From shadows to slices

Before diving in, it helps to see the whole route. A clinical CT scanner, a dental cone-beam (CBCT) unit and a laboratory microCT look very different, but they all run the same four stages. Hardware turns photons into numbers. Preprocessing turns those numbers into line integrals, sorted into a sinogram. Reconstruction turns the sinogram into a map of how strongly each point attenuates X-rays. Packaging rescales that map and writes it into DICOM.

Notice where the word “projection” lives. The physical scanner projects the object into shadows: that produces the sinogram. The computer backprojects those shadows into an image: that produces the slices. So the DICOM volume is not a projection of the sinogram; it is the result of undoing the projection. Chapters 3 and 4 make that precise.

Picture a thin pencil of X-ray photons crossing the body. Each photon travels in a straight line until it is either absorbed or scattered away, and each centimetre of material has a fixed probability of stopping it. That probability depends on what the material is (its density and atomic number) and on the photon energy. Bone stops more photons per centimetre than soft tissue, and metal stops far more than bone.

“A fixed fraction lost per centimetre” is the recipe for exponential decay. If 17% of photons are stopped in the first centimetre of tissue, 17% of the remaining ones are stopped in the next, and so on. Run the beam below and watch the photons thin out as they pass each slab.

Two observations matter for everything that follows. First, the slabs multiply: the fraction surviving the whole path is the product of the fractions surviving each slab. Second, the detector cannot tell you where along the ray the absorption happened. Swap the slabs’ order and the reading is identical. One ray gives one number, and that number mixes everything along the path.

A product is awkward to work with, but its logarithm is a sum. That is why every CT system takes \(-\ln(I/I_0)\) of each reading: it turns “fraction of photons that survived” into “total attenuation accumulated along the ray”, which is a plain sum over the path. Reconstruction algorithms are designed to unscramble sums, not products.

Now line up many parallel rays side by side, all hitting a row of detector elements. One exposure gives a whole projection: a 1D profile of total attenuation, like the shadow of the slice on a wall. Then the source and detector rotate a little and take another projection, and another, until they have gone half or all the way around.

Stack those profiles as rows of an image, one row per angle, and you get the sinogram. It is literally the raw data of CT. It looks nothing like the patient, and that is expected: the horizontal axis is “position on the detector” and the vertical axis is “angle of the gantry”.

Real scanners use a fan or a cone of rays from a single source rather than parallel rays. Chapter 5 shows that this only changes the bookkeeping, so we start with the parallel case where the geometry is cleanest.

Here is the origin of the name. Follow a single point while the scanner turns. When the rays are perpendicular to the line joining the point to the rotation centre, its shadow falls far to one side of the detector; half a turn later it falls on the other side; in between it sweeps across. Its shadow’s position traces a sine wave. The amplitude of that wave is the point’s distance from the centre, and its phase is the point’s angular position. A point exactly at the centre draws a straight vertical line.

A real slice is a continuum of points, each drawing its own sine wave with a brightness proportional to its attenuation. The sinogram is the sum of all those waves stacked on top of each other. Reconstruction asks the reverse question: which set of sine waves, with which brightness, adds up to exactly this picture?

Take one detector reading. You know the total attenuation along its ray, but not where along the ray it came from. The most neutral guess is to spread it evenly along the whole line. Do that for every detector element of one projection and you get a smear: the shadow, stretched back across the image in the direction the rays came from.

Now do the same for every angle and add the smears up. Where the object really is, smears from all directions overlap and pile up. Elsewhere they mostly scatter. The object emerges. That is backprojection.

Try it below. Start with the filter set to “None”, one angle, then increase the number of angles.

With “None”, even 180 angles give a blurry, washed-out slice: every point is surrounded by a halo that fades with distance. The reason is a sampling imbalance. Every smear passes through the centre of whatever it hits, so each point collects some signal from all its neighbours, and the closer a neighbour, the more of its smears overlap. Fine detail (high spatial frequencies) is under-represented; coarse blobs (low frequencies) are over-represented.

