1-D Fourier Transform Pairs

Transcription

1 1-D Fourier Transform Pairs

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3 The concept of the PSF is most easily explained by considering a very small point source being placed in the imaging field-of-view The relationship between the image, I, and the object, O, can be represented by: I x,y,z = O x,y,z *h x,y,z ( ) ( ) ( ) where * represents a convolution, and h(x,y,z) is the three-dimensional PSF. Ideally, the PSF would be a delta function in all three dimensions, and the image and object would be identical. Object y z x Image (left) the object to be imaged consists of a small sphere. (right) the image obtained is larger than the actual object, and may be blurred asymmetrically in the x, y and z dimensions. In this particular case the PSF is shown as being broad in the x direction and relatively narrow in the z and y directions.

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7 Object with noise 2-D Fourier Transform

8 Low-Pass Filter Inverse Fourier Transform

9 High-Pass Filter Inverse Fourier Transform

10 Bandpass Filter Inverse Fourier Transform

11 Two-Dimensional o Fourier Transform Pairs

12 2-D Fourier Transform Pairs

13 Sampling and Quantization

14 Sampling and Quantization

15 Sampling and Quantization

16 Sampling and Quantization Histograms

17 Image Filtering Original Original + Noise

18 Image Filtering Noisy Image Low-Pass Filtered

19 Image Filtering Original Low-Pass Filtered

20 Image Filtering Blurred Image High-Pass Filtered

21 Image Filtering Original Edge-Detection

22 Chapter 1: Medical Imaging: X-rays, CT Note: The class will be taught using the black board instead of using multiple power point slides. However, there is often the time where pictures are just too difficult to draw (e.g. images) and therefore power point will be used. These slides are taken from information directly from your text and will not take the place of taking notes in class. Therefore, it is NOT necessary to print these slides for the lecture.

23 X-ray and Computed Tomography (CT) Conventional x-ray: chest x-ray, dental x- ray Fluoroscopy, Angiography, Mammography Conventional and computed tomography Spiral or helical CT

24 X-rays were discovered by a German physicist, W. K. Roentgen, in 1895 and received the Nobel Prize in University of Wuerzburg One of his earliest photographic plate from his experiments was a film of his wife, Bertha's hand with a ring, was produced on Friday, November 8, 1895.

25 x-ray source low attenuation, low ρ collimator high attenuation, high ρ anti-scatter grid x-ray film (left) Basic set-up for x-ray imaging. The collimator restricts the beam of x-rays, and an anti-scatter grid increases image contrast by reducing the contribution from x-rays that have been scattered, rather than absorbed, by tissue. (right) Contrast: A typical chest x-ray radiograph in which the highly attenuating regions of bone appear white.

26 X-ray wave properties Wavelength, propagation speed, and frequency f λ = c f = frequency Hz (or ν) Particle: E = hf, hν (or photon) λ = wavelength, m c = light speed =3x10 8 m/s 18 h = Plank constant = kev s 19 1 ev = ( J) Joule

27 Major components of an x-ray tube. The tube is typically surrounded by an oil bath and lead housing. The magnified view of the target illustrates the line focus principle, whereby the focal spot size (F) is smaller than the electron beam (L) because of the anode angle.

28 X-ray Production -Diagnostic x-rays are produced when electrons with energies of 20 to 150 kev are stopped in matter. -The kinetic energy of the electron is transformed into heat and x-rays when the electrons strike the anode (tungsten). -Electrons rapidly lose their energy by ionization (loose e - ) and excitation (add energy, low to hi nrg system) of electrons in the anode material. -X-rays are generated by two different processes known as bremsstrahlung and characteristic x-ray production.

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31 Bremsstrahlung (breaking, auf Deutsch, or general) x-rays are produced when incident electrons interact with electric fields, which slow them down and change their direction. -Bremsstrahlung x-rays produce a continuous spectrum of radiation, up to a maximum energy determined by the maximum kinetic energy of the incident electron. -Bremsstrahlung x-ray production increases with the accelerating voltage (kv) and the atomic number (Z) of the anode. tungsten Bremsstrahlung radiation is produced when an energetic electron (1) (with initial energy E1,) passes close to an atomic nucleus. The attractive force of the positively charged nucleus causes the electron to change direction and lose energy. The electron (2) now has a lower energy (E2). The energy difference (E1 -E2) is released as an x-ray photon (3).

