Medical Physics. Image Quality 1) Ho Kyung Kim. Pusan National University

Transcription

1 Medical Physics Prince & Links 3 Image Quality 1) Ho Kyung Kim hokyung@pusan.ac.kr Pusan National University 1) The degree to which an image allows medical professionals to accomplish their goals (e.g., diagnosis)

2 Digitalization = sampling + quantization Intensity Pixel pitch Space 2

3 Digital images 8 bits/pixel 4 bits/pixel Sampling, quantization (integer) Dynamic range: the set of possible gray levels Contouring: an artificial looking height How many gray values are needed to produce a continuous-looking image? 3

4 Sampling The conversion from a continuous function to a discrete function retaining only the values at the grid points

5 Quantization The conversion from analog samples to discrete-value samples 8 bits 7 bits 6 bits 5 bits 4 bits 3 bits 2 bits 1 bit 5

6 Comparative tomograms Modality CT MRI PET Radiation X-ray RF g-ray Role Anat Anat/F/M Funct/Mole Resolution Very high High Relatively low Contrast Medium Medium Very high Radiation exposure High None Low WB scan time few min 10~20 min 10~20 min Taken from Dr. JS Lee s Slides 6

7 Anatomical vs. functional imaging Taken from Dr. K. Mueller s Slides 7

8 Six factors determining image quality: 1 Contrast The difference btwn image characteristics (e.g., shades of gray => i.e., intensity) of an object (or feature within an object) & surrounding objects or background 2 Resolution 3 Noise The ability of a medical imaging system to depict details Random fluctuations in image intensity that do not contribute to image quality 4 Artifacts Reducing object visibility by masking image features Image features that do not represent a valid object or characteristics of the patient 5 Distortion Obscuring important features or being falsely interpreted as abnormal findings Inaccurate impression of shape, size, position, & other geometric characteristics of features 6 Accuracy Should be corrected to improve the diagnostic quality Conformity to truth & clinical utility 8

9 Contrast Differences btwn the image intensity of an object & surrounding objects or background Inherent object contrast within the patient The goal of medical imaging system is to accurately portray or preserve the true object contrast Modulation of a periodic signal f(x, y) 0 Amplitude (or difference) m f = (f max f min )/2 (f max + f min )/2 = f max f min f max + f min Average (or background) An accurate way to quantify contrast for a periodic signal The contrast of the periodic signal relative to its average value The ratio of the amplitude (or difference) of f(x, y) to its average value (or background) 0 m f 1 for nonnegative values of f(x, y) Nonzero background intensity reduces image contrast (note that m f = 1 only when f min = 0) No contrast when f min = f max 9

10 Modulation transfer function (MTF) For a sinusoidal object f x, y = A + B sin(2πu 0 x) A, B = nonnegative constants (A B) f max = A + B, f min = A B Therefore, m f = B A See the examples when m f = 0, 0.2, 0.5, 1: As m f, contrast [much easier to distinguish differences in shades of gray in f x, y ] 10

11 How does an LSI imaging syst. with PSF h(x, y) affect the modulation of f(x, y)? Input: f x, y = A + B sin(2πu 0 x) = A + B 2j ej2πu0x e j2πu 0x Output: g x, y = AH(0,0) + B H(u 0, 0) sin(2πu 0 x) g max = AH(0,0) + B H(u 0, 0) g min = AH 0,0 B H(u 0, 0) m g = B H(u 0,0) AH(0,0) = m f H(u 0,0) H(0,0) Depending on the spatial frequency of input object u 0 Scaled version of m f w/ the scaling factor (= H(u 0, 0), magnitude spectrum) m g < m f if H(0,0) = 1 & H(u 0, 0) < 1 Less contrast of g(x, y) than the input f(x, y) because both f(x, y) & g(x, y) have the same average value [= (max + min) / 2 = A] 11

