lec14

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Output image
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Lecture 14: Geometric Transformation
Detail see function scal.m.
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Spatial transformations#1: Resizing
When requiring magnification to a non_integer times, pixel replication is not applicable, spatial transformation method is utilized. General transformation relationship (x‘, y‘)
Lecture 14: Geometric Transformation
Today’s lecture objective
1. Understand the concept of spatial transformation and gray level interpolation. 2. Be able to perform pixel replication, image resizing to any scale, and rotation, include techniques such as nearest neighbor interpolation and bilinear interpolate concepts. 3. Be able to use related Matlab functions to solve real world geometric transformation problems.
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Geometric transformations
Definition: geometric transformations modify the spatial relationships between pixels in an image. It is also called rubber-sheet transformations. In terms of digital image processing, a geometric transformation consists of two basic operations:
(x, y)
x‘ = r(x, y) and y‘ = s(x, y) For resizing:
r(x,y)=c1*x; s(x,y)=c2*y
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Gray-level interpolation (nearest neighbor)
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Mystery to be solved
The right side is a famous picture cal‘ painted by Hans Holbein in the year1533. What is the odd shaped object at the bottom of the painting? Solve the mystery using geometric transformation.
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Magnify digital images
We already discussed image contraction by simply deleting some rows and columns in chapter 2. Now we discuss a more complex operation—magnification.
Magnification requires two steps: 1) the creation of new pixel locations; 2) the assignment of gray levels to those new locations. Pixel replication is the simplest way of magnification, it is applicable when we want to increase the size of an image an integer number of times.
Output image
Input image
For noninteger (x‘,y‘), interpolation is needed, the nearest neighbor interpolation is simple to implement but has the drawback of producing undesirable artifacts.
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Pixel replication example
Task#1: Double the size of an image.
We perform the task in following steps:
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Demonstration of uncovering the mystery using geometric transformation
First, we are going to reveal the mystery, then followed by the explanation on how to do it. Matlab demo: anamorphosis.m
1) Duplicate each column, this doubles the image size in the horizontal direction. 2) Duplicate each row of the enlarged image to double the size in vertical direction. The same procedure is used to enlarge the image by any integer number of times (triple, quadruple, and so on).
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Nearest neighbor interpolation example Task#2: 2.5 times the size of an image.
We perform the task in following steps:
1) Use the transformation expression x‘=x*2.5 and y‘=y*2.5 to do a mapping from (x‘,y‘) to (x,y), ie, for each pair of (x‘,y‘), find (x,y). 2) If mapped (x,y) is integer, then assign the value at (x,y) to (x‘,y‘); otherwise, find the nearest neighbor to (x,y) and assign the value at this neighbor to (x‘,y‘).
For detail, see nearest_scal.m
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Gray-level interpolation: Method#2
To get a smoother result, we need to use bilinear interp. method.
1) a spatial transformation, which defines the ―rearrangement‖ of pixels on the image plane;
2) interpolation of gray-level, which deals with the assignment of gray-levels to pixels in the spatially transformed image.
The value at Input this location image (x,y) is calculated using formula
(x‘,y‘)
v(x,y) = ax + by + cxy + d