8. Geometric Operations
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Transcript of 8. Geometric Operations
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