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13 Half output contrast for a sinusoidal object at freq. ~0.6 mm -1 Zero output contrast for any sinusoidal input w/ freq. larger than 0.8 mm -1 MTF The ratio of the output modulation to the input modulation as a function of spatial frequency MTF u = m g m f = H(u, 0) H(0,0) Frequency response of the system Directly obtained from the Fourier transform of the PSF of the system Characterizing contrast; characterizing blurring (or resolution) because it is related to the PSF Blurring reduces contrast Degradation of contrast as a function of spatial frequency 0 MTF u MTF 0 = 1 for every u We can think of loss of contrast as the result of the blurring action of a medical imaging system 13

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15 For a nonisotropic system; Orientation-dependent resolution (e.g., ultrasound imaging systems: range vs. lateral resolutions) MTF u, v = m g m f = H(u, v) H(0,0) 0 MTF u, v = H(u,v) H(0,0) MTF 0,0 = 1 for every u, v 15

16 Local contrast Target: an object of interest (e.g., a tumor in the liver) Background: other objects surrounding the target (e.g., the liver tissue) Obscuring our ability to see or detect the target Local contrast C = f t f b f b 16

17 Example Consider an image showing an organ w/ intensity I 0 & a tumor w/ intensity I t > I 0. What is the local contrast of the tumor? If we add a constant intensity I c > 0 to the image, what is the local contrast? Is the local contrast improved? 17

18 Resolution The ability of a medical imaging system to accurately depict two distinct events in space, time, or frequency as separate spatial, temporal, or spectral resolution, respectively The degree of smearing or blurring of a single event in space (e.g., a point), time, or frequency Can be described by the PSF (i.e., impulse response function, IRF) 18

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20 Line spread function (LSF) Alternative to the PSF Consider an LSI medical imaging system w/ isotropic PSF h(x, y) that is normalized to 1 Line impulse f x, y = δ l x, y = δ(x) since the system is isotropic g x, y = = h ξ, η f x ξ, y η dξdη h ξ, η δ x ξ dξ dη = h x, η dη lsf(x) Relationship btwn the LSF & the PSF lsf(x) is symmetric [i.e.,lsf x = lsf( x)] if the PSF h(x, y) is isotropic» lsf x dx = 1 because the PSF is normalized to 1 Relation btwn the LSF & the transfer function:» L u = F 1D lsf(x) = lsf x e j2πux dx =» MTF u = L(u) L(0) for every u h x, η e j2πux dxdη = H(u, 0) 20

21 Full width at half maximum (FWHM) The (full) width of the LSF (or the PSF) at one-half its max value (usually in units of mm) The min distance that two lines (or points) must be separated in space in order to appear as separate in the recorded image 21

22 Resolution & MTF Consider the output of a medical imaging system for the input of f x, y = B sin(2πux); The separation btwn two adjacent maxima (or minima) of f(x, y) is 1/u g x, y = MTF u H 0,0 B sin(2πux) = MTF u B sin(2πux) The separation btwn two adjacent maxima (or minima) of g(x, y) is 1/u as well The magnitude of g x, y = amplitude of f(x, y) MTF(u) The resolution of the system = 1/u c when g x, y = 0 for every u > u c because MTF(u) 0 for every u u c & MTF u = 0 for every u > u c MTF Can be used to compare two competing medical imaging systems in terms of their contrast & resolution If the MTFs are of a similar shape but have a different u c ; Better system w/ higher MTF value in terms of contrast & resolution 22

23 Example What is the resolution of this system? 23

24 Complicated if the MTF curves are of different shapes: Better low-freq. contrast, better for imaging coarse details Better high-freq. contrast, better for imaging fine details Contrast is a function of spatial freq. (i.e., MTF) frequency-by-frequency comparison e.g., better low-freq. contrast of SYSTEM1 & high-freq. contrast of SYSTEM2 Resolution is not frequency-dependent difficult to directly compare MTFs FWHM of the PSF or LSF is the most direct metric of resolution Further understanding of a system's resolution from its MTF usually comes from MTF values at higher u, and the cutoff freq. u c 24

25 Example Sometimes, the PSF, LSF, or MTF can be described by an analytical function. Such a function can arise by either fitting observed data or by making simplifying assumptions about its shape. Assume that the MTF of a medical imaging system is given by What is the FWHM of this system? MTF u = e πu2. 25

26 Subsystem cascade The recorded image g(x, y) can be modeled as the successive convolutions of the input object f(x, y) w/ the PSFs of the corresponding the subsystems because medical imaging systems are often modeled as a cascade of LSI subsystems For K subsystems; g x, y = h K x, y h 2 x, y h 1 x, y f(x, y) Using the FWHM; R = R R R K where R k = FWHM of kth subsystem Overall FWHM R is dominated by the largest (i.e., the poorest resolution) term Small improvement in R k does not often yield improvements in R 26

27 Example Consider a 1D medical imaging system w/ PSF h(x) composed of two subsystems w/ Gaussian PSFs of the form h 1 (x) = 1 2πσ 1 e x2 /2σ 1 2 and h2 (x) = 1 2πσ 2 e x2 /2σ 2 2. What is the FWHM of this system? 27

28 Using the MTF MTF u, v = MTF 1 (u, v)mtf 2 (u, v) MTF K (u, v) MTF of the overall system MTF k MTF(u, v) MTF k (u, v) for every u, v The overall quality of a medical imaging system, in terms of contrast & resolution, will be inferior to the quality of each subsystem 28

29 Spatial resolution & image contrast are tightly linked Spatial resolution can be thought of as the ability of an imaging system to preserve object contrast in the image Orientation-dependent resolution Nonisotropic system The range resolution is better than the lateral resolution in the ultrasound imaging Spatially-dependent resolution Linear but not shift-invariant systems e.g., ultrasound imaging systems, nuclear medicine systems 29

30 Resolution tool Resolution tool or bar phantom Line pairs per millimeter (lp/mm) 6 8 lp/mm for a projection radiography system 2 lp/mm for a CT scanner Temporal & spectral resolution Temporal resolution The ability to distinguish two events in time as being separate Spectral resolution The ability to distinguish two different frequency (or, equivalently, energies) The concepts of PSF, LSF, and FWHM can be equally applied to the temporal and spectral resolutions 30

31 Gopal and Samant, Med. Phys. 35:1, ,

32 Noise An unwanted characteristic of medical imaging systems Random fluctuation in an image Image quality as noise In projection radiography: Quanta or photons: discrete packets of energy arriving at the detector from the x-ray source Quantum mottle: random fluctuation due to the discrete nature of their arrival Textured or grainy appearance in an x-ray image In magnetic resonance imaging: RF pulses generated by nuclear spin systems are sensed by antennas connected to amplifiers Competing w/ signals being generated in the antenna from natural unpredictable (i.e., random) thermal vibrations 32

33 The source of noise in a medical imaging system depends on the physics & instrumentation of the particular modality Consider the noise as the numerical outcome of a random event or experiment Think of the noise as the deviation from a nominal value predicted from purely deterministic arguments e.g., Random nature radioactive emissions in nuclear medicine where gamma ray photons are emitted at random times in random directions 33

34 Random variables The numerical quantity associated w/ a random number or experiment Probability distribution function (PDF) P N η = Pr N η The probability that random variable N will take on a value less than or equal to η 0 P N η 1 P N = 0, P N = 1 P N η 1 P N η 2 for η 1 η 2 34

35 Continuous random variables N is a continuous random variable if P N η is a continuous function of η Probability density function (pdf) p N η = dp N η dη p N η 0 pn η dη = 1 η P N η = pn u du μ N = E N = ηpn η dη expected value or mean» Average value of the RV σ 2 N = Var N = E (N μ N ) 2 = (η μn ) 2 p N η dη variance σ N = 2 σ N standard deviation» Average variation of the values of the RV about its mean» The larger σ N, the more random the RV 35

36 Uniform random variable over the interval [a, b] pdf p N η = 1, b a for a η b 0, otherwise distribution function P N η = 0, for η < a η a, b a for a η b 1, for η > b expected value μ N = a+b variance σ N 2 = (b a)

