sparse image representation via combined transforms - Convex ...

26 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES
1-D signal into a distribution on the time-frequency plane (which is 2-D). Since this thesis
mainly emphasizes decomposition, from now on “transform” means only the one in the
sense of decomposition.
Some statements could be seemingly subjective. Given the same fact (in our case,
for example, it could be a picture), different viewers may draw different conclusions. We
manage to be consistent with the majority’s point of view. (Or at least we try to.)
Key message
Before we move into the detailed discussion, we would like to summarizes the key results.
Readers will see that they play an important role in the following chapters. Table 3.1
summarize the correspondence between three transforms, three image features, and orders
of complexity of their fast algorithms. Note that the 2-D DCT is good for an image with
homogeneous components. For an N by N image, its fast algorithm has O(N 2 log N)
order of complexity. The two-dimensional wavelet transform is good for images with point
singularities. For an N by N image, its fast algorithm has O(N 2 ) order of complexity. The
edgelet transform is good for an image with line singularities. For an N by N image, its
order of complexity is O(N 2 log N), which is the same as the order of complexity for the
2-D DCT.
Transform Image Feature Complexity of Discrete Algorithms
(for N × N image)
DCT Homogeneous Components N 2 log N
Wavelet Transform Point Singularities N 2
Edgelet Transform Line Singularities N 2 log N
Table 3.1: Comparison of transforms and the image features that they are good at processing.
The third column lists the order of computational complexity for their discrete fast
algorithms.
Why so many figures?
In the main body of this chapter, we show many figures. The most important reason is that
this project is an image processing project, and showing figures is the most intuitive way
to illustrate points.