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As described in Images in MATLAB, MATLAB^{®} stores most images as two-dimensional arrays (i.e., matrices), in which
each element of the matrix corresponds to a single *pixel* in the
displayed image. To access locations in images, the Image Processing
Toolbox™ uses several different image coordinate systems as conventions for
representing images as arrays.

Pixel Indices—Because images are arrays, you can use standard MATLAB indexing.

Spatial Coordinates—You can consider locations in images as positions on a plane using Cartesian coordinates.

Often, the most convenient method for expressing locations in an image is to use pixel indices. The image is treated as a grid of discrete elements, ordered from top to bottom, and left to right, as illustrated by the following figure.

**Pixel Indices**

For pixel indices, the row increases downward, while the column increases to the right. Pixel indices are integer values, and range from 1 to the length of the row or column.

There is a one-to-one correspondence between pixel indices and subscripts for the first two matrix dimensions in MATLAB. For example, the data for the pixel in the fifth row, second column is stored in the matrix element (5,2). You use normal MATLAB matrix subscripting to access values of individual pixels. For example, the MATLAB code

I(2,15)

returns the value of the pixel at row 2, column 15 of the image
`I`

. Similarly, the MATLAB code

RGB(2,15,:)

returns the `R`

, `G`

, `B`

values of the pixel at row 2, column 15 of the image `RGB`

.

The correspondence between pixel indices and subscripts for the first two matrix dimensions in MATLAB makes the relationship between an image's data matrix and the way the image is displayed easy to understand.

Another method for expressing locations in an image is to use a system of
continuously varying coordinates rather than discrete indices. This lets you
consider an image as covering a square patch, for example. In a *spatial
coordinate system* like this, locations in an image are positions on a
plane, and they are described in terms of *x* and
*y* (not row and column as in the pixel indexing system). From
this Cartesian perspective, an (*x*,*y*) location
such as (3.2,5.3) is meaningful, and is distinct from pixel (5,3).

The Image Processing Toolbox defines two types of spatial coordinate systems

Intrinsic Coordinates—A spatial coordinate system that corresponds to pixel indices

World Coordinates—A spatial coordinate system that relates the image to some other coordinate space

By default, the toolbox uses a spatial coordinate system for an image that
corresponds to the image’s pixel indices. It is called the intrinsic coordinate
system and is illustrated in the following figure. Notice that
*y* increases downward, because this orientation is
consistent with the way in which digital images are typically viewed.

**Intrinsic Coordinate System**

The intrinsic coordinates (*x*,*y*) of
the center point of any pixel are identical to the column and row indices for
that pixel. For example, the center point of the pixel in row 5, column 3 has
spatial coordinates *x* = 3.0, *y* = 5.0.
This correspondence simplifies many toolbox functions considerably. Be aware,
however, that the order of coordinate specification (3.0,5.0) is reversed in
intrinsic coordinates relative to pixel indices (5,3).

Several functions primarily work with spatial coordinates rather than pixel
indices, but as long as you are using the default spatial coordinate system
(intrinsic coordinates), you can specify locations in terms of their columns
(*x*) and rows (*y*).

When looking at the intrinsic coordinate system, note that the upper left
corner of the image is located at (0.5,0.5), not at (0,0), and the lower right
corner of the image is located at (`numCols`

+ 0.5,
`numRows`

+ 0.5), where `numCols`

and
`numRows`

are the number of rows and columns in the image.
In contrast, the upper left pixel is pixel (1,1) and the lower right pixel is
pixel (`numRows`

, `numCols`

). The center of
the upper left pixel is (1.0, 1.0) and the center of the lower right pixel is
(`numCols`

, `numRows`

). In fact, the
center coordinates of every pixel are integer valued. The center of the pixel
with indices (*r*, *c*) — where
*r* and *c* are integers by definition
— falls at the point *x* = *c*,
*y* = *r* in the intrinsic coordinate
system.

In some situations, you might want to use a world coordinate system (also called a nondefault spatial coordinate system). For example, you could shift the origin by specifying that the upper left corner of an image is the point (19.0,7.5), rather than (0.5,0.5). Or, you might want to specify a coordinate system in which every pixel covers a 5-by-5 meter patch on the ground.

There are several ways to define a world coordinate system: