Medical Imaging: 20-Bio-B4 Robotics, December 2016, Q4

Question 4: Medical Imaging (20 marks)

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(a) What Makes Medical Imaging Distinct

Medical images are usually indirect measurements of an internal physical/physiological quantity (X-ray attenuation, proton relaxation time, acoustic reflectivity) reconstructed through a specific acquisition physics — the Radon transform/CT reconstruction in part (b) is the clearest example — rather than direct photographs of a visible surface. The consequences of an algorithmic error are also categorically different: a missed tumour or a mis-registered pre/post-treatment scan has direct clinical/patient-safety consequences, which drives far more conservative validation, quantitative accuracy requirements, and regulatory scrutiny than a consumer photo application would ever need.

(b) The Radon Transform

The Radon transform at angle $\theta$: a bank of parallel line integrals through $f(x,y)$ produces one 1-D projection $p(t,\theta)$; sweeping $\theta$ builds the sinogram.

The Radon transform maps a 2D function $f(x,y)$ to the set of its line integrals along every possible line, parameterized by angle $\theta$ and perpendicular offset $t$:

$$p(t,\theta)=\int\!\!\int f(x,y)\,\delta(x\cos\theta+y\sin\theta-t)\,dx\,dy$$

It is significant to medical imaging because it is exactly the physical measurement an X-ray CT scanner makes: each detector reading is the line integral of tissue attenuation along the ray path, and a full CT acquisition collects $p(t,\theta)$ over many angles (the resulting 2D array is the sinogram). Reconstructing the cross-sectional image $f(x,y)$ from the measured projections is an inverse Radon transform problem, solved in practice by filtered back-projection or iterative reconstruction — making the Radon transform the mathematical foundation of tomographic image formation.

(c) Anomaly Detection, Segmentation, and Denoising in Medical Imaging

i. Anomaly detection. Automatically flagging regions that deviate from expected normal anatomy/tissue appearance (a tumour, lesion, or micro-calcification) directly supports diagnosis and screening, particularly where the target is small, subtle, or easily missed by a time-pressured human reader. ii. Segmentation. Delineating organ, tumour, or tissue boundaries is a prerequisite for quantitative measurement (tumour volume/growth over time, organ volume, radiotherapy treatment planning) and for surgical planning/navigation, where the segmented structure directly drives a clinical decision or a physical intervention. iii. Denoising. Medical images are frequently acquired under a hard physical constraint that trades signal for safety or time (limiting radiation dose in CT/X-ray, limiting acquisition time in MRI), which pushes the raw image toward a low-SNR regime that denoising must recover from without blurring away the fine diagnostic detail (a small lesion, a thin vessel) that the whole acquisition exists to reveal.

(d) Speckle Noise

Speckle is a granular, multiplicative interference pattern that arises whenever coherent radiation (ultrasound, laser, synthetic-aperture radar) reflects from a surface or medium rough at the scale of the wavelength: many sub-resolution scatterers within one resolution cell return waves that interfere constructively or destructively depending on their random relative phases, producing a signal-dependent granular texture rather than additive noise. It is characteristic of ultrasound B-mode images, laser speckle, and SAR imagery, and is best modelled as $g=f\cdot n$ (multiplicative), not the additive model $g=f+n$ appropriate for incoherent-imaging sensor noise — a distinction that matters directly for which denoising filter is appropriate (a filter designed for additive Gaussian noise is not optimal for multiplicative speckle).

(e) Registration: Rigid, Affine, and Deformable

Registration means finding the spatial transformation that aligns one image (the "moving" image) onto another (the "fixed" image) so that corresponding anatomical points coincide; it is used whenever images from different times (tracking disease progression, pre/post-treatment comparison), different modalities (fusing a CT's bone/anatomy detail with a PET's functional signal), or different subjects (mapping a patient onto a standard anatomical atlas) must be directly compared pixel-for-pixel. Rigid registration allows only translation and rotation (6 degrees of freedom in 3D), appropriate when the imaged structure genuinely does not deform between acquisitions (e.g. the skull in two brain scans of the same patient). Affine registration adds scaling and shearing to rigid's translation/rotation, still a single global linear transformation, useful when overall size/aspect differs slightly (different scanner calibration) but the internal anatomy is still rigid. Deformable (non-rigid) registration allows a spatially-varying transformation (a dense displacement field, often regularized/smoothed), necessary whenever the anatomy itself genuinely changes shape between acquisitions — breathing/cardiac motion, tumour growth/shrinkage, or mapping between two different patients' inherently different anatomy onto a common atlas.

Registration type	Degrees of freedom	Typical use
Rigid	Translation + rotation	Same rigid structure (e.g. skull), same subject
Affine	+ scaling + shearing	Different scanner/calibration, still rigid anatomy
Deformable	Spatially-varying displacement field	Genuine anatomical change (motion, growth) or inter-subject/atlas mapping

Concept & Theory: Medical Image Formation and Analysis

Topic: the physical acquisition principles (tomographic reconstruction, speckle) and the core analysis tasks (anomaly detection, segmentation, denoising, registration) that recur across imaging modalities.
Key relations: Radon transform $p(t,\theta)=\iint f(x,y)\delta(x\cos\theta+y\sin\theta-t)\,dx\,dy$; speckle model $g=f\cdot n$ (multiplicative); registration degrees of freedom increase rigid $\subset$ affine $\subset$ deformable.
Why this works: CT/MRI/ultrasound each measure a different physical quantity through indirect acquisition physics, so the reconstruction and analysis tasks (anomaly detection, segmentation, denoising, registration) are all, at root, inference problems that must respect the acquisition's specific noise model and geometry rather than treating the image as an arbitrary photograph.
Common pitfall: applying an additive-noise denoiser to speckle (which is multiplicative), and choosing rigid registration for anatomy that genuinely deforms between scans (motion/growth), which silently leaves residual misalignment a rigid transform cannot correct.
Source: Gonzalez & Woods, Digital Image Processing, ch. 2 (image sensing/acquisition) and ch. 5 (restoration, noise models); standard medical-image-analysis references for CT reconstruction, segmentation, and registration.

Question 4 of 6: Medical Imaging