We propose an algorithm for lossy compression of computer-aided design models in Z-map representation. Our method employs Daubechies wavelet functions, which are smoother than those of the Haar wavelet used in a previous work for the lossy compression of shape models. A significant reduction in the amount of data of the compressed shape model was achieved using the proposed lossy in which the least significant coefficients of the wavelet synopsis were deleted. The nonlinear filtering of coefficients was based on the quickselect algorithm, which was seven to ten times faster than a normal quicksort algorithm and allowed us to accelerate the entire process. We conducted a series of experiments using shape models with 512 × 512–8192 × 8192 resolutions to evaluate our technique using various wavelet functions. The proposed method performed the process in 50–90 ms for the models at 1024 × 1024 resolution and reduced the output binary size by 75%–90% compared with those compressed using a previous method. Some Daubechies wavelets, such as D4 and D6, were found superior in lossy compression using nonlinear filtering based on the order of magnitude of wavelet coefficients.

1. [1] M. Inui and R. Ishizuka, “Data compression method of Z-map model representing milling result shape,” Proc. of Int. Symp. on Flexible Automation (ISFA), pp. 343-348, 2006.
2. [2] M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, “Image coding using vector quantization in the wavelet transform domain,” Int. Conf. on Acoustics, Speech, and Signal Processing, Vol.4, pp. 2297-2300, 1990. https://doi.org/10.1109/ICASSP.1990.116036
3. [3] M. W. Marcellin, M. J. Gormish, A. Bilgin, and M. P. Bollek, “An overview of JPEG-2000,” Proc. of Data Compression Conf. 200 (DCC), pp. 523-541, 2000. https://doi.org/10.1109/DCC.2000.838192
4. [4] A. Lucero, S. D. Cabrera, and E. Vidal, “On the use of JPEG 2000 to achieve minimum L-infinity error when specifying a compression ratio,” 2007 IEEE Int. Conf. on Image Processing, pp. II-361-II-364, 2007. https://doi.org/10.1109/ICIP.2007.4379167
5. [5] M. O. Benouamer and D. Michelucci, “Bridging the gap between CSG and Brep via a triple ray representation,” Proc. of the 4th ACM Symp. on Solid Modeling and Applications (SMA’97), pp. 68-79, 1997. https://doi.org/10.1145/267734.267755
6. [6] M. Garofalakis and P. B. Gibbons, “Wavelet synopses with error guarantees,” Proc. of the 2002 ACM SIGMOD Int. Conf. on Management of Data, pp. 476-487, 2002. https://doi.org/10.1145/564691.564746
7. [7] M. Garofalakis and A. Kumar, “Deterministic wavelet thresholding for maximum-error metrics,” Proc. of the 23rd ACM SIGMOD-SIGACT-SIGART Symp. on Principles of Database Systems, pp. 166-176, 2004. https://doi.org/10.1145/1055558.1055582
8. [8] P. Karras and N. Mamoulis, “One-pass wavelet synopses for maximum-error metrics,” Proc. of the 31st Int. Conf. on Very Large Data Bases, pp. 421-432, 2005.
9. [9] R. A. DeVore, B. Jawerth, and B. J. Lucier, “Image compression through wavelet transform coding,” IEEE Trans. on Information Theory, Vol.38, No.2, pp. 719-746, 1992. https://doi.org/10.1109/18.119733
10. [10] S. G. Chang, B. Yu, and M. Vetterli, “Adaptive wavelet thresholding for image denoising and compression,” IEEE Trans. on Image Processing, Vol.9, No.9, pp. 1532-1546, 2000. https://doi.org/10.1109/83.862633
11. [11] A. Alecu, A. Munteanu, P. Schelkens, J. Cornelis, and S. Dewitte, “On the optimality of embedded deadzone scalar-quantizers for wavelet-based L-infinite-constrained image coding,” Proc. of Data Compression Conf., p. 413, 2003. https://doi.org/10.1109/DCC.2003.1194032
12. [12] S. Guha and B. Harb, “Approximation algorithms for wavelet transform coding of data streams,” IEEE Trans. on Information Theory, Vol.54, No.2, pp. 811-830, 2008. https://doi.org/10.1109/TIT.2007.913569
13. [13] A. J. Pinho and A. J. R. Neves, “L-infinity progressive image compression,” Proc. of the Picture Coding Symp., 2007.
14. [14] A. J. Pinho and A. J. R. Neves, “Progressive lossless compression of medical images,” 2009 IEEE Int. Conf. on Acoustics, Speech and Signal Processing, pp. 409-412, 2009. https://doi.org/10.1109/ICASSP.2009.4959607
15. [15] J. Beerten, I. Blanes, and J. Serra-Sagristà, “A fully embedded two-stage coder for hyperspectral near-lossless compression,” IEEE Geoscience and Remote Sensing Letters, Vol.12, No.8, pp. 1775–1779, 2015. https://doi.org/10.1109/LGRS.2015.2425548
16. [16] X. Zhang and X. Wu, “Ultra high fidelity deep image decompression with l_∞-constrained compression,” IEEE Trans. on Image Processing, Vol.30, pp. 963-975, 2021. https://doi.org/10.1109/TIP.2020.3040074
17. [17] M. Bertram, M. A. Duchaineau, B. Hamann, and K. I. Joy, “Generalized B-spline subdivision-surface wavelets for geometry compression,” IEEE Trans. on Visualization and Computer Graphics, Vol.10, No.3, pp. 326-338, 2004. https://doi.org/10.1109/TVCG.2004.1272731
18. [18] S. Valette and R. Prost, ”Wavelet-based progressive compression scheme for triangle meshes: Wavemesh,” IEEE Trans. on Visualization and Computer Graphics, Vol.10, No.2, pp. 123-129, 2004. https://doi.org/10.1109/TVCG.2004.1260764
19. [19] F. Payan and M. Antonini, “Temporal wavelet-based compression for 3D animated models,” Computers & Graphics, Vol.31, No.1, pp. 77-88, 2007. https://doi.org/10.1016/j.cag.2006.09.009
20. [20] N. Umezu, K. Asai, and M. Inui, “Wavelet transform data compression with an error level guarantee for Z-map models,” Int. J. Automation Technol., Vol.10, No.2, pp. 201-208, 2016. https://doi.org/10.20965/ijat.2016.p0201
21. [21] N. Umezu, K. Yokota, and M. Inui, “2D wavelet transform data compression with error level guarantee for Z-map models,” J. of Computational Design and Engineering, Vol.4, No.3, pp. 238-247, 2017. https://doi.org/10.1016/j.jcde.2017.04.002
22. [22] J.-L. Gailly and M. Adler, “zlib.” https://www.zlib.net/ [Accessed January 31, 2024]
23. [23] J. Seward, “bzip2 and libbzip2.” https://sourceware.org/bzip2/index.html [Accessed January 31, 2024]
24. [24] Institute of Electrical and Electronics Engineers (IEEE), “754-2008: IEEE standard for floating-point arithmetic,” 2008. https://doi.org/10.1109/IEEESTD.2008.4610935
25. [25] M. Galassi et al., “GSL Scientific Library Release 2.7,” 2021. https://www.gnu.org/software/gsl/doc/latex/gsl-ref.pdf [Accessed January 31, 2024]
26. [26] C. Li, S. Bedi, and S. Mann, “Flank millable surface design with conical and barrel tools,” Computer-Aided Design and Applications, Vol.5, Nos.1-4, pp. 461-470, 2008. https://doi.org/10.3722/cadaps.2008.461-470
27. [27] P.-Y. Pechard, C. Tournier, C. Lartigue, and J.-P. Lugarini, “Geometrical deviations versus smoothness in 5-axis high-speed flank milling,” Int. J. of Machine Tools and Manufacture, Vol.49, No.6, pp. 454-461, 2009. https://doi.org/10.1016/j.ijmachtools.2009.01.005
28. [28] J. Senatore, S. Segonds, W. Rubio, and G. Dessein, “Correlation between machining direction, cutter geometry and step-over distance in 3-axis milling: Application to milling by zones,” Computer-Aided Design, Vol.44, No.12, pp. 1151-1160, 2012. https://doi.org/10.1016/j.cad.2012.06.008
29. [29] A. Calleja, P. Bo, H. González, M. Bartoň, and L. N. López de Lacalle, “Highly accurate 5-axis flank CNC machining with conical tools,” The Int. J. of Advanced Manufacturing Technology, Vol.97, No.5, pp. 1605-1615, 2018. https://doi.org/10.1007/s00170-018-2033-7
30. [30] P. Bo, M. Bartoň, and H. Pottmann, “Automatic fitting of conical envelopes to free-form surfaces for flank CNC machining,” Computer-Aided Design, Vol.91, pp. 84-94, 2017. https://doi.org/10.1016/j.cad.2017.06.006
31. [31] M. Inui, K. Kaba, and N. Umezu, “Fast dexelization of polyhedral models using ray-tracing cores of GPU,” Computer-Aided Design and Applications, Vol.18, No.4, pp. 786-798, 2021. https://doi.org/10.14733/cadaps.2021.786-798