C09 · Change and measure
Noise and repeated measurements
Error mechanism matters as much as strength. Similar grain does not mean the same problem.
Background if needed
Theory
Five different causes
Gaussian noise adds to signal; impulses replace samples with outliers. Poisson event counts have equal mean and variance; more photons increase absolute spread while reducing relative spread. Multiplicative noise scales signal.
Regular stripes are structured interference, not independent errors. Fixed Seed keeps one input for comparison. Strength below is model-specific, not a universal instrument unit.
Five mechanisms of error
Keep Seed and compare models in dark and bright regions. Fixed absolute noise and signal-dependent noise differ.
Exact values are in the point inspector
Grayscale -0.5…1.5: black to white.
- Signal + realized disturbance
Exact values are in the point inspector
Grayscale -0.5…1.5: black to white.
Inspect the selected point
What the plot shows
Outliers, random grain and regular stripes require different assumptions. Strength has meaning within its model.
- Mean squared error
- 0.0274
Parameters and details
Seed fixes the noise; increasing the frame count preserves earlier frames. Values are not clipped before display. Gaussian noise adds, impulses replace samples, Poisson noise counts photons, multiplicative noise scales the signal. Periodic interference is identical in every frame and does not average away.
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.
Repeat one point instead of blurring neighbours
Averaging N independent zero-mean errors with equal variance reduces variance by N and standard deviation by about . This is a distribution property, not a per-point guarantee.
Spatial smoothing replaces a point with neighbours and may erase detail. Registered frames repeatedly measure the same point. Motion, shared bias and identical stripes violate the simple averaging assumptions.
Average repeated frames
Compare 1, 4 and 16 frames with one Seed. Then choose periodic stripes: why do they remain?
Exact values are in the point inspector
Grayscale -0.5…1.5: black to white.
- The same point in successive frames
- Divide by the number of measurements
Exact values are in the point inspector
Grayscale -0.5…1.5: black to white.
Inspect the selected point
What the plot shows
The curve measures whole-image error after each added frame. Independent noise usually decreases overall, not necessarily at every step. Identical stripes remain.
- Observed error
- Current: 4; 0.0072
Independent zero-mean errors partly cancel. Shared offsets and identical stripes remain.
- Mean squared error
- 0.0072
Parameters and details
Seed fixes the noise; increasing the frame count preserves earlier frames. Values are not clipped before display. Gaussian noise adds, impulses replace samples, Poisson noise counts photons, multiplicative noise scales the signal. Periodic interference is identical in every frame and does not average away.
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.
Smoothness is not accuracy
Mean error is bias; spread around the mean is variance. A smooth image can displace an edge. MSE here uses a known clean reference; ordinary photos often lack one. Keep coordinate, Seed and scale fixed to avoid confounding causes.
The MSE graph subtracts the clean value from every result, squares each difference and averages across the image. Errors +0.1 and −0.1 therefore do not cancel: both contribute 0.01. Select one point in the frame experiment to inspect its actual observations; the graph evaluates the whole scene. These two views answer different questions.
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 do more frames not help?
These frames share identical stripes. Increase the count from 1 to 16: error does not decrease. Switch to Gaussian noise without changing strength: independent errors now partly cancel. Use the graph to explain why more frames are not a universal repair.
Exact values are in the point inspector
Grayscale -0.5…1.5: black to white.
- The same point in successive frames
- Divide by the number of measurements
Exact values are in the point inspector
Grayscale -0.5…1.5: black to white.
Inspect the selected point
What the plot shows
The curve measures whole-image error after each added frame. Independent noise usually decreases overall, not necessarily at every step. Identical stripes remain.
- Observed error
- Current: 1; 0.0128
Independent zero-mean errors partly cancel. Shared offsets and identical stripes remain.
- Mean squared error
- 0.0128
Parameters and details
Seed fixes the noise; increasing the frame count preserves earlier frames. Values are not clipped before display. Gaussian noise adds, impulses replace samples, Poisson noise counts photons, multiplicative noise scales the signal. Periodic interference is identical in every frame and does not average away.
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
Why can averaging aligned frames preserve a thin line better than averaging neighbours?
Exact values are in the point inspector
Grayscale -0.5…1.5: black to white.
Exact values are in the point inspector
Grayscale -0.5…1.5: black to white.
Exact values are in the point inspector
Grayscale -0.5…1.5: black to white.
Exact values are in the point inspector
Grayscale -0.5…1.5: black to white.
Choose one answer. It is checked immediately.
Limitations and references
Limits of this example
Five explicit models, up to 16 small frames. Illustrative photons and reproducible non-cryptographic Seed; no registration of moving photos or promise of improving every pixel.