.5: One dimensional profiles of two hypothetical noise signals. The two signals have the same variance but look very different. The noise in b) is correlated over a longer distance than the noise in a).

A random process having at least the expectation value and the au- tocorrelation function stationary in (x, y) (i.e. space invariant) is called a wide-sense stationary (WSS) process. If d(x, y) is a WSS process its correlation function will depend only on (x, y), but not on the position (x

, y

):

R

(x

, x

+ x, y

, y

+ y) = R

(x, y) (2.10) The expression of the autocorrelation function may be further simplified for ergodic systems, whereas ergodic means that expected values can be de- termined equivalently from ensamble averages or spatial averages. Likely, many random processes responsible for noise in medical imaging systems are ergodic or can be approximated as ergodic [20]. If d(x, y) is an ergodic WSS process, the autocorrelation function may be expressed as:

R

(x, y) = lim

X,Y →∞

1 X

1 Y

Z

X

Z

Y

d(x

, y

)d(x

+ x, y

+ y)dx

dy

(2.11) In practice, for the noise characterization is more useful to work in terms of the autocorrelation function of the fluctuations (also called autocovariance C

), defined as

C

(x, y) = lim

X,Y →∞

1 X

1 Y

Z

X

Z

Y

∆d(x

, y

)∆d(x

+ x, y

+ y)dx

dy

(2.12)

It is noted that the scale value of the autocovariance function is simply equal to the signal variance:

C

(0, 0) = lim

X,Y →∞

1 X

1 Y

Z

X

Z

Y

∆d(x

, y

)

dx

dy

= σ

(2.13) Since the autocorrelation function includes the variance, it completely specifies the first and second order statistics of the measured density fluctu- ations, provided that the process is Gaussian, and so this function defines the whole random process.

Although the autocorrelation function of the fluctuations is exhaustive in describing a random process as the image noise, it is customary to represent the same information in the spatial-frequency domain by means of the Noise Power Spectrum (NPS), also called the Wiener spectrum, defined for a WSS process by Dainty and Shaw [21], as

N P S(u, v) = lim

X,Y →∞

 1 X

1 Y

    Z

X

Z

Y

∆d(x, y)e

dxdy

2

(2.14) where the symbol <> denotes the ensamble average. This equation de- scribes the contribution to the variance from spatial frequencies between (u, v) and (u + du, v + dv). The Wiener-Khintchin theorem states that the NPS and the autocovariance function are Fourier transform pairs (see Fig.

2.6)

N P S(u, v) =

ZZ

−∞

C

(x, y)e

dxdy (2.15)

C

(x, y) =

ZZ

−∞

N P S(u, v)e

dudv (2.16) From Equations 2.13 and 2.16 is possible to connect the signal variance with the volume under the two-dimensional NPS:

σ

= C

(0, 0) =

ZZ

−∞

N P S(u, v)dudv (2.17) The autocorrelation function and the NPS are essentially equivalent mea- sures of image noise in much the same way as the PSF and the MTF are equivalent ways of describing image resolution.

The NPS is an useful descriptor of the noise transfer properties of an

imaging system. If we present at input of the system a flat signal (as a

2.4. The Noise Power Spectrum 43

Figure 2.6: The autocorrelation function (on the left) and the power spectrum (on the right) are Fourier pairs.

spatial uniform quantic distribution), the system noise properties can be analyzed by measuring the NPS of the output signal. This function will then represents as the system transfers the noise as a function of the spatial frequency.

2.4.1 NPS for digital imaging systems

Likewise in the case of the MTF, when one tires to apply the concepts of transfer functions to digital imaging systems, the problems of undersam- pling have to be considered.

We are now considering a digital system that produces at the output a discrete sampled signal d

. For this process we can define, analogously to the 2.14, the digital Noise Power Spectrum (N P S

) as [20]:

N P S

(u, v) = lim

Nx,Ny→∞

∆x∆y N

N

Nx

X

k=1 Ny

X

j=1

∆d

e

2

(2.18)

where ∆

∆

represents the sampling distances along the two direction, N

N

the number of sampling point, and the expression in the angled brackets is the bi-dimensional Fourier transform of a discrete signal, that is [19]:

F

=

Nx

X

k=1 Ny

X

j=1

∆d

e

(2.19)

The Eq. 2.18 is the same quantity expressed in the formula 2.14 for

analog signals in the limit

X

k

X

j

∆x∆y →

Z Z

dxdy (2.20)

For digital undersampled systems the interpretation of the N P S

is not so straightforward as in the analog case, due to the fact that the NPS has been defined only for ergodic WSS random process. A digital system consists of an array of discrete elements which does not represent a WSS random process. This problem is easily overcame by considering that the digital signal d

may be viewed as the sampling of the analog signal d(x, y) (the presampled signal), which can be usually approximated by an ergodic WSS process.

In Fig. 2.7 is represented the sampling of a one-dimensional analog signal d(x) (in a) in both the spatial and frequency domain. The sampled signal (c) is described by a multiplication of d(x) with a comb of delta functions spaced at the sampling distance x

(b). In the frequency domain the spectrum of the sampled signal is equal to infinite copies of the analog signal spectrum (a), shifted by multiples of the sampling rate and combined by addition (sub-figure c in Fig. 2.7). Only the frequencies below the Nyquist one are selected by multiplying the spectrum of the digital signal (in c) with a BOX function in the frequency domain (d). The Power Spectrum is calculated on the final signal (shown in e) and the relative N P S

is also shown. It is possible to demonstrate that the N P S

and the NPS of the presampled signal (N P S

) are connected by the following expression:

N P S

(u, v) = N P S

(u, v) +

∞

X

i=1

∞

X

j=1

N P S

(u ± i

∆x , v ± j

∆y ) (2.21) A mathematically rigorous derivation of 2.21 is reported in Ref. [18]. Eq.

2.21 explicitly states the undesirable effects of noise aliasing with the sec- ond term. Unfortunately, the aliasing artifacts in the N P S

, which have arisen from the undersampling are not eliminable.

In the experimental measurements of the N P S

, the Eq. 2.18 needs

to be further modified to take into account the consequences of the finite

2.4. The Noise Power Spectrum 45

Figure 2.7: The derivation of the N P S

is shown in the spatial and frequency

domain. The presampled signal (a) is multiplied by a comb of delta

(b) to obtain the sampled signal (c). Only the frequencies below

the Nyquist one are selected by multiplying the digital signal with a

BOX function in the frequency domain (d). In e) the final N P S

is shown.