# Oriented Wavelet Transform

 Oriented Wavelet Transform

contact: V. ChappelierC. Guillemot

During the course of my PhD at TEMICS, I have designed a new transform for image compression and denoising. The idea behind this transform simply consists in applying the lifting steps of a 1D wavelet transform in the direction of the image contours. Using quincunx multiresolution sampling, the image is filtered along horizontal and vertical or diagonal and antidiagonal directions. This results in better energy compaction in the lowest subband compared to the separable wavelet. The figure below illustrates which part of the spectrum is conserved in the low-pass subband (of same size) for both transform:

## Image compression

This transform has similar complexity as the separable wavelet transform while providing better energy compaction and staying critically sampled, which makes it a good candidate for compression applications. The increased decorrelation offers relatively good performance compared to state-of-the-art coder, even without complex adaptive high order models. To illustrate this, two simple separable wavelet and oriented wavelet codecs have been designed, using a non-adaptive first-order generalized Gaussian model for each subband. While the performance of this coder could be improved by using rate-distorsion optimization or better quantization techniques like TCQ, it performs within 1 dB from the state-of-the-art coders. Besides, the model is simple enough to apply efficient error resilient entropy coding, which is another research topic of the team.

The figure below illustrate images obtained from this coder combined with either the separable wavelet or oriented wavelet transform. To view the full size images, click on the thumbnails. The corresponding compressed files are also available for download by clicking on the corresponding PSNR result.

 Original (512x512 8bpp)

 JPEG Compression [The GIMP] 0.25 bpp 0.50 bpp PSNR 31.8 dB PSNR 25.4 dB PSNR 22.5 dB PSNR 35.2 dB PSNR 28.5 dB PSNR 24.1 dB

 1st-order Subband Compression 0.25 bpp 0.50 bpp separable wavelets (5 levels) [wavelet_codec] PSNR 33.5 dB PSNR 26.6 dB PSNR 23.0 dB PSNR 36.6 dB PSNR 29.9 dB PSNR 25.2 dB oriented wavelets (10 levels) [owavelet_codec] PSNR 34.3 dB PSNR 27.7 dB PSNR 23.3 dB PSNR 37.3 dB PSNR 31.2 dB PSNR 25.6 dB

Of course, this transform can also be combined with more advanced subband coders such as EBCOT, which is used in the JPEG2000 standard. The following figure compares the separable wavelet transform with the standardized oriented wavelet tarnsform. The codec for these experiments is based on OpenJPEG 0.95, which is a JPEG2000 compliant codec.

 EBCOT Subband Compression 0.25 bpp 0.50 bpp separable wavelets (5 levels) [OpenJPEG] PSNR 34.3 dB PSNR 28.3 dB PSNR 23.2 dB PSNR 37.5 dB PSNR 32.1 dB PSNR 25.5 dB oriented wavelets (10 levels) [owavelet_ebcot_codec] PSNR 34.7 dB PSNR 28.8 dB PSNR 23.5 dB PSNR 37.6 dB PSNR 32.6 dB PSNR 25.7 dB

## Denoising

Filtering along the image contours allows to remove the noise more efficiently than anisotropic techniques like the ones based on separable wavelets. The following figure illustrates the denoising results obtained from separable and oriented wavelet tranforms for additive white Gaussian noise of standard deviation 25. In this experiment, the orientation map is modeled as an Hidden Markov Field. The results for separable wavelets (DWT) and oriented wavelets (OWT) are for 5 and 10 levels of decomposition respectively.

noisy DWT OWT FOE BLS-GSM
PSNR = 20.24 dB PSNR = 29.86 dB PSNR = 30.41 dB PSNR = 30.81 dB PSNR = 31.73 dB

## Codecs

Only Linux binaries are currently provided here. The owavelet+EBCOT coder is based on OpenJPEG, which is provided under a BSD licence (reproduced in the README file) by the Communications and remote sensing Laboratory from the UCL. The programs provided herein are for scientific use only. Commercial use of these softwares, in particular, is prohibited without authorization from the author. These softwares are provided in the hope that they will be useful, and without any warranty of any kind.

encoderdecodercodecusage
separable wavelet wavelet_encoder wavelet_decoder wavelet_codec README
oriented wavelet owavelet_encoder owavelet_decoder owavelet_codec README
oriented wavelet + EBCOT owavelet_ebcot_encoder owavelet_ebcot_decoder owavelet_ebcot_codec README
 Webmaster: temics_web (at) irisa.fr Last time modified: 2016-02-14

## References

• Vivien Chappelier (Oct. 02 - Sept. 05), Codage progressif d'images par ondelettes orientees,PhD thesis, Univ. of Rennes 1, Dec. 2005, (pdf)
• V. Chappelier and C. Guillemot, Oriented wavelet transform for image compression and de-noising, IEEE Trans. on Image Processing, Vol. 15, No. 10, pp. 2892-2903, Oct. 2006. (pdf)
• V. Chappelier, C. Guillemot, Oriented 1D wavelet transform on a quincunx pyramid for image compression, Proc. IEEE Int. conf. on Image Processing, ICIP, Genoa, Sept. 2005. (pdf)
• V. Chappelier, C. Guillemot, and S. Marinkovic, Image coding with iterated contourlet and wavelet transforms, Proc. IEEE International Conference on Image Processing, ICIP, Singapore, Oct. 2004. (pdf)
• V. Chappelier, C. Guillemot, et S. Marinkovic, Turbo Trellis Coded Quantization, International symposium on turbo codes, Brest, Sept. 2003 (pdf)