physics

Mach bands

Mach bands is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Mach bands rather than just read about it. In short: Mach bands is an optical illusion named after the physicist Ernst Mach. It exaggerates the contrast between edges of the slightly differing shades of gray, as soon as they contact one another, by triggering edge-detection in the human visual system.

Key takeaways

Mach bands belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Mach bands to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Mach bands from memory before moving on to harder problems.

Reference excerpt

Mach bands is an optical illusion named after the physicist Ernst Mach. It exaggerates the contrast between edges of the slightly differing shades of gray, as soon as they contact one another, by triggering edge-detection in the human visual system. The Mach band illusion is sometimes called the Chevreul illusion.

Explanation The Mach bands effect is due to the spatial high-boost filtering performed by the human visual system on the luminance channel of the image captured by the retina. Mach reported the effect in 1865, conjecturing that filtering is performed in the retina itself, by lateral inhibition among its neurons. This conjecture is supported by observations on other (non-visual) senses, as pointed out by Georg von Békésy. The visual pattern is often found on curved surfaces subject to a particular, naturally occurring illumination, so the occurrence of filtering can be explained as the result of learnt image statistics. The effect of filtering can be modeled as a convolution between a trapezoidal function that describes the illumination and one or more bandpass filters. A tight approximation is obtained by a model employing 9 even-symmetric filters scaled at octave intervals. The effect is independent of the orientation of the boundary.

In radiology

This visual phenomenon is important to keep in mind when evaluating dental radiographs for evidence of decay, in which grayscale images of teeth and bone are analyzed for abnormal variances of density. A false-positive radiological diagnosis of dental caries can easily arise if the practitioner does not take into account the likelihood of this illusion. Mach bands manifest adjacent to metal restorations or appliances and the boundary between enamel and dentin. Mach bands may also result in the misdiagnosis of horizontal root fractures because of the differing radiographic intensities of tooth and bone. Mach effect can also lead to an erroneous diagnosis of pneumothorax by creating a dark line at the lung periphery (whereas a true pneumothorax will have a white pleural line).

In computer graphics

Mach bands can also appear when there is a discontinuity in the derivative of a gradient, a visual effect common when intensities are linearly interpolated such as in Gouraud shading. Computer image processing systems use edge-detection in a way analogous to the brain, using unsharp masking to clarify edges in photos for example.

See also Acutance Cornsweet illusion Hermann grid illusion Lateral inhibition Optical illusions Watercolour illusion Gibbs phenomenon

References

Further reading Lotto RB, Williams SM, Purves D (1999). "Mach bands as empirically derived associations". Proceedings of the National Academy of Sciences. 96 (9): 5245–50. Bibcode:1999PNAS...96.5245L. doi:10.1073/pnas.96.9.5245. PMC 21849. PMID 10220451. Eagleman, DM (2001) "Visual Illusions and Neurobiology." Nature Reviews Neuroscience. 2(12): 920–6.

External links

Demonstration of the Mach band was at NIST until July 2016 Mach Band figure and explanatory hypothesis, Laboratory of Dale Purves

Illustrations

Mach bands: Exaggerated contrast between edges of the slightly differing shades of gray appears as soon as they touch — Exaggerated contrast between edges of the slightly differing shades of gray appears as soon as they touch

Example of Mach bands at the ends of gradients where the derivative of the luminance is discontinuous Actual luminance profilePerceived luminance profileSmooth luminance profile for comparison

Worked examples

Example 1 — a first encounter with Mach bands

Start with the simplest possible case. Write down what Mach bands claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Mach bands before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Mach bands ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Mach bands

In research: Mach bands appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Mach bands in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Mach bands is common in secondary-school and first-year university syllabi. It links to neighbouring topics Ernst Mach, Optical illusions, so understanding it makes those chapters shorter.
In everyday life: Look for Mach bands outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Mach bands in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Mach bands means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Mach bands out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Mach bands in simple terms?

Why does Mach bands matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Mach bands?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Mach bands.