37 Gaussian random variable over the interval [a, b] pdf p N η = 1 2πσ 2 e (η μ)2 2σ 2 distribution function P N η = erf η μ expected value μ N = μ variance σ 2 N = σ 2 σ where the error function erf x = 1 x 2π 0 e u 2 2 du Noise in medical imaging systems is the result of a summation of a large number of independent noise sources Central limit theorem of probability A random variable that is the sum of a large number of independent causes tends to be Gaussian Often natural to model noise in medical imaging system by means of a Gaussian random variable 37

38 Discrete random variables Specified by the probability mass function (PMF) Pr N = η i for i = 1, 2,, k Probability that random variable N will take on the particular value η i 0 Pr N = η i 1 for i = 1, 2,, k k i=1 Pr N = η i = 1 P N η = Pr N η = allηi η Pr N = η i k μ N = E N = i=1 η i Pr N = η i expected value or mean σ 2 N = Var N = E (N μ N ) 2 = k i=1 (η i μ N ) 2 Pr N = η i variance Poisson random variable Pr N = k = ak k! e a for k = 0,1, 2, a > 0 (real-valued parameter) μ N = a σ N 2 = a 38

39 Example In x-ray imaging, the Poisson random variable is used to model the number of photons that arrive at a detector in time t, which is a random variable referred to as a Poisson process and given the notation N(t). The PMF is given by Pr N(t) = k = (λt)k e λt. k! where is called the average arrival rate of the x-ray photons. What is the probability that there is no photon detected in time t? 39

40 Example For the Poisson process N(t), the time that the first photon arrives is a random variable, say T. What is the pdf p T t of random variable T? 40

41 Independent random variables Consider the collection of random variables N 1, N 2,, N m having the pdf s p 1 η, p 2 η,, p m η, respectively The sum of these random variables S is another random variable having another pdf, p S η : μ S = μ 1 + μ μ m When the random variables are independent; σ S 2 = σ σ σ m 2 p S η = p 1 η p 2 η p m η 41

42 Signal-to-Noise Ratio Let G a random variable describing the output of a medical imaging system & is composed of: 1 Signal f (deterministic or nonrandom) true value of G 2 Noise N random fluctuation or error component How close is an observed value of g of G to its true value f? Signal-to-noise ratio (SNR) The relative strength of signal f w.r.t. that of noise N Higher SNR g is a more accurate representation of f Lower SNR g is less accurate Signal = modulation or contrast in the image Blurring reduces contrast; thus SNR Noise = unwanted random fluctuation Noise reduces SNR 42

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44 Amplitude SNR SNR a = Amplitude(f) Amplitude(N) What is the amplitude SNR in an x-ray projection image in which the number of photons G counted per unit area (or fluence) follows the Poisson distribution? SNR a = μ σ = μ μ = μ intrinsic SNR of x rays The greater the average number of μ of photons, the larger the amplitude SNR or the smaller the relative amplitude of random fluctuations in G (i.e., SNR a 1 ) Higher x-ray exposure improves the quality of radiographic images (but, more risk of radiogenic cancer) 44

45 Power SNR SNR p = Power(f) Power(N) Power SNR of a system w/ PSF h(x, y) & noise variance σ N 2 (x, y) for the input f(x, y) SNR p = h x,y f(x,y) 2 dxdy σ N 2 with the white noise assumption White noise: no correlation btwn noise values in space μ N x, y = 0 (i.e., zero mean) & σ N x, y = σ N for every (x, y) Correlated noise w/ assumption that μ N & σ N 2 do not depend on (x, y) wide-sense stationary noise SNR p = h x, y f(x, y) 2 dxdy NPS(u, v) dudv NPS u, v = lim 1 E x 0,y 0 4x 0 y 0 x0» Noise power spectrum (NPS) x 0 y 0 y N x, y μn e j2π(ux+vy) dxdy

46 From the Parseval's theorem SNR p = H u, v 2 F u, v 2 dudv NPS(u, v) dudv = SNRp (u, v)nps(u, v) dudv NPS(u, v) dudv where frequency-dependent power SNR SNR p u, v = H u,v 2 F u,v 2 NPS u,v = MTF2 (u,v) NPS u,v H 0,0 2 F u, v 2 Quantifying, at a given freq., the relative strength of signal to that noise at the output of the LSI system Providing a relationship btwn contrast, resolution, noise, and image quality 46

