Is the centroid always inside the object?

No. A concave shape or separated regions can have a centroid outside its foreground pixels.

Are moments only used with binary images?

No. Intensities or other nonnegative weights can also be used, with a different interpretation.

Does one moment identify a shape?

Usually not. Moments summarize selected properties and are commonly combined with other measurements.

AI Glossary / Computer vision and multimodal AI / Intermediate

Image moment

Definition

An image moment is a numerical summary formed by adding pixel values or region weights after multiplying them by powers of their coordinates. Different moments describe properties such as total weight, center of mass, and spatial spread, making them useful building blocks for shape analysis and object measurement.

What it means

A picture contains many individual pixel values. Moments compress selected aspects of that information into numbers. For a binary object mask, the foreground pixels can each contribute a weight of one, so the simplest moment represents the number of foreground pixels.

For a grayscale image, pixel intensities can instead act as weights. The resulting center describes the distribution of brightness, which is not necessarily the geometric center of the object.

Raw moments depend on the chosen coordinate origin. Central moments measure coordinates relative to the weighted center. Normalized central moments adjust for scale under the assumptions behind their definition. These distinctions matter when comparing objects that move or change apparent size.

How it works

Choose the image region and decide what its weights mean. A segmentation mask measures the geometry of a detected region; an intensity image measures a distribution of intensity. Record whether the background is excluded and whether pixel spacing matters.

Calculate raw moments by summing each weight multiplied by horizontal and vertical coordinate powers. The zeroth moment is the sum of weights. Dividing the first horizontal and vertical moments by that sum gives the weighted centroid, provided the total is nonzero.

Subtract the centroid before calculating central moments. Second-order central moments describe spread and cross-direction relationships. Appropriate combinations can help estimate an elongated object's orientation, although nearly symmetric objects may have an unstable preferred direction.

Some descriptors combine normalized moments to obtain particular invariance properties. Such properties do not mean immunity to segmentation mistakes, image noise, perspective changes, or every possible transformation.

Three practical examples

1. Conveyor inspection: The centroid of a segmented component estimates its location so a handling system can align a pickup operation.

2. Document processing: Moments summarize the distribution of a connected ink region as one component of a character recognition pipeline.

3. Object monitoring: Changes in a region's area and spatial spread help flag a silhouette that has split, merged, or been partially obscured.

Think of it this way

Think of balancing a thin cardboard shape on a point. Its balancing point summarizes where its material is distributed, while other measurements describe how widely that material spreads around the point.

The analogy works well for a binary region with equal pixel weights. It becomes less literal for intensity-weighted images, where bright pixels effectively carry more weight. Neither the balancing point nor a handful of moments reconstructs every detail of the original shape.

Worked example

Consider three foreground pixels at coordinates (0,0), (2,0), and (1,3), each with weight one. The zeroth moment is 3. The first horizontal moment is 0 + 2 + 1 = 3, and the first vertical moment is 0 + 0 + 3 = 3.

The centroid is therefore (3/3,3/3) = (1,1). Relative to that center, the coordinates are (-1,-1), (1,-1), and (0,2).

The horizontal second central moment is 1 + 1 + 0 = 2. The vertical second central moment is 1 + 1 + 4 = 6. The cross moment is 1 - 1 + 0 = 0. In this example, the pixels are more spread vertically than horizontally.

Moving every pixel five units to the right changes the raw horizontal moments and centroid, but leaves these central moments unchanged. Enlarging the shape requires a separate treatment of scale.

Illustrative case study

Fictional case: A packaging workshop uses a camera to locate flat labels before placement. Its first version uses the brightness-weighted center of each crop. Glossy printing shifts that center even when the label itself stays still.

The team changes the measurement to a binary mask of the label boundary and checks the mask against manually reviewed images. It also flags masks that touch the crop border because part of the label may be missing.

The lesson is that the definition of the weights controls what the moment measures. Switching formulas without checking the input representation would not have solved the underlying problem.

Common misconceptions and mistakes

Do not confuse geometric area with intensity sum. Both can be zeroth moments under different input conventions.

Do not divide by a zero total weight. Empty masks need an explicit missing-measurement outcome.

Do not assume two shapes with similar moments are identical. A finite descriptor loses information.

Do not treat normalized moments as universally invariant. Rotation, reflection, scale, and perspective require different considerations, and numerical precision can matter for high-order moments.

Business use

Image moments offer compact, interpretable measurements for inspection, tracking, and preprocessing. They can be cheaper to compute and easier to explain than a large learned representation when the task depends on simple geometry.

Their value depends on reliable region extraction. Before deployment, test lighting changes, occlusion, merged objects, and boundary noise. Preserve units: pixel-based spread cannot automatically be interpreted as physical dimensions without calibration and appropriate pixel geometry.

Check your understanding

A binary mask moves ten pixels right without changing shape. Which changes: its centroid or its second central moments?

Answer: Its horizontal centroid increases by ten pixels. Its second central moments remain the same because they use coordinates relative to that centroid.

Related terms

Learn next

Calculate area and centroid for a small hand-drawn binary mask, then translate the mask and compare raw and central moments. Next study Hu moment invariants, paying attention to exactly which transformations their construction addresses.

Centroid

Sources

OpenCV, Moments class reference, gives the raw, central, and normalized moment definitions and centroid relationships: https://docs.opencv.org/4.x/d8/d23/classcv_1_1Moments.html . The reference was consulted for the mathematical definitions. All numerical examples and the workshop case here are original teaching illustrations.

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