This paper presents an efficient texture synthesis based on wavelet packet tree (TSWPT). It has the advantage of using a multiresolution representation with a greater diversity of bases functions for the nonlinear time series applications such as fractal images. The input image is decomposed into wavelet packet coefficients, which are rearranged and organized to form hierarchical trees called wavelet packet trees. A 2-step matching, that is, coarse matching based on low-frequency wavelet packet coefficients followed by fine matching based on middle-high-frequency wavelet packet coefficients, is proposed for texture synthesis. Experimental results show that the TSWPT algorithm is preferable, especially in terms of computation time.
Texture modeling can be effectively applied to a wide variety of natural surfaces such as plants, furs, skins, minerals, terrains, and fractal materials [
Given an example texture, the goal of texture synthesis is to produce a visually similar image of any size. One may easily tile small textures to synthesize a larger image; however, there are some blocking effects near the tile edges [
Wavelet transform provides an efficient multi-resolution representation [
For images with textures, lots of wavelet coefficients are likely to be significant in the middle-high frequency subbands, which surely demand further decompositions for a more compact representation [
The remainder of this paper proceeds as follows. In Section
Wavelet-transform- (WT-) based multiresolution analysis/synthesis has drawn a lot of attention to the signal/image/video applications. The extension of WT known as wavelet packet transform (WPT) provides a much larger family of bases functions with a more compact representation. In this section, a brief review of WT and WPT is given. For a more complete survey, we refer readers to [
WT has a variety of desirable properties, for example, joint space-spatial frequency localization, self-similarity across subbands of the same orientation, and energy clustering within each subband. For a discrete signal at resolution
For the image applications, 2D WT can be obtained by using a tensor product of two 1D WTs, horizontally followed by vertically or vice versa. Specifically, let
2-level wavelet transform (solid boundaries), wavelet packet transform (dashed boundaries), and a wavelet packet tree with wavelet packet coefficients connected by arrows.
2D WT is only to decompose the lowest-frequency subband of an image in an iterative manner. More specifically, only the scaling coefficients are decomposed from higher to lower resolutions. However, for the texture applications, wavelet coefficients in the middle- and high-frequency subbands are likely to be significant, which needs to be taken into account to improve the multiresolution representation.
As one can see, both the low-frequency scaling coefficients and high-frequency wavelet coefficients of a signal, at any resolution, can be decomposed, which leads to wavelet packet transform (WPT), and a much larger family of bases functions can be produced [
Wavelet packet transform provides more bases functions, which leads to a more compact representation in comparison with wavelet transform. For the image-coding applications, we had proposed an efficient scheme to organize the wavelet packet (WP) coefficients of an image into hierarchical trees called WP trees [
The WP coefficients of a sequence of wavelet coefficients
The key idea behind the construction of WP trees is based on the spatial relationships of WP coefficients. It has the same structure, that is, quad-tree structure, as the conventional wavelet trees. Furthermore, the number of high-energy wavelet coefficients can be significantly reduced through the use of wavelet packet transform. Take the rice image shown in Figure
Cumulative energy distributions (CEDs) of the wavelet coefficients (solid line) and wavelet packet coefficients (dashed line) of the rice image shown in Figure
As noted, the low-frequency WP coefficients retain the global information of an image, and the high-frequency WP coefficients contain the local detail. It is desirable to coarsely synthesize an image based on the low-frequency WP coefficients and then tune the intermediate synthesis result based on the high-frequency WP coefficients. Motivated by the fact above, we propose an efficient WP-tree-based texture synthesis algorithm using a two-step process: a coarse searching followed by a fine tuning. Figure
Decompose the source image by wavelet packet transform, rearrange the high-frequency WP coefficients, and construct the WP trees. Randomly select a WP tree from the source image, which is replicated in the upper left corner of the synthesis image.
For every WP tree to be synthesized, starting from the upper left corner to the lower right corner of the synthesis image, search the candidate WP trees from the source image by using a coarse matching with a tolerance as follows:
After the coarse matching in Step
Repeat Step
Finally, the synthesis image is obtained by taking the inverse wavelet packet transform of the synthesis WP trees.
To reduce the synthesis time, one can easily modify Steps
Flowchart of the TSWPT algorithm.
In the first experiment, the size of source images is
Visual comparison. Source images (1st column), synthesis images using Efros’ algorithm [
Synthesis images at double the size of the source image (top) using Efros’ algorithm [
Two widely used algorithms, Efros’ algorithm [
Comparison of computation times.
Efros and freeman [ |
Cui et al. [ | TSWPT | |
---|---|---|---|
Image 1 | 3.784 s | 0.703 s | 0.416 s |
Image 2 | 3.893 s | 0.854 s | 0.425 s |
Image 3 | 3.939 s | 0.810 s | 0.404 s |
Image 4 | 3.914 s | 0.737 s | 0.372 s |
Image 5 | 3.891 s | 0.731 s | 0.392 s |
In the second experiment, the sizes of source image and synthesis image are
Synthesis times of the texture image (Figure
The multi-resolution approach is suitable for texture synthesis in terms of computation time. Wavelet packet transform provides more bases functions than wavelet transform and therefore produces a more compact representation. We adopt wavelet packet transform to analyze the input texture and organize wavelet packet coefficients to form hierarchical trees called wavelet packet trees. The low-frequency nodes of wavelet packet trees contain the global characteristics of an image; the high-frequency nodes contain the local details. Thus, we propose texture synthesis based on wavelet packet tree (TSWPT). It has the advantage of saving computation time dramatically, and moreover, no training process is needed. Given a
The National Science Council of Taiwan, under Grants NSC100-2628-E-239-002-MY2, supported this work.