Lernzettel: Medical Imaging Fundamentals

📋 Course Outline

1. Sampling model and aliasing
2. Nyquist theorem and anti-aliasing
3. Image quality measures
4. Contrast and modulation transfer function
5. Resolution and spread functions
6. Noise and random variables
7. Signal-to-noise ratio
8. Artifacts and distortions
9. Diagnostic accuracy metrics
10. ROC analysis

📖 1. Sampling model and aliasing

🔑 Key Concepts & Definitions

• Discrete sampled function : A discrete representation obtained by evaluating a continuous image at grid points spaced by sampling periods in each direction.
• Sampling periods : The spatial spacings 9x and 9y between neighboring sampled points along $x$ and $y$.
• Sampling frequencies : The quantities $1/\Delta x$ and $1/\Delta y$ that set how densely samples are taken along each axis.
• Aliasing : An under-sampling artefact where higher spatial frequency components appear as falsely lower frequencies in the sampled image.

📝 Essential Points

• In 2D rectangular sampling, the discrete image is $f_d(m,n)=f(m\Delta x,n\Delta y)$ for integers $m,n\ge 0$.
• Sampling too coarsely produces artefacts because the sampled representation cannot preserve the original frequency content.

💡 Memory Hook

Aliasing happens when “too few samples steal high frequencies and disguise them as low ones.”

📖 2. Nyquist theorem and anti-aliasing

🔑 Key Concepts & Definitions

• Nyquist-Shannon theorem : A sampling criterion stating when shifted replicas of the original spectrum do not overlap, preventing aliasing.
• Nyquist frequencies : The maximum sampling frequencies determined by half the original band-limits along each axis.
• Nyquist periods : The minimum sampling periods $\Delta x_{\max}=\frac{1}{2U}$ and $\Delta y_{\max}=\frac{1}{2V}$ that avoid overlap.
• Anti-aliasing filters : Low-pass operations applied to limit bandwidth before sampling so overlapping spectral replicas never occur.

📝 Essential Points

• If the original spectrum is limited to cut-offs $(U,V)$, non-overlap requires $\Delta x_{\max}=\frac{1}{2U}$ and $\Delta y_{\max}=\frac{1}{2V}$.
• When aliasing would occur, anti-aliasing reduces the bandwidth by low-pass filtering, but it increases blurring.
• For point-like sampling, the low-pass filter is applied before sampling (equivalently, before detection) with cut-off set by the sampling grid.

💡 Memory Hook

Nyquist: “Half the bandwidth” so spectrum copies don’t overlap.

📖 3. Image quality measures

🔑 Key Concepts & Definitions

• Physics-oriented image quality : An image-quality view that judges how faithfully the measured image reproduces the real signal.
• Medical-task oriented image quality : An image-quality view that judges whether the image enables clear discrimination between healthy and diseased states.
• Contrast : A measure of intensity difference between a target and a local background.
• Resolution : A measure of the system’s ability to distinguish close objects in space.
• Noise : Random fluctuations in the image arising from different physical origins.

📝 Essential Points

• The course links image-quality evaluation to contrast, resolution, noise, artefacts, and distortions, with diagnostic accuracy as a task-level metric.
• X-ray imaging is described as limited in contrast but high in resolution, while nuclear medicine has high contrast, low resolution, and high noise.
• Diagnostic accuracy is treated as a key measure separate from the physics quantities that affect image appearance.

💡 Memory Hook

Quality has two lenses: physics (contrast/resolution/noise) and task (discrimination/diagnostic accuracy).

📖 4. Contrast and modulation transfer function

🔑 Key Concepts & Definitions

• Local contrast : A contrast concept comparing a target intensity to a local background intensity.
• Modulation : An effective contrast measure for periodic signals based on the ratio of sine amplitude to offset.
• Modulation transfer function : A system frequency-response quantity that characterizes how modulation at spatial frequency $(u,v)$ is transmitted relative to zero frequency.
• Modulation at output : The amplitude of the sinusoidal component after the system’s impulse response acts on the input.

📝 Essential Points

• For a periodic signal with input $f(x)=A+B\sin(2\pi u_0 x)$, the modulation is $m_f=\frac{B}{A}$ given $A\ge B\ge 0$.
• The output modulation becomes $m_g=\frac{H(u_0,0)}{H(0,0)}\,\frac{B}{A}$, so $\mathrm{MTF}(u,v)=\frac{H(u,v)}{H(0,0)}$.
• For most medical systems, $0\le \mathrm{MTF}(u,v)\le 1$ with $\mathrm{MTF}(0,0)=1$, and poorer MTF reduces contrast at higher frequencies.

💡 Memory Hook

MTF = “contrast gain per frequency”: how much of $B/A$ survives at $(u,v)$.

📖 5. Resolution and spread functions

🔑 Key Concepts & Definitions

• Line spread function : The 1D spatial spread obtained as the system output to a line impulse input.
• Line impulse input : A test input modeled as $\delta_l(x,y)$ (a delta function along a chosen line) to probe spreading.
• Point spread function : The 2D impulse response that describes how a point is imaged, whose frequency content links to system resolution.

