Image moment

Key takeaways

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

Reference excerpt

In image processing, computer vision and related fields, an image moment is a certain particular weighted average (moment) of the image pixels' intensities, or a function of such moments, usually chosen to have some attractive property or interpretation. Image moments are useful to describe objects after segmentation. Simple properties of the image which are found via image moments include area (or total intensity), its centroid, and information about its orientation.

Raw moments For a 2D continuous function f(x,y) the moment (sometimes called "raw moment") of order (p + q) is defined as

M p q = ∫ − ∞ ∞ ∫ − ∞ ∞ x p y q f ( x , y ) d x d y {\displaystyle M_{pq}=\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty }x^{p}y^{q}f(x,y)\,dx\,dy}

for p,q = 0,1,2,... Adapting this to scalar (grayscale) image with pixel intensities I(x,y), raw image moments Mij are calculated by

M i j = ∑ x ∑ y x i y j I ( x , y ) {\displaystyle M_{ij}=\sum _{x}\sum _{y}x^{i}y^{j}I(x,y)\,\!}

In some cases, this may be calculated by considering the image as a probability density function, i.e., by dividing the above by

∑ x ∑ y I ( x , y ) {\displaystyle \sum _{x}\sum _{y}I(x,y)\,\!}

A uniqueness theorem states that if f(x,y) is piecewise continuous and has nonzero values only in a finite part of the xy plane, moments of all orders exist, and the moment sequence (Mpq) is uniquely determined by f(x,y). Conversely, (Mpq) uniquely determines f(x,y). In practice, the image is summarized with functions of a few lower order moments.

Examples Simple image properties derived via raw moments include:

Area (for binary images) or sum of grey level (for greytone images): M 00 {\displaystyle M_{00}}

Centroid: { x ¯ , y ¯ } = { M 10 M 00 , M 01 M 00 } {\displaystyle \{{\bar {x}},\ {\bar {y}}\}=\left\{{\frac {M_{10}}{M_{00}}},{\frac {M_{01}}{M_{00}}}\right\}}

Central moments Central moments are defined as

μ p q = ∫ − ∞ ∞ ∫ − ∞ ∞ ( x − x ¯ ) p ( y − y ¯ ) q f ( x , y ) d x d y {\displaystyle \mu _{pq}=\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty }(x-{\bar {x}})^{p}(y-{\bar {y}})^{q}f(x,y)\,dx\,dy}

where x ¯ = M 10 M 00 {\displaystyle {\bar {x}}={\frac {M_{10}}{M_{00}}}} and y ¯ = M 01 M 00 {\displaystyle {\bar {y}}={\frac {M_{01}}{M_{00}}}} are the components of the centroid. If ƒ(x, y) is a digital image, then the previous equation becomes

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Image moment

Start with the simplest possible case. Write down what Image moment 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 Image moment 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 Image moment 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 Image moment

In research: Image moment 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 Image moment 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: Image moment is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computer vision, Digital imaging, Image processing, so understanding it makes those chapters shorter.
In everyday life: Look for Image moment 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.

How to study Image moment in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Image moment 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 Image moment out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Image moment in simple terms?

In image processing, computer vision and related fields, an image moment is a certain particular weighted average (moment) of the image pixels' intensities, or a function of such moments, usually chosen to have some attractive property or interpretation. Image moments are useful to describe objects…

Why does Image moment 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 Image moment?

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 Image moment.