Image Geometry

Continuous Image Domain

We consider that the discrete image originates from an underlying continuous signal

\[F:\Omega \subset \mathbb{R}^{2} \to \mathbb{R}\]

defined over a rectangular domain \(\Omega\).

The coordinates \((x, y) \in \Omega\) are continuous spatial coordinates, with \(x\) increasing to the right and \(y\) increasing downward.

The function \(F\) represents the ideal, infinitely-resolved image intensity at any real-valued location in \(\Omega\).

The digital image is a sampling of this continuous signal.

Discrete Image

../_images/numeric_image.svg

A discrete image \(I\) of width \(W\) and height \(H\) is the finite regularly sampled function from the underlying continuous signal \(F\).

\[I:\{0, ...,H-1 \} \times \{0, ...,W-1 \} \to \mathbb{R}\]

with elements \(I[i,j] \in \mathbb{R}\) called pixels.

Pixel model

A discrete coordinate \((i, j)\) refers to the pixel at row index \(i\) and column index \(j\), with \((0, 0)\) referring the upper-left pixel.

Each pixel corresponds to a rectangular region \(R_{i,j}\) in the continuous domain. GridR uses the pixel-centered sampling convention :

\[R_{i,j} = \left[ j - \frac{1}{2}, j + \frac{1}{2} \right[ \times \left[ i - \frac{1}{2}, i + \frac{1}{2} \right[\]

Consequences :

  • pixel centers lie on integer coordinates

  • pixel edges lie on half-integer coordinates

  • the center-to-center spacing is 1 in both directions.

Sampling Grid

To avoid ambiguity between array indices and geometric coordinates we define the sampling grid (the sampling lattice) explicitly :

\[\mathscr{G} = \{ (x,y) \in \mathbb{R}^{2} | x \in \mathbb{Z}, y \in \mathbb{Z} \}\]

In the pixel-centered sampling convention used in GridR, each integer coordinate \((j, i) \in \mathbb{Z}^{2}\) corresponds to the center of pixel \((i, j)\) in the discrete image \(I\). We therefore use the mapping :

\[(i, j) \longleftrightarrow (x, y) = (j, i) \in \mathscr{G}\]

All geometric operations and transformation are expressed in this coordinate system.

Image Footprint

Given image width \(W\) and height \(H\), the continuous domain covered by the discrete image \(I\) — the footprint — is

\[\Omega_{I} = \left[ - \frac{1}{2}, H + \frac{1}{2} \right] \times \left[ - \frac{1}{2}, W + \frac{1}{2} \right]\]