📝 Essential Points

• Using a line impulse input, the output profile is $g(x)=\int_{-\infty}^{\infty}h(x,\eta)\,d\eta$, which defines the line spread function LSF.
• If the PSF is normalized, the associated LSF is symmetric and its 1D Fourier transform is related to the 2D transfer function through $H(u,0)$.
• A practical resolution criterion is that, when the object separation is at least the LSF/PSF FWHM, the separated maxima are resolved.

💡 Memory Hook

FWHM: “how wide one point looks,” so if objects are farther apart than that width, they separate.

📖 6. Noise and random variables

🔑 Key Concepts & Definitions

• Probability distribution function : A CDF $P_N(\eta)=\Pr(N\le \eta)$ describing how probability accumulates up to a value $\eta$.
• Probability density function : A continuous random-variable density $p_N(\eta)$ where $\int_{-\infty}^{\infty} p_N(\eta)\,d\eta=1$.
• Mean and variance : Numerical summaries of a random variable: expected value and spread around the mean.

📝 Essential Points

• For continuous variables, the mean and variance are given by $\mu_N=E[N]=\int_{-\infty}^{\infty}\eta\,p_N(\eta)\,d\eta$ and $\sigma_N^2=\mathrm{Var}(N)=E[(N-\mu_N)^2]$.
• Uniform random variables on $[a,b]$ have $\mu_N=\frac{a+b}{2}$ and $\sigma_N^2=\frac{(b-a)^2}{12}$.
• Poissonian random variables model photon counting with PMF $\Pr(N=k)=\frac{a^k e^{-a}}{k!}$ for $k=0,1,2,\dots$ and satisfy $\mu_N=a$ and $\sigma_N^2=a$.
• For sums of independent variables, means add while variances add, giving $\mu_S=\sum\mu_i$ and $\sigma_S^2=\sum\sigma_i^2$.

💡 Memory Hook

Stats add for independent sums: mean adds, variance adds, so noise grows predictably.

📖 7. Signal-to-noise ratio

🔑 Key Concepts & Definitions

• Signal-to-noise ratio : A scalar image-quality quantity defined as a ratio of signal strength to noise variability using random-variable moments.
• Differential SNR : A local form of SNR computed for an area of interest using signal and background densities.
• Poissonian statistics : A counting model where both mean and variance scale together for the number of detected events.
• Decibel scale : A logarithmic unit conversion used to express SNR as $\mathrm{SNR}_{\mathrm{dB}}=20\log_{10}(\mathrm{SNR})$.

📝 Essential Points

• In the noise model with output random variable $G=f+N$, the course expresses $\mathrm{SNR}_a=\frac{\mu_G}{\sigma_G}=\frac{\mu}{\mu}=\mu$ for Poisson-based signal quality.
• For area of interest $A$, the differential SNR is $\mathrm{SNR}_a=\frac{A\,\mathrm{mpqlitude}(f)}{A\,\mathrm{mpqlitude}(N)}=\frac{A(f_t-f_b)}{\sigma_b(A)}\,\frac{C}{f_b}$ as written in the lecture.
• Increasing noise and/or decreasing resolution reduces SNR, and SNR in dB uses $\mathrm{SNR}_{\mathrm{dB}}=20\log_{10}(\mathrm{SNR})$.

💡 Memory Hook

SNR in dB is a log “contrast over clutter”: louder means larger ratio.

📖 8. Artifacts and distortions

🔑 Key Concepts & Definitions

• Artefacts : Image features produced by the imaging process that do not correspond to valid structural or functional objects.
• Motion artefacts : Artefacts caused by patient motion during acquisition.
• Beam-hardening artefacts : Artefacts caused by preferential absorption of low-energy photons, producing shadows and streak-like effects.
• Distortions : Systematic incorrect reproduction of an object’s shape, size, or position.
• Magnification distortion : A distortion in projection radiography caused by a diverging x-ray beam leading to position-dependent magnification.

📝 Essential Points

• Non-random artefacts can include motion-induced changes, black/white bands, dark spots, and local loss of resolution that can be misread as real structures.
• Beam-hardening artefacts create shadows beneath ribs, in the mediastinum, or in the skull due to preferential absorption of lower-energy photons.
• Star artefacts arise when objects with exceptionally high or low attenuation generate streaking artefacts.
• Size distortion and shape distortion both originate from diverging-beam geometry, changing how projections map to the original object geometry.

💡 Memory Hook

Artefacts are “wrong features,” distortions are “wrong geometry.”

📖 9. Diagnostic accuracy metrics

🔑 Key Concepts & Definitions

• Diagnostic accuracy : A measure of how good the diagnostic conclusion is when a medical task is formulated as classification from an imaging output.
• Sensitivity : The fraction of diseased patients correctly diagnosed positive.
• Specificity : The fraction of normal patients correctly diagnosed negative.
• Positive predictive value : The fraction of diagnosed-positive patients that truly have the disease.
• Negative predictive value : The fraction of diagnosed-negative patients that truly are normal.

