4. Proposed Bit Allocation Framework
4.1. Analysis on curvelet transform coefficients T
Although curvelet transform provides decomposed frequency components at different resolutions, the coefficients at the coarsest and the finest resolutions do not arrange frequency components according to the direction of the transform windows. In this thesis, these coefficients are referred to as the first group of coefficients. The second group contains coefficients at other intermediate resolutions. These coefficients are arranged according to the direction of the transform windows. In section 4.1.1, we will describe the detail information of the curvelet coefficients in the two groups. In section 4.1.2, we will show the spatial
mapping between the original image positions and the corresponding curvelet coefficients.
4.1.1. The composition of the curvelet transform T
We have glanced at some features in section 3.5. In this section, we will put more emphasis on the meaning of coefficients in a curvelet domain image and their relation to the original image.
First, we explain the components in the first group that do not contain the information of directional decomposition. For the coarsest-level coefficients, they contain the low frequency components and the size is diminished into fixed size of 32×32 pixels. Therefore, the position mapping between the original image and the transformed coefficients is a direct down-scale.
Figure 4.1 Position mapping of curvelet coefficients in the coarsest level resolution
Let’s take the simple edge image that the only edge starts from right-up corner to left-bottom corner as an instance. XFigure 4.1X shows the way of position mapping between curvelet coefficients in the coarsest level resolution and the original image. The image of normalized coarsest coefficients is listed left whose size is 32×32 pixels, and the original image is listed right whose size is 352×288 pixels. We can easily find out the original spatial properties are remained in the curvelet coarsest level coefficients. Similarly, XFigure 4.2X is the image of normalized finest level coefficients. Curvelet coefficients in the finest level resolution are the collection of high frequency components of the original image, and their total size is the same as the original image. Therefore, the positional mapping is direct
one-to-one mapping. In the proposed bit-allocation method, neither coefficient in the coarsest nor finest level of resolutions is used.
Figure 4.2 Curvelet coefficients in the finest level resolution
For the second group of coefficients that contain the information of directional decomposition, the number of levels depends on the original image size. For example, if the original image has 352×288 pixels, we can obtain three middle levels of resolution which are the 2PndP level, 3PrdP level and 4PthP level respectively, see XFigure 4.3X.
Figure 4.3 Curvelet coefficients in the 2PndP, 3PrdP and 4PthP level resolution
For the 2PndP and 3PrdP level, coefficients are separated into 32 angles. However, there are 64 angular wedges in the 4PthP level. That is to say, the total number of separated angles in one
As XFigure 4.3X shows, because the original image contains only an edge starts from the right-up corner to the left-down corner, we can easily find out the coefficients are centralized in the region of left-up and right-down. In short, curvelet coefficients centralize the energy into the orthogonal angles of direction of the curves.
More over, the position mapping function of the middle levels is different from that of the first group. In section 3.5.1, as XFigure 3.11X shows, we have glanced at the relation of positions between the curvelet coefficients and the original image. Because the directional decomposition process re-samples the coefficients in the frequency domain, and the direction of re-sampling varies according to the specified angles, the actual positions of the final coefficients are shifted in some direction. We can classify the directions of shifting into two groups. First, for East and West quadrants, coefficients in the vertical direction are simply proportional, but coefficients are shifted by the angle of the orientation in the horizontal direction. Secondly, for North and South quadrants, coefficients in the horizontal direction are simply proportional, but coefficients are shifted by the angle of the orientation in the vertical direction.
4.1.2. Image type and the presentation of the related coefficient
As XFigure 4.3X shows, we can easily understand the distribution of the curvelet coefficients. Since the original image only contains a simple edge, the positions where the coefficient appears are simple, too.
Let’s see a more complex example. If the original image contains multiple multi-directional edges, such as the image in XFigure 4.4X(a), the distribution of its coefficients is much more complicated.
(a) (b)
Figure 4.4 Curvelet coefficients in the 2PndP, 3PrdP and 4PthP level resolution
In XFigure 4.4X(b), we can see that the coefficients are distributed in multiple angular wedges. Furthermore, it is natural that the angular wedges which the coefficient appears are different in each resolution. Therefore, curvelet transform can determine whether the curves in an image are complicated or not according to the directional decomposition in the middle levels of resolutions.
In our proposed scheme, we take each angular wedge in each resolution scale as a data unit. The actual process is to calculate the magnitude of one angular wedge in one resolution, and the magnitude becomes the representative value of the energy in the orientation in the resolution. Secondly, since we can get three resolution scales with directional decomposition, the coordinate formed by the magnitudes can be shown as in XFigure 4.5X.
Figure 4.5 The coordinate formed by curvelet coefficients
It is a three dimensional coordinate which is formed by the scale, magnitude, and angle.
The angle indicates the angle of the orientation which starts at 0∘and ends at 360∘in the direction of clockwise. However, the total angles in each resolution scale are different. To be more specific, the 2PndP, 3PrdP level resolution contains 32 angles respectively, but the 4PthP level resolution contains 64 angles. In the plane of angle θand magnitude, we can see the figure as histogram. Naturally, value of the magnitude is according to its quantity of coefficient in the orientation. If the value of magnitude is larger, it means more data in this direction. Therefore, we can analyze the composition of histogram to see whether the direction in the image is structured image or not.
As a result, we can take advantage of the property in curvelet transform to analyze the input video data in our proposed bit-allocation scheme. In next section, we will introduce the Otsu algorithm to help analyzing the curvelet coefficients for the sake of determine whether a small area in an image contains complicated curves or not.