8. Geometric Operations

• Geometric operations change image geometry by moving pixels around in a carefully constrained way.

• We might do this to remove distortions inherent in the imaging process, or to introduce a deliberate distortion that matches one image with another.

• In this chapter, we will consider simple operations such as scaling and rotation.

Simple techniques• There are some simple techniques for

manipulating image geometry.

• An image can be enlarged by an integer factor, n, simply by copying each pixel to an n x n block of pixels in the output image.

• This technique is fast, and has been a standard feature of specialised image processing hardware.

• An obvious disadvantage is that it cannot be used to expand an image by some arbitrary, non‑integer factor.

• Another problem is that greatly enlarged images have a very 'blocky' appearance.

• This may not bother us in applications where we merely wish to examine pixels more closely, but is of serious concern otherwise.

• There are similar problems with the technique of shrinking an image by subsampling its array of pixels.

• First, the technique cannot be used to reduce image dimensions by an arbitrary factor.

• Second, subsampling can eliminate information from the image completely.

• One solution to this latter problem is to turn an n x n block of pixels in the input image into a single pixel in the output image.

• The value of each output pixel must be representative of the corresponding block in the input image.

• The median and mean grey level of the block can be used.

Fig. 8.1 Shrinking. (a) Original image. (b) Subsampling (c) Mean of n x n block. (d) Median of n x n block.

• Figure 8.1 compares subsampling with the n x n mean and n x n median approaches for a real image of a face.

• In this case, there is little to choose between the mean and median images.

• The subsampled image is clearly inferior, with data loss leading to an apparent change in facial expression.

Affine transformation• An arbitrary geometric transformation will

move a pixel at coordinates (x, y) to a new position, (x’, y'), given by a pair of transformation equations,

X’ = Tx (X, Y), (8.1)

Y’ = TY (X, Y) (8.2)• Tx and Ty are typically expressed as

polynomials in x and y. • In their simplest form, they are linear in x

and y, giving us an affine transformation,

x' = a0x + a1y + a2 (8.3)

y' = b0x + b1y + b2 (8.4)

This can be expressed in matrix form as

• Under an affine transformation, straight lines are preserved and parallel lines remain parallel.

• Translation, scaling, rotation and shearing are all special cases of Equations 8.3 and 8.4.

• For example, a translation of 3 pixels down and 5 pixels to the right is

x' = x + 5,

y' = y + 3.

The corresponding aftine transformation matrix is

• Table 8.1 specifies how the elements of the transformation matrix are computed for selected special cases of affine transformation.

Table 8.1 Transformation coefficients for some simple affine transformations.

Rotation

• Suppose, for example, that we wish to rotate an image by an angle θ about the origin.

• This is accomplished with the transformation matrix

Algorithm 8.1 Image rotation by forward mapping