PE2-2.3 Stochastic Modelling & Compression
You can build AR-1 and Markov chain models and solve the Yule-Walker equations, and apply correlation, predictive coding, transform domain compression and multiresolution coding to images.
The Yule-Walker equations are the practical payoff of AR modelling: measured autocorrelations in, model coefficients out, by solving a linear system. Predictive coding is the idea underneath most compression — send the error of a prediction rather than the value, because the error is smaller and less correlated. Transform and multiresolution coding are where JPEG and its successors come from, and seeing them derived rather than described makes image compression stop being a black box.
Work through these
Stochastic models; the AR-1 process
The simplest model where the present depends on its own past, simulated rather than derived. Seeing the sample paths makes the parameter's meaning obvious.
Yule-Walker equations
The equations that recover the model parameters from the measured correlation. It is the practical estimation method and it is a linear system, which is why this model dominates.
Markov chains
Processes that move between states with fixed probabilities. They reappear in the speech and coding material and are worth solid attention.
Analog to digital conversion
Turning a continuous signal into numbers, revisited here as a source coding problem rather than a circuit one. The quantisation error becomes a distortion measure.
K means
Grouping data by nearness, used here to design a quantiser. It is the same algorithm as in machine learning, which is a connection worth noticing.
Correlation and predictive coding
Using the correlation from earlier to predict the next sample and send only the difference. It is the whole idea behind the compression that follows.
Image compression
Applying the ideas above to images, where the correlation is two-dimensional. It is the first real application of the course's machinery.
Transform domain compression
Compressing in a transformed domain instead, which concentrates the energy into fewer coefficients. This is what actual image standards do.
Multi resolution coding
Coding at several scales at once, which allows progressive transmission. It closes the compression sequence and connects to wavelet ideas.
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Links last checked 30 Aug 2026.
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