47 Differential SNR SNR d = A(f t f b ) σ b (A) σ b (A) = the standard deviation of image intensity values from their mean over an area A of the background SNR d = CAf b σ b (A) C = f t f b local contrast f b Relating the differential SNR to contrast 47

48 Rose model SNR d = CAf b = CAλ b = C Aλ σ b (A) Aλ b = SNR Rose λ b = SNR 2 Rose b C 2 A λ b = mean number of background photons counted per unit area (= f b ) σ b A = Aλ b To maintain good image quality, high radiation dose is required when viewing small, low-contrast object Decibels (db) When the SNR is the ratio of amplitudes, such as with SNR a or SNR d : SNR (in db) = 20 log 10 SNR (ratio of amplitudes) When the SNR is the ratio of powers, such as with SNR p : SNR (in db) = 10 log 10 SNR (ratio of powers) 48

49 SNR Rose = 5 Image Courtesy of MJ Yaffe 49

50 Nonrandom Effects Artifacts Image features representing non-valid anatomical or functional objects Obscuring important targets Being falsely interpreted as valid image features Impairing correct detection & characterization of features of interest by adding clutter to images Due to a variety of reasons & at any step of imaging process Nonuniformities in x-ray detectors, x-ray source CT a. Motion artifacts (appeared as streak artifacts) b. Star artifacts by the presence of metallic materials (resulting in incomplete projections) c. Beam-hardening artifacts (appeared as broad dark bands or streaks) due to significant beam attenuation by certain materials d. Ring artifacts by detectors that go out of calibration 50

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53 Distortion Geometrical problems of a system resulting in inaccurate impression of the shape, size, and/or position of objects of interest Examples a. Size distortion due to magnification b. Shape distortion due to unequal magnification of the object being imaged or beam divergence Very difficult to determine & correct 53

54 Accuracy It should be noted that image quality ultimately must be judged in the context of a specific clinical application Clinical applications with medial images (i.e., clinical utility): 1 Diagnosis Is the disease present? 2 Prognosis How will the disease progress, & what is the expected outcome? 3 Treatment planning Which treatment will work best? 4 Treatment monitoring Is the treatment reversing the disease, & to what extent? Accuracy = conformity to truth (i.e., freedom from error) & clinical utility Quantitative accuracy Diagnostic accuracy 54

55 Quantitative accuracy Quantification of a given anatomic or functional feature w/i an image in the numerical values e.g., Anatomic quantity: tumor dimensions from a radiograph Functional quantity: glucose metabolic rate from a nuclear medicine image Error or difference from the true value Bias systematic, reproducible difference from the truth can be corrected through the use of a calibration standard Imprecision a random, measurement-to-measurement variation In practice, never error-free! 55

56 Diagnostic accuracy Define two parameters following Gaussian distributions in a clinical setting: 1 Sensitivity (= true-positive fraction) The fraction of patients with disease who the test* calls abnormal 2 Specificity (= true-negative fraction) The fraction of patients without disease who the test* calls normal *Test = the medical image 56

57 Making a 2 2 contingency table after evaluating images from a group of patients a b c d the number of diseased patients who the test calls abnormal the number of normal patients who the test calls abnormal the number of diseased patients who the test calls normal the number of normal patients who the test calls normal Then, we can calculate, sensitivity = a a + c specificity = d b + d 57

58 Diagnostic accuracy: the fraction of patients that are diagnosed correctly DA = a+d a+b+c+d To maximize DA, both sensitivity & specificity must be maximized Threshold t 0 Sensitivity but specificity as t 0 Sensitivity but specificity as t 0 t 0 should be chosen as a balance btwn sensitivity & specificity Dependent upon the prevalence or proportion of all patients who have disease Prevalence 1 Positive predictive value (PPV), PPV = a a+b The fraction of patients called abnormal who actually have the disease 2 Negative predictive value (NPV), NPV = d c+d The fraction of patients called normal who do not have the disease Prevalence (PR), PR = a+c a+b+c+d 58