Part 1 — Vectors and Dot Product

Learning Objective¶

Understand how image patches become vectors and how the dot product measures agreement between two patches — the foundation for all normalised template matching.

Roadmap¶

1. Pixels as Vectors¶

Intuition¶

An image patch is a 2D grid of pixel values. But for math, we flatten it into a 1D vector — just a list of numbers. A 3×3 patch with 9 pixels becomes a point in 9-dimensional space.

This is not an abstraction — it is literally how OpenCV’s template matching works internally. It flattens the template and each scene patch into vectors, then compares them.

Why this matters¶

Once we see patches as vectors, all of linear algebra applies:

• Comparing two patches = measuring the distance or angle between two vectors

• Brightness change = adding a vector to shift the point in space

• Contrast change = scaling the vector (stretching it longer or shorter)

For visualization, we use 3-pixel patches (3D space) so we can actually plot them. Everything generalises to any number of pixels.

2. Dot Product¶

Intuition¶

The dot product of two vectors answers: how much do these two vectors point in the same direction?

For image patches, the dot product tells you how much two patches “agree” — pixel by pixel, do they go bright in the same places and dark in the same places?

The formula¶

For two vectors $\vec{a} = [a_1, a_2, \ldots, a_n]$ and $\vec{b} = [b_1, b_2, \ldots, b_n]$:

Multiply corresponding elements, then sum.

The geometric definition¶

The dot product has an equivalent geometric definition:

where $\theta$ is the angle between the two vectors.

This connects the algebraic formula (multiply and sum) to geometry (angles). We will use this in Part 2 to derive cosine similarity.

Key observation¶

A higher dot product means more agreement between two patches. However, the raw dot product also depends on the magnitude of both vectors — a brighter patch produces a larger dot product even with the same pattern. Normalising fixes this (covered in Part 2).