📝 Essential Points

• With a threshold $t_{th}$, decisions become binary, producing false negatives $F_N$ and false positives $F_P$ in the contingency table.
• Sensitivity equals $\frac{TP}{TP+FN}$ and specificity equals $\frac{TN}{TN+FP}$.
• Diagnostic accuracy is $\frac{TP+TN}{TP+TN+FP+FN}$.
• Predictive values depend on prevalence $PV$ via $PPV=\frac{TP}{TP+FP}$ and $NPV=\frac{TN}{TN+FN}$, so changing prevalence changes PPV/NPV.

💡 Memory Hook

Sensitivity/specificity are disease-relative; PPV/NPV are patient-prevalence-relative.

📖 10. ROC analysis

🔑 Key Concepts & Definitions

• Receiver operating characteristic curve : A curve that plots true positive fraction against false positive fraction as the decision criterion varies.
• True positive fraction : The sensitivity-like quantity on the ROC axes, equal to $\frac{TP}{TP+FN}$ in the course notation.
• False positive fraction : The ROC x-axis quantity, equal to $\frac{FP}{TN+FP}$ in terms of the contingency table.
• Area under the curve : A single-number ROC summary quantifying overall classification performance.
• Index of detectability : A ROC-derived parameter $d'$ that increases with detectability and separates performance levels.

📝 Essential Points

• The ROC curve is generated by varying the decision criterion from lenient to strict, yielding a family of $(TPF, FPF)$ pairs.
• The specific operating point (SOP) is defined by a specified $FPF$, and the corresponding $TPF$ is used to quantify detectability at that bias.
• The area under the ROC curve (AUC) summarizes overall performance, and the index of detectability $d'$ increases for better detection.

💡 Memory Hook

ROC sweeps bias: move threshold, trace $TPF$ vs $FPF$, and read performance from AUC or $d'$.

⚠️ Common Pitfalls & Confusions

1. Confusing sampling frequency with sampling period: the course uses $\Delta x$ and $\Delta y$ directly in $f_d(m,n)=f(m\Delta x,n\Delta y)$ and frequencies are their reciprocals.
2. Thinking anti-aliasing removes all blur: it prevents aliasing by low-pass filtering but increases blurring because bandwidth is reduced.
3. Using FWHM as an absolute “best resolution” without context: the criterion depends on comparing separation to the LSF/PSF FWHM or on MTF-based cut-off.
4. Mixing LSF and PSF: LSF is a 1D spread from a line impulse, while PSF is the 2D impulse response linked via transfer functions.
5. Forgetting that PPV and NPV depend on prevalence: unlike sensitivity and specificity, prevalence changes predictive values strongly.
6. Dropping the distinction between artefacts and distortions: artefacts create false features, distortions shift/reshape/scale the reproduced geometry.
7. Mixing ROC axes: ROC uses false positive fraction vs true positive fraction, not sensitivity vs prevalence or accuracy alone.

✅ Exam Checklist

1. Write the 2D discretization formula for rectangular sampling: $f_d(m,n)=f(m\Delta x,n\Delta y)$.
2. Explain what aliasing is and why it appears when sampling is too coarse.
3. State the Nyquist-Shannon criterion in terms of non-overlapping shifted spectra and give $\Delta x_{max}=1/(2U)$ and $\Delta y_{max}=1/(2V)$.
4. Describe how anti-aliasing avoids aliasing by bandwidth reduction and what trade-off it causes for blur.
5. List the main physics-oriented quality quantities: contrast, resolution, noise, artefacts, and distortions.
6. Define modulation for a periodic input and compute it as $m_f=B/A$ for $A\ge B\ge 0$.
7. State the MTF definition as frequency response normalized by $H(0,0)$ and interpret $\mathrm{MTF}(0,0)=1$ and $0\le \mathrm{MTF}\le 1$.
8. Use the FWHM resolution criterion: resolution limit occurs when separation is at least the LSF/PSF FWHM so maxima are resolved.
9. Convert from a line impulse input to the output profile integral defining the LSF and relate LSF Fourier transform to $H(u,0)$.
10. Write the continuous random-variable PDF normalization condition $\int_{-\infty}^{\infty}p_N(\eta)d\eta=1$ and give the mean/variance formulas used.
11. Give the uniform distribution mean and variance ($\mu_N=(a+b)/2$, $\sigma_N^2=(b-a)^2/12$) and the Poisson PMF with $\mu_N=a$ and $\sigma_N^2=a$.
12. State how mean and variance combine for sums of independent random variables.
13. Compute SNR expressions as given for the course (including the Poisson-based form and the dB conversion $20\log_{10}(\mathrm{SNR})$).
14. Distinguish non-random artefacts from distortions and name at least four artefacts/distortion mechanisms from the lecture.