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A colorization-based anisotropic variational model for vector-valued image compression

  • *Corresponding author: Dazhi Zhang

    *Corresponding author: Dazhi Zhang 
Abstract Full Text(HTML) Figure(11) / Table(5) Related Papers Cited by
  • Image compression is an important technology in digital image processing. In this paper, a novel colorization-based codec for vector-valued images is proposed. In compression, we first define the concept of "structure image", which contains rich geometric structure information of the vector-valued image. Then, to extract representative pixels from the original vector-valued image, a "one-iteration method" is proposed. It can tremendously improve compression efficiency. In decompression, starting from colorizing the structure image, an anisotropic variational model is proposed. The existence and uniqueness of minimizers for the proposed variational model are established. Besides, we develop a fast and efficient algorithm for solving the model numerically by employing the scaled form of the alternating direction method of multipliers (ADMM). Numerical experiments on natural color images demonstrate that the proposed method outperforms the state-of-art colorization-based image compression method. Compared with the transform-based approaches, experiments on satellite multispectral images illustrate that the proposed method is superior to the JPEG and JPEG2000 standards.

    Mathematics Subject Classification: Primary: 68U10, 94A08; Secondary: 32A70.


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  • Figure 1.  $ \rm (a) $ Original images in the $ \rm RGB $ color space. $ \rm (b) $ Structure images obtained by taking $ (\alpha_1, \alpha_2, \alpha_3) = (0.299, 0.587, 0.114) $. $ \rm (c) $ Structure images obtained by taking a random set, $ \left(\alpha_1, \alpha_2, \alpha_3 \right) = (0.05, 0.3, 0.65) $. $ \rm (d) $ The difference between structure images (b) and (c). Note that the pixel values have been rearranged to [0, 255]

    Figure 2.  $ \rm (a) $ When $ (\alpha_1, \alpha_2, \alpha_3) = (0.299, 0.587, 0.114) $, the geometric structure information in the structure image. $ \rm (b) $ The geometric structure information contained in the structure image, when $ \left(\alpha_1, \alpha_2, \alpha_3 \right) = (0.05, 0.3, 0.65) $. $ \rm (c) $ The difference of geometric structure information between (a) and (b). Pixel values have been rearranged into $ [0, 255] $ for better visibility

    Figure 3.  Results of image change rate for Kodak images [19] $ (07, 23, 24) $ calculated by different methods

    Figure 4.  Extracting representative pixels from the $ 96\times96 $ $ 24 $ bit depth "Cap" image

    Figure 5.  The Kodak images $ 06, 09, 15 $ of $ 384 \times 256 $ pixels used in the experimentation. The second line is the enlarged images

    Figure 6.  Decompression results for the Kodak images $ \rm 06, 09, $ and $ \rm 15 $. The even rows are the enlarged images

    Figure 7.  Decompression results for Kodak image $ \rm 15 $ using different methods

    Figure 8.  Plots of energy and PSNR values for decompressing the Kodak image $ \rm 15 $

    Figure 9.  $ \rm (a) $ Original false-color image for the Los Angeles multispectral image set, which is generated by assigning the spectral band $ \rm B4 $ to red, $ \rm B3 $ to green, and $ \rm B2 $ to blue. $ \rm (b)–(f) $ Original images for the spectral bands $ \rm B1-B5 $. The size of these images is $ 512\times512 $ $ 8 $ bit depth. $ \rm (g)–(h) $ Original images for the thermal bands $ \rm B61 $ and $ \rm B62 $. The size of these two images is $ 512\times512 $ $ 8 $ bit depth

    Figure 10.  $ \rm (a) $ Structure image for the Los Angeles image thermal bands (B61 and $ \rm B62) $. $ \rm (b) $ Structure image for the Los Angeles image spectral bands $ \rm B1-B5 $ with $ \alpha $ = (0.2040, 0.2028, 0.2012, 0.2097, 0.1823). $ \rm (c) $ Structure image for the Los Angeles image spectral bands $ \rm B1-B5 $ with $ \alpha = (0.1912, 0.3681, 0.1305, 0.2244, 0.0858) $

    Figure 11.  Decompression results for the Los Angeles images. {(a), (c) and (e)}: Decompressed thermal band $ \rm B62 $ using the JPEG standard, the JPEG2000 standard and the proposed method, respectively. {(b), (d) and (f)}: Decompressed spectral band $ \rm B3 $ using the JPEG standard, the JPEG2000 standard and the proposed method, respectively

    Table 1.  The computational time (sec) of the proposed "one-iteration method" is compared with the local-optimal strategy in [43] for extracting representative pixels

    Image Size Number of representative pixels Time of One-iteration method (sec) Time of Local-optimal strategy (sec)
    05 $ 384 \times 256 $ 546 6.8 18.3
    23 $ 768 \times 512 $ 1404 15.9 62.5
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    Table 2.  The computational time (sec) taken by different methods

    Number of representative pixels Time of reaction-diffusion equations (21) (sec) Time of Algorithm 2 (sec))
    184 15.7 3.3
    394 21.6 3.6
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    Table 3.  Decompression results of different methods. The PSNR values are shown in Table 3(a). Table 3(b) gives the SSIM values of the decompressed images. For the image to be tested, we first calculate the index value of each channel and then average the values of all channels to obtain the PSNR or SSIM values of the decompressed image. Here the notation $ \rm CR $ denotes the compression ratio

    (a) PSNR
    Image CR JPEG JPEG2000 Shi et al. Proposed
    06 20:1 29.5755 31.5652 30.5245 30.6496
    09 41:1 28.0773 30.3243 29.2732 29.5357
    15 61:1 25.7227 28.3352 26.8622 27.4383
    (b) SSIM
    Image CR JPEG JPEG2000 Shi et al. Proposed
    06 20:1 0.8387 0.8702 0.8556 0.8566
    09 41:1 0.8173 0.8816 0.8704 0.8751
    15 61:1 0.7247 0.8086 0.7985 0.8112
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    Table 4.  PSNR values of the two thermal bands (B61 and B62) using different decompression methods. Here the notation $ \rm CR $ denotes the compression ratio

    Thermal bands CR JPEG JPEG2000 Proposed
    B61 10:1 44.5696 44.7990 47.0500
    B62 10:1 40.8826 41.3790 42.9845
    Avg. 42.7261 43.0890 45.0172
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    Table 5.  PSNR values of the spectral bands $ \rm B1-B5 $ using different decompression methods. Here the notation $ \rm CR $ denotes the compression ratio

    Spectral bands CR JPEG JPEG2000 Proposed ($\alpha$ _rand) Proposed
    B1 27:1 32.5552 32.8792 33.5227 33.4983
    B2 27:1 31.6115 31.8138 34.4564 33.8015
    B3 27:1 28.9824 29.4833 31.1467 31.4802
    B4 27:1 30.5670 31.1113 28.8501 29.5729
    B5 27:1 27.7357 28.1368 27.6438 28.6967
    Avg. 30.2903 30.6848 31.1239 31.4103
     | Show Table
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