M04 · Mathematics alongside
Probability and statistical models
Repeated measurement of one region explains noise, averages, uncertainty and the limits of averaging.
Mathematical support for the textbook chapters: start with a small calculation, then an interactive example and precise definitions.
Work through a small example
Three equally likely measurements are 2, 4 and 6. How spread out are they around the mean?
- Mean
- Mean squared deviation
This is the variance of a finite distribution with three equal probabilities. Estimating an unknown variance from a sample is a different question: an unbiased estimator divides by n−1.
The observation is random, not the formula
A noise model describes a distribution of possible measurements. With and , , but one frame need not equal s. A Gaussian model specifies mean and variance; a Poisson model describes a nonnegative event count with equal mean and variance before gain and other noise.
For N independent identically distributed measurements, the mean's variance decreases N-fold. Correlation adds covariance terms: identical fixed-pattern noise does not disappear by repeated averaging. Independence and zero bias are assumptions to test, not properties of all noise.
Increase the number of observations of one scene. Different noise realizations are averaged, not copies of one corrupted frame. At a selected pixel compare the source, one observation and the mean: averaging usually helps, but a particular value need not improve at every increase in N.
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.
Conditional probability changes after observation
Definitions and details
Bayes connects prior p(x), measurement likelihood p(y|x) and posterior p(x|y). A likelihood is a function of x for fixed observed y, not itself a distribution over x. MAP selects a posterior maximum but does not describe all uncertainty.
For densities, the normalizing denominator must be positive and conditional densities and units compatible. Discrete class probabilities use a sum; continuous intensity uses an integral. Softmax normalizes numbers but does not prove real-data calibration.
Pixels can depend on one another
Definitions and details
A random field is a spatially indexed collection of random variables. A Markov assumption limits conditional dependence to a neighborhood; it does not assert unconditional independence of distant pixels. A Gibbs form p(x)∝exp(−E(x)) expresses preferred configurations through energy when normalization exists.
A probabilistic model may prefer smooth regions while allowing edges. Fine regular texture can also matter: a model treating it as noise erases real signal. A plausible prior is part of a method's assumptions, not a property of all images.
Bias and variability are different errors
Definitions and details
For estimator θ̂, mean squared error decomposes into variance plus squared bias when second moments exist. Strong smoothing often reduces variability but shifts an edge or erases detail. Output smoothness alone cannot select a method.
Reusing fixed noise realizations makes parameter comparisons fairer. Claims about average quality need varied realizations and independent examples, not one favorable frame.