The fix is to sharpen each projection before smearing it. The filter keeps every bump but adds small negative dips on either side of it (look at the blue curve). When the filtered projections are smeared back, those negative dips land exactly where the halos of neighbouring smears would be, and cancel them. The result is filtered backprojection (FBP), the workhorse of CT for fifty years.

Different filters trade sharpness against noise. The pure ramp gives the sharpest edges but amplifies noise the most; Shepp-Logan and Hann roll off the highest frequencies to produce smoother images. In a clinical scanner these choices are called reconstruction kernels, with names like “soft”, “standard”, “bone” or “lung”.

The filter is not a heuristic. It comes out of one striking fact, best seen as a picture. Take one projection and compute its 1D Fourier transform. The result is exactly one line through the centre of the 2D Fourier transform of the slice, at the same angle. So each angle hands you one spoke of the slice’s 2D spectrum, and all angles together fill the spectrum like the spokes of a wheel.

An X-ray tube is close to a point source: photons leave a tiny focal spot and spread out. In a clinical scanner the beam is collimated into a thin fan that hits a curved strip of detector rows. In CBCT and microCT, the beam is a full cone that hits a flat panel detector, covering a whole volume at once. A fan-beam ray at angle γ inside a fan whose centre points at angle β is the same line as a parallel-beam ray at angle β + γ. So fan data can be re-sorted (“rebinned”) into a parallel sinogram, and everything from chapter 4 applies.

In a cone beam, an object close to the source casts a large shadow and one close to the detector casts a shadow of its own size. MicroCT exploits this as a magnifying glass. Slide the sample below and watch the trade-off: more magnification means smaller voxels, but also a smaller field of view, and a larger blur from the finite focal spot.

The raw output of FBP is \(\mu\), which depends on the tube voltage, the filtration and the detector. To make numbers comparable between scanners, Godfrey Hounsfield pinned the scale to two materials everyone has: water reads 0 and air reads −1000. Every other tissue lands on that ruler according to how much more or less it attenuates than water. Fat sits slightly below zero, muscle and blood slightly above, bone in the hundreds to over a thousand, and metal goes off the chart.

A monitor shows about 256 gray levels, but the ruler spans several thousand units. So viewers apply a window: they choose a range (centre and width), stretch it across the gray levels and clip everything outside to black or white. The window changes only what you see, never the stored numbers. Hover over the slice to read the values at each pixel, and try the narrow window to find the faint lesion near the spine.

FBP assumes four things: every ray is a perfect line integral, every detector element is calibrated, there is no noise, and the angles are dense enough. Below, each switch breaks one assumption on the simulated scanner, and the reconstruction pipeline from chapter 4 runs on the damaged data. The profile plot follows one row of the image (click the reconstruction to choose another row).

Two artifacts cannot be simulated in a single static slice but matter just as much for segmentation:

• Partial volume. A voxel straddling bone and soft tissue gets an intermediate value. Thin structures (trabeculae, cortical shells, small vessels) look less dense than they are and can vanish under a threshold, while thick slices blend neighbouring anatomy. This is a property of every voxel grid, not a fault.
• Motion. If the object moves during the rotation, projections disagree with each other, producing streaks, doubled edges and blur. Breathing and heartbeat in clinical CT, patient movement in long CBCT scans, and sample drying or shifting in long microCT scans.

DICOM organises data in a four-level hierarchy: a patient has studies (one visit), a study has series (one reconstruction: one kernel, one slice thickness, or one scout image), and a series has instances. In classic CT, each instance is one file holding one slice. Newer “enhanced” objects, common in CBCT, put every slice of the series in a single multi-frame file.

Inside a file, everything is a data element: a tag written as two hexadecimal numbers (group, element), a two-letter type code (the VR, e.g. DS for decimal string, US for unsigned short), a length, and the value. The pixels are just one more element, the last one, with tag (7FE0,0010).

Each slice is a flat sheet floating in a patient-attached coordinate system called LPS: x increases towards the patient’s left, y towards the back (posterior), z towards the head (superior). The header tells you where the sheet’s first pixel sits and which way its rows and columns point. Everything else follows.