A bright constant patch can have zero gradient. The derivative describes change, not brightness itself.
Retain directional change
The Sobel operator approximates directional derivatives of image intensity using defined discrete kernels. Horizontal and vertical responses can be combined into a gradient magnitude. Their sign and scale belong to the selected kernel convention, and the result should remain distinct from the original intensity values. Primary reference: OpenCV Sobel directional gradients.
For an original worksheet, retain the input neighborhood and signed responses before any absolute-value display conversion. State whether the combination uses Euclidean magnitude or another definition. A bright image location does not necessarily imply a large local derivative.
Calculate fictional patches
Take a three-by-three patch whose three rows are all [10,20,30]. Applying the stated horizontal weights [−1,0,1], [−2,0,2], [−1,0,1] by correlation gives 20 + 40 + 20 = 80. The corresponding vertical response is zero because the rows match.
For a constant three-by-three patch of value 200, both responses are zero under these weights. That brighter patch therefore has smaller gradient than the lower-valued ramp patch. These are invented unscaled, signed calculations; output storage, kernel flipping or display conversion is not silently included.
Preserve the response convention
Prepare an original comparison with neighborhood values, derivative directions and the chosen magnitude calculation. Keep negative responses if their direction matters. If a display takes absolute values or clips results, record that operation separately instead of treating the displayed brightness as the signed derivative.
This guide promises no HM edge accuracy and selects no universal threshold. Its reader decision is whether directional image change has been distinguished from local intensity. Physical edge interpretation needs an appropriate feature definition and evaluation beyond the numerical response of a chosen kernel.
Customer Questions
What is the fictional horizontal response?
Eighty under the stated correlation weights.
What is the constant-patch response?
Zero in both directions.
Does greater brightness require greater gradient?
No. A constant bright neighborhood has no local change.
Primary References
These references support the technical principles discussed in this guide. The worked examples and review questions are educational.
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