C14 · Coordinates, frequencies, scales
Scales, wavelets and textures
Coarse structure and fine details live at different scales. They can be stored separately if analysis and reconstruction agree.
Theory
A coarse level plus a residual
A Gaussian pyramid smooths before downsampling, suppressing frequencies the new grid cannot retain. Fine details are lost. Expanding the coarse level back will not match the original; their difference is the Laplacian detail for that expansion rule.
The saved residual can be added back exactly. This example has one 16×16→8×8 transition and a fixed binomial filter. A multilevel pyramid repeats the rule. It is useful for alignment, blending and scale analysis, but a downsampled image alone cannot replace the source.
Coarse layer and missing details
Downsampling gives 8×8 samples. Restore details by moving their contribution from zero to one.
Exact values are in the point inspector
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Exact values are in the point inspector
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Inspect the selected point
What the plot shows
At contribution 1 the residual exactly fills the difference. The coarse layer alone does not contain vanished fine details.
Parameters and details
Smooth, then retain every second sample. Expand bilinearly, repeating edge samples beyond the boundary. The Laplacian detail is the exact residual of this expansion; strength 1 reconstructs the input despite information lost in the coarse level.
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.
Haar: average and three difference directions
A 2×2 block becomes four orthogonal coefficients. Top left stores LL sums; right stores x changes; bottom stores y changes; bottom right stores the diagonal combination. This is our explicit axis convention; library LH/HL names may use another convention.
Small details can be zeroed by a hard threshold or shrunk by a soft threshold. The live example uses hard thresholding only. Removing weak coefficients can remove noise and useful texture together. A one-pixel image shift changes the block grouping, so ordinary DWT is not shift-invariant.
Four pixels become a sum and differences
Increase the threshold for removing small detail coefficients. Compare coefficients and reconstruction.
Exact values are in the point inspector
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Exact values are in the point inspector
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What the plot shows
A high threshold leaves averaged blocks. A zero threshold reconstructs the input up to numerical precision.
Parameters and details
Orthonormal Haar on 2×2 blocks: the sum and three differences are divided by 2. LL is retained; the threshold zeros small details. At zero threshold the transform is reversible.
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.
Scale and direction are different properties
Reference and further methods
Daubechies wavelets use longer filters and different smoothness properties with a matched analysis/synthesis pair. A Gabor filter is a local wave under a Gaussian window, with a selectable direction and frequency. A bank describes where texture occurs and in which direction.
LBP compares a center with neighbors; a co-occurrence matrix counts intensity pairs at a chosen offset. These are texture features, not automatic material understanding. Tiny windows give unstable statistics; large windows mix objects. Segmentation uses such features only after defining the task.
Practice
Test the idea under another condition. Follow the task above the images; point inspection helps explain the result. This exploration is not scored.
Recover the missing detail
A high threshold has removed weak differences. Lower it until a fine detail returns. Select a pixel on that detail and compare the four coefficients reconstructing it.
Exact values are in the point inspector
Blue − · white 0 · red +; scale -2…2
Exact values are in the point inspector
Grayscale 0…1: black to white.
Inspect the selected point
What the plot shows
The threshold changes retained detail, not the source sample count. At zero, reconstruction matches the source; a nonzero threshold cannot identify noise by itself.
Parameters and details
Orthonormal Haar on 2×2 blocks: the sum and three differences are divided by 2. LL is retained; the threshold zeros small details. At zero threshold the transform is reversible.
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
What is needed for exact Laplacian-level reconstruction?
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…1: black to white.
Exact values are in the point inspector
Blue − · white 0 · red +; scale -2…2
Exact values are in the point inspector
Blue − · white 0 · red +; scale -1…1
Exact values are in the point inspector
Grayscale 0…1: black to white.
Choose one answer. It is checked immediately.
Limitations and references
Limits of this example
Live models are a one-level pyramid and orthonormal 2×2 Haar. Other families are explained by purpose, without relabeling a generic filter as their implementation.