C20 · Recovering information
Shape, voting and tomography
Line-based sums have different meanings: points vote for geometric hypotheses, while tomography measurements sum a quantity along rays.
Theory
One point permits many lines
Many lines pass through one point. In ρ,θ space it votes along . Several collinear points produce a shared peak. Here image coordinates are centered and ρ is rounded to a pixel.
Angular resolution affects precision and separation of nearby lines. Circles and ellipses need other parameters and more costly voting. A strong peak can arise by accidental alignment, so support and task constraints still matter.
Points vote for a line
Select a white point on the left: its vote is marked in each row on the right. Then select a bright voting peak to draw its corresponding line on the left. Read angle and distance in the inspector without converting row indices.
Exact values are in the point inspector
Grayscale 0…1: black to white.
Exact values are in the point inspector
Grayscale 0…16: black to white.
Inspect the selected point
What the plot shows
A peak appears where many points agree on one line. θ is the normal angle, not the direction along the line.
Parameters and details
Each white point votes at 24 angles for rounded ρ. θ is the line normal angle; a voting peak is a line hypothesis, not a proven object boundary.
Select an image point. Arrows move it; Home and End reach row edges. Viewing does not change the calculation.
Changing the example requires JavaScript and Canvas. Computed images remain available above.
A sinogram stores measurements, not votes
In this discrete Radon example every phantom pixel contributes its intensity to the nearest detector at each angle. A sinogram row is one angle's projection profile. A point traces a sinusoidal path. Values are computed from the displayed phantom, not drawn independently.
Unfiltered backprojection spreads and sums projections along lines, producing blur. Filtered backprojection corrects spectral weighting by filtering projections; ART/SART iteratively fit measurements. Few angles or wrong geometry cause artifacts even with exact measurements.
Measure sums, then spread them back
Select a sinogram cell on the right: the left outlines every pixel contributing to that measurement. Trace one ray first, then select backprojection and compare 2 with 24 angles.
Exact values are in the point inspector
Grayscale 0…1: black to white.
Exact values are in the point inspector
Grayscale 0…16: black to white.
Inspect the selected point
What the plot shows
More angles reduce directional artifacts, but without a frequency filter the backprojection stays blurred.
Parameters and details
Each pixel adds intensity to its nearest detector bin; backprojection spreads the sum along a ray. /16 is a fixed display scale, not inversion. Even many angles leave blur: a filter or iterative solver is needed.
Select an image point. Arrows move it; Home and End reach row edges. Viewing does not change the calculation.
Changing the example requires JavaScript and Canvas. Computed images remain available above.
Shape measurements depend on the grid
Reference and further methods
Binary area is foreground-cell count times pixel area; the centroid is a suitably weighted coordinate mean. Perimeter depends on contour tracing and diagonal handling. Central moments describe distribution around the center; normalized invariants reduce translation, scale or rotation dependence under stated models.
Chain codes describe contour step sequences. Curvature needs a smoothing scale; Fourier descriptors expand contour shape into frequencies. A stair-step pixel boundary does not become an exact continuous object by renaming its metric. This phantom is educational, not a medical diagnostic tool.
Practice
Test the idea under another condition. Follow the task above the images; point inspection helps explain the result. This exploration is not scored.
Why two directions are not enough
Only two measurement directions are available. Select one detector and trace its strip on the phantom. Inspect backprojection, then add directions without changing the phantom.
Exact values are in the point inspector
Grayscale 0…1: black to white.
Exact values are in the point inspector
Grayscale 0…16: black to white.
Inspect the selected point
What the plot shows
One measurement cannot locate material along its strip. Other directions add constraints; simply spreading sums back still leaves blur.
Parameters and details
Each pixel adds intensity to its nearest detector bin; backprojection spreads the sum along a ray. /16 is a fixed display scale, not inversion. Even many angles leave blur: a filter or iterative solver is needed.
Select an image point. Arrows move it; Home and End reach row edges. Viewing does not change the calculation.
Changing the example requires JavaScript and Canvas. Computed images remain available above.
Games
Test your explanation
Question 1 of 5
How do Hough and Radon differ here?
Exact values are in the point inspector
Grayscale 0…1: black to white.
Exact values are in the point inspector
Grayscale 0…1: black to white.
Exact values are in the point inspector
Grayscale 0…16: black to white.
Exact values are in the point inspector
Grayscale 0…1: black to white.
Exact values are in the point inspector
Grayscale 0…1: black to white.
Exact values are in the point inspector
Grayscale 0…16: black to white.
Choose one answer. It is checked immediately.
Limitations and references
Limits of this example
Computed: 24-angle Hough voting, nearest-bin phantom projections and unfiltered backprojection. No clinical CT or FBP/ART/SART implementation is claimed.