14. Separate averages and differences
All examples · Glossary: Haar pair · Papers, proofs, Lean, and code
Explore the 16-sample interactive Haar lab for four levels of decomposition, selectable details, and exact reconstruction errors.
Take the real signal \((2,6,10,14)\). For each neighboring pair \((u,v)\), define
\[a=(u+v)/2,\qquad d=(u-v)/2.
\]
We use the averaging normalization throughout this example.
Analyze the first level
For \((2,6)\), the average is \(4\) and the detail is \(-2\). For \((10,14)\), they are \(12\) and \(-2\). The averaging transform and its inverse are
\[H=\frac12\begin{pmatrix}1&1\\1&-1\end{pmatrix},\qquad
H^{-1}=\begin{pmatrix}1&1\\1&-1\end{pmatrix}.
\]
Multiplying gives \(H^{-1}H=HH^{-1}=I\). The inverse rule is \(u=a+d\) and \(v=a-d\). It recovers \((4-2,4+2)=(2,6)\) and \((12-2,12+2)=(10,14)\).
Analyze the averages once more
The pair \((4,12)\) has average \(8\) and detail \(-4\). Store coefficients in the declared order
\[(a_{\mathrm{top}},d_{\mathrm{top}},d_{\mathrm{left}},d_{\mathrm{right}})
=(8,-4,-2,-2).
\]
To invert, first reconstruct \((8-4,8+4)=(4,12)\). Then combine each of these averages with its saved detail. The result is the original four samples.
Track the effect of the normalization
For one pair, \(u^2+v^2=2(a^2+d^2)\). Applying this at both levels gives
\[2^2+6^2+10^2+14^2=336
=4(8^2+(-4)^2)+2((-2)^2+(-2)^2).
\]
The unweighted sum of the four stored coefficient squares is only \(88\). This \(H\) is not an orthogonal matrix: \(H^{\mathsf T}H=I/2\). Using \(\sqrt2H\) for a single pair gives an orthogonal matrix instead. Do not mix the two normalization conventions in reconstruction or energy checks.
Drop fine details and measure the loss
Set the two first-level details to zero, while keeping \(8\) and \(-4\). Reconstruction gives \((4,4,12,12)\). The residual is \((-2,2,-2,2)\) and its squared Euclidean norm is \(16\). The reconstructed squared norm is \(320\), so \(336=320+16\).
Try it
Transform \((1,3,5,7)\) in the same coefficient order.
Answer: The first averages are \((2,6)\) and the details are \((-1,-1)\). The stored coefficients are \((4,-2,-1,-1)\). Their weighted energy is \(4(16+4)+2(1+1)=84\), the original squared norm.
Papers, proofs, Lean, and code
Terms in this example: Haar pair · Detail coefficient · Normalization · Orthogonal matrix · Refinement
Paper V supplies signal background. Information Realization includes a finite Haar contract within its recorded formal core.
Paper V: Polyphase Signal Calculus
Read · Written proofs · Lean coverage · Code and setup · Evidence
Information Realization
Read · Written proofs · Lean coverage · Code and setup · Evidence
Related library guide and code examples. The guide covers related constructions and their software interfaces.
Python arithmetic check for this example: haar_pairs(). This checks the displayed rational, modular or finite-set calculations exactly.
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