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).

By default, the toolbox uses a spatial coordinate system for
an image that corresponds to the image's pixel indices. It's
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.

One way to define a world coordinate system for an image is
to specify the `XData`

and `YData`

image
properties for the image. The `XData`

and `YData`

image
properties are two-element vectors that control the range of coordinates
spanned by the image. When you do this, the MATLAB axes coordinates
become identical to the world (nondefault) coordinates. If you do
not specify `XData`

and `YData`

,
the axes coordinates are identical to the intrinsic coordinates of
the image. By default, for an image `A`

, `XData`

is ```
[1
size(A,2)]
```

, and `YData`

is ```
[1
size(A,1)]
```

. With this default, the world coordinate system
and intrinsic coordinate system coincide perfectly.

For example, if `A`

is a 100 row by 200 column
image, the default `XData`

is [1 200], and the default `YData`

is
[1 100]. The values in these vectors are actually the coordinates
for the center points of the first and last pixels (not the pixel
edges), so the actual coordinate range spanned is slightly larger.
For instance, if `XData`

is [1 200], the interval
in `X`

spanned by the image is [0.5 200.5].

It's also possible to set `XData`

or `YData`

such
that the *x*-axis or *y*-axis is
reversed. You'd do this by placing the larger value first.
(For example, set the `YData`

to [1000 1].) This
is a common technique to use with geospatial data.

These commands display an image using nondefault `XData`

and `YData`

.

A = magic(5); x = [19.5 23.5]; y = [8.0 12.0]; image(A,'XData',x,'YData',y), axis image, colormap(jet(25))

To specify a world (nondefault spatial) coordinate system for
an image, use the spatial referencing objects `imref2d`

and `imref3d`

.
Spatial referencing objects let you define the location of the image
in a world coordinate system and specify the image resolution, including
nonsquare pixel shapes. These objects also support methods for converting
between the world, intrinsic, and subscript coordinate systems. Several
toolbox functions accept or return spatial referencing objects: `imwarp`

, `imshow`

, `imshowpair`

, `imfuse`

, `imregtform`

, and `imregister`

.

This example creates a spatial referencing object associated
with a 2-by-2 image where the world extent is 4 units/pixel in the **x** direction
and 2 units/pixel in the `y`

direction. The example
creates the object, specifying the pixels dimensions as arguments
but does not specify world limits in the **x** and **y** directions.
You could specify other information when creating an object, see `imref2d`

for
more information.

I = [1 2; 3 4] R = imref2d(size(I),4,2)

R = imref2d with properties: XWorldLimits: [2 10] YWorldLimits: [1 5] ImageSize: [2 2] PixelExtentInWorldX: 4 PixelExtentInWorldY: 2 ImageExtentInWorldX: 8 ImageExtentInWorldY: 4 XIntrinsicLimits: [0.5000 2.5000] YIntrinsicLimits: [0.5000 2.5000]

The `imref2d`

object contains information about
the image, some of it provided by you and some of it derived by the
object. The following table provides descriptions of spatial referencing
object fields.

Field | Description |
---|---|

`XWorldLimits` | Upper and lower bounds along the X dimension
in world coordinates (nondefault spatial coordinates) |

`YWorldLimits` | Upper and lower bounds along the Y dimension
in world coordinates (nondefault spatial coordinates) |

`ImageSize` | Size of the image, returned by the `size` function. |

`PixelExtentInWorldX` | Size of pixel along the X dimension |

`PixelExtentInWorldY` | Size of pixel along the Y dimension |

`ImageExtentInWorldX` | Size of image along the X dimension |

`ImageExtentInWorldY` | Size of image along the Y dimension |

`XIntrinsicLimits` | Upper and lower bounds along X dimension
in intrinsic coordinates (default spatial coordinates) |

`YIntrinsicLimits` | Upper and lower bounds along Y dimension
in intrinsic coordinates (default spatial coordinates). |

The following figure illustrates how these properties map to elements of an image.

You can also use the `XData`

and `YData`

properties
to define a world (nondefault spatial) coordinate system. Several
toolbox functions accept this data as arguments and return coordinates
in the world coordinate system: `bwselect`

, `imcrop`

, `impixel`

, `roipoly`

, and `imtransform`

.

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