32 -Characteristic radiation is produced when inner shell electrons of the anode target are ejected by the incident electrons. -The resultant vacancies are filled by outer shell electrons, and the energy difference is emitted as characteristic radiation as shown below. -Each anode material emits characteristic x-rays of a given energy. -K-shell electrons are ejected only if incident electrons have energies greater than the K-shell binding energy. M L K nucleus Atomic structure showing the maximum number of electrons that can occupy the K (2), L (8) and M (18) shells.

33 Characteristic radiation (cont) -L-shell radiation also normally accompanies K-shell radiation. L-shell characteristic x-rays have very low energies and are absorbed by the glass of the x-ray tube. Only K-shell characteristic x-rays are important in diagnostic radiology. -Most incident electrons interact with outer shell electrons and produce heat but not x-rays. Characteristic radiation is produced when an incoming electron (1) interacts with an inner shell electron (2) and both are ejected (3). When one of the outer shell electrons moves to fill the inner shell vacancy, the excess energy is emitted as characteristic radiation (4).

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36 Interaction of x-ray and matter Scattering and absorption X-ray photon E Nucleus Electrons X-ray photon E E < E Absorption = Photoelectric effect Scattered = Compton scatter

37 X-ray interactions When passing through matter photons may: pass through (i.e. penetrate) absorbed (and x-fer nrg) scattered (change direction and loose nrg) Compton scatter and photoelectric (PE) effect are the important interaction in diagnostic radiology. Others are coherent scatter pair production photodisintegration

38 X-ray interactions Photoelectric (PE) effect occurs between tightly bound (inner shell) electrons and the incident x-ray photons. The PE effect occurs when an incident x-ray (1) is totally absorbed by an inner shell electron, which is ejected as a photoelectron (2). The vacancy (3) is filled by an outer shell electron, and the energy difference is emitted as characteristic radiation (4) or as an Auger electron (5).

39 X-ray interactions In Compton scatter, incident photons interact with loosely bound (outer shell) electrons. -A Compton interaction results in a scattered photon that has less energy than the incident photon and travels in a new direction. - A scattered electron carries the energy lost by the incident photon. (4) -This electron loses energy by ionizing other atoms in the tissue, thereby contributing to the patient dose. incoming x-ray photon (1) interacts with outer shell electron (2) x-ray photon loses energy and changes direction (3) Compton electron (4) carries away energy lost by scattered photon

40 X-ray interactions -As a result of the Compton interaction, a positive atomic ion, which has lost an outer shell electron, remains. -Compton interactions occur most commonly with electrons with a low binding energy. -Outer shell electrons have binding energies of only a few electron volts, which is negligible compared to the high energy (30 kev) of a typical diagnostic energy x-ray photon. -Compton interactions account for most scattered radiation encountered in diagnostic radiology.

41 Intensity of x-ray beam Intensity of the power per unit area of the beam. Intensity ~ energy/time/area ~ function of number of photons (E) /time/area Two Units associated with X-rays 1) Roentgen - R: Internationally accepted unit of measurement of exposure to x- and gamma radiation. One roentgen is the photon exposure that produces (under standard temperature and pressure) a total positive or negative charge of 2.58x10-4 coulomb/kg in 1 ml of air. -ortotal number of ion pairs produced due to the radiation in 1 ml of air under standard conditions equivalent to 2.58x10-4 coulomb/kg in 1 ml of air 2) Radiation absorbed dose (rad): 0.01 joule of radiation absorbed by a 1 kg of material (1 Gy (gray) = 100 rad) 1 rad / R for soft tissues, 4 rad / R in bone at 30 kev

42 Attenuation of the x-ray beam I = I o e -µx where µ is defined as the linear attenuation coefficient with a unit of 1/cm (or np/cm, db/cm). Linear attenuation coefficient is the fraction of photons "lost" from the beam when traveling a unit distance. Two sources for attenuation: Compton, coherent scattering and absorption (photoelectric effect) attenuation = scattering + absorption

43 Attenuation of the x-ray beam -The linear attenuation coefficient normally depends on the density of the absorbing material. -For any absorbing medium, however, the attenuation is the same with only half the thickness but double the density. The propagation length required to reduce the intensity of the beam by 1/2 is given by Half value layer thickness (HVL) = 0.693/µ=ln2/ µ

44 Mass attenuation coefficient (cm 2g -1) 10 1 Bone Muscle 0.1 Fat X-ray energy (kev)

45 Subject contrast decreases with increasing photon energy. As energy increases, so does the ability of the x-ray to penetrate, resulting in less difference in the x-ray attenuation between the air and bone at high energies.

46 Hard and soft x-ray soft x-ray = Low energy x-ray, long λ hard x-ray = High energy x-ray, shorter λ Filtration removes low-energy photons (long-wavelength or "soft" x-rays) from the beam by absorbing them and permits higher energy photons to pass through. This reduces the amount of radiation received by a patient.

47 X-ray detectors Photographic film, digital or solid state detectors Conventional radiography Digital or computed radiography Intensifying screen and fluorescent screen For film For human eye Purpose of an intensifying screen is to maximize for light photons at λ's that are optimal for photographic film. Process of converting X-ray photons into visible photons is called fluorescence.

48 Intensifying screen - emits light photons when struck with X-ray photons X-ray photons Substrate, 0.5 mm Reflecting layer, mm Phosphor, 0.2 mm Protective layer, 0.02 mm Phosphor's: CaWO 4, (calcium tungstate) emits light at 430 nm blue ( range) or Gd 2 O 2 S, (gadolinium lanthanum) 410 nm blue Speed of a screen (ability of photons to escape from screen) ~ thickness Exposure ~ energy, same amount of energy produces more visible photons with a thicker screen which can darken the film in a shorter period of time

49 Image intensifier for fluoroscopy Increase brightness for human visualization Film Optical density (OD) = log 10 (I i /I t ) I i I t

50 Film fog, characteristic curve, film gamma γ, measure of the film contrast Characteristic curve between film density and exposure Optical density OD 2 OD 1 loge 1 loge 2 Log exposure Film gamma = ( OD 2 OD 1 ) - loge 2 - loge 1 / ( )

51 X-ray diagnostic methods Conventional x-ray is a map of the intensity distribution of the x-ray beam that has traversed the medium being interrogated. Higher intensity results in a darker image. The 3-D information is compressed into a 2-D image. No depth info in the image. Lesion density smaller lighter Darker, high intensity

52 Penumbra (shadow) and Geometric Unsharpness (object size distortion) S o f S 1 P f = effective focal spot size, t =distance from object to film, S = distance from F to film P = f (S 1 -S o )/ S o P is the geometric penumbra or geometric unsharpness

53 X-ray is most useful for objects which differ greatly in density from surrounding structures e.g. lung and bone. It is not good for soft tissue differentiation. Fluoroscopy, angiography, mammography In fluoroscopy, a contrast agent is taken by the patient and the movement of the contrast agent in the body is followed via a fluorescent screen or an image intensifier tube. Radiation exposure is high (50 R/min)

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55 Angiography is an x-ray method for imaging the lumen of a blood vessel where a contrast agent is injected either intravenously or arterially. Venography, arteriography, multi-view angiography Lesion Vessel wall X-ray Vessel lumen

56 Mammography Low energy x-ray photons for better soft tissue differentiation and high resolution (0.1 mm) for diagnosing microcalcifications are needed. Breast compression, short exposure time, single emulsion film, Mo target yielding energy levels between 20 and 30 kev, 0.1 to 0.4 R radiation Alternatives: MR and ultrasound

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58 Disadvantage The major disadvantage of both x-ray and CT imaging is the fact that the technique uses ionizing radiation, in the energy range kev. Since this ionizing radiation can cause tissue damage, there is a limit on the number of x- ray examinations per year that can be performed on a patient.

59 Computed tomography (CT) (def) tomography (to-mòg re-fê) noun tomo - Greek, to cut Any of several techniques for making detailed x-rays of a predetermined plane section of a solid object while blurring out the images of other planes.

60 CT reconstruction Parallel beam projections Fan beam projections Cone beam projections (Multi-slice Spiral CT) Spatial coordinate system transformation must be implemented for fan beam and cone beam reconstruction

61 x-ray source x-ray detectors Principle of computed tomography with the x-ray source and detector unit rotating synchronously around the patient. (right) An example of a brain CT.

62 CT Scanners: hardware GE: Prospeed, LightSpeed, Synergy Philips: Tomoscan Siemens: Somatom Patient Fan beam Ring of detectors X-ray source

63 Computed tomography (CT) I 0 I 0 x y µ 1 µ 2 µ 3 µ 4 I 1 =I 0 e -(µ 1+µ 2 ) x I 2 =I 0 e -(µ 3+µ 4 ) x I 3 =I 0 e -(µ 1+µ 3 ) y I 4 =I 0 e -(µ 2+µ 4 ) y 4 equations solved for 4 unknown µ s

64 (a) (b) (c) (d) P 4 P 3 P 1 P 2

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66 (a) (b) (c) (d) (e)

67 H(k) k

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69 φ p(r,φ)

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