{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:26:05Z","timestamp":1760235965699,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,10,16]],"date-time":"2021-10-16T00:00:00Z","timestamp":1634342400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Block compressed sensing (BCS) is a promising technology for image sampling and compression for resource-constrained applications, but it needs to balance the sampling rate and quantization bit-depth for a bit-rate constraint. In this paper, we summarize the commonly used CS quantization frameworks into a unified framework, and a new bit-rate model and a model of the optimal bit-depth are proposed for the unified CS framework. The proposed bit-rate model reveals the relationship between the bit-rate, sampling rate, and bit-depth based on the information entropy of generalized Gaussian distribution. The optimal bit-depth model can predict the optimal bit-depth of CS measurements at a given bit-rate. Then, we propose a general algorithm for choosing sampling rate and bit-depth based on the proposed models. Experimental results show that the proposed algorithm achieves near-optimal rate-distortion performance for the uniform quantization framework and predictive quantization framework in BCS.<\/jats:p>","DOI":"10.3390\/e23101354","type":"journal-article","created":{"date-parts":[[2021,10,17]],"date-time":"2021-10-17T23:13:32Z","timestamp":1634512412000},"page":"1354","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A General Rate-Distortion Optimization Method for Block Compressed Sensing of Images"],"prefix":"10.3390","volume":"23","author":[{"given":"Qunlin","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mechatronical Engineering, Beijing Institute of Technology, Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Derong","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mechatronical Engineering, Beijing Institute of Technology, Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiulu","family":"Gong","sequence":"additional","affiliation":[{"name":"School of Mechatronical Engineering, Beijing Institute of Technology, Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1109\/TIT.2005.862083","article-title":"Robust Uncertainty Principles: Exact Signal Frequency Information","volume":"52","author":"Romberg","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TIT.2006.871582","article-title":"Compressed sensing","volume":"52","author":"Donoho","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1109\/MSP.2007.4286571","article-title":"Compressive sensing","volume":"24","author":"Baraniuk","year":"2007","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Wakin, M.B., Laska, J.N., Duarte, M.F., Baron, D., Sarvotham, S., Takhar, D., Kelly, K.F., and Baraniuk, R.G. (2006, January 8\u201311). An architecture for compressive imaging. Proceedings of the International Conference on Image Processing, Atlanta, GA, USA.","DOI":"10.1109\/ICIP.2006.312577"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1109\/MSP.2007.914729","article-title":"Imaging via compressive sampling","volume":"25","author":"Romberg","year":"2008","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1109\/MSP.2007.915001","article-title":"Compressive sampling and lossy compression: Do random measurements provide an efficient method of representing sparse signals?","volume":"25","author":"Goyal","year":"2008","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"23398","DOI":"10.3390\/s141223398","article-title":"Efficient lossy compression for compressive sensing acquisition of images in compressive sensing imaging systems","volume":"14","author":"Li","year":"2014","journal-title":"Sensors"},{"key":"ref_8","unstructured":"Gan, L. (2007, January 1\u20134). Block compressed sensing of natural images. Proceedings of the 15th International Conference on Digital Signal Processing, Wales, UK."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/j.jvcir.2017.01.028","article-title":"Block compressive sensing: Individual and joint reconstruction of correlated images","volume":"44","author":"Unde","year":"2017","journal-title":"J. Vis. Commun. Image Represent."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.image.2018.03.019","article-title":"Rate\u2013distortion analysis of structured sensing matrices for block compressive sensing of images","volume":"65","author":"Unde","year":"2018","journal-title":"Signal Process. Image Commun."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Fletcher, A.K., Rangan, S., and Goyal, V.K. (2007, January 15\u201320). On the rate-distortion performance of compressed sensing. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, Honolulu, HI, USA.","DOI":"10.1109\/ICASSP.2007.366822"},{"key":"ref_12","unstructured":"Mun, S., and Fowler, J.E. (2012, January 27\u201331). DPCM for quantized block-based compressed sensing of images. Proceedings of the 20th European Signal Processing Conference, Bucharest, Romania."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2980","DOI":"10.1109\/TIP.2012.2188810","article-title":"Binned progressive quantization for compressive sensing","volume":"21","author":"Wang","year":"2012","journal-title":"IEEE Trans. Image Process."},{"key":"ref_14","first-page":"1610","article-title":"Efficient and Robust Image Coding and Transmission Based on Scrambled Block Compressive Sensing","volume":"20","author":"Chen","year":"2018","journal-title":"IEEE Trans. Multimed."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1109","DOI":"10.1109\/TCSVT.2019.2898908","article-title":"Compressive Sensing Multi-Layer Residual Coefficients for Image Coding","volume":"30","author":"Chen","year":"2020","journal-title":"IEEE Trans. Circuits Syst. Video Technol."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Li, X., Lan, X., Yang, M., Xue, J., and Zheng, N. (2013, January 17\u201320). Universal and low-complexity quantizer design for compressive sensing image coding. Proceedings of the 2013 Visual Communications and Image Processing, Kuching, Malaysia.","DOI":"10.1109\/VCIP.2013.6706403"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Zhang, J., Zhao, D., and Jiang, F. (2013, January 15\u201318). Spatially directional predictive coding for block-based compressive sensing of natural images. Proceedings of the 2013 IEEE International Conference on Image Processing, Melbourne, Australia.","DOI":"10.1109\/ICIP.2013.6738211"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"5789","DOI":"10.1109\/TSP.2013.2280445","article-title":"Analysis-by-synthesis quantization for compressed sensing measurements","volume":"61","author":"Shirazinia","year":"2013","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1109\/LSP.2017.2770018","article-title":"Optimizing Quantization for Lasso Recovery","volume":"25","author":"Gu","year":"2018","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3496","DOI":"10.1109\/TSP.2012.2194710","article-title":"Regime change: Bit-depth versus measurement-rate in compressive sensing","volume":"60","author":"Laska","year":"2012","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Chen, Q., Chen, D., Gong, J., and Ruan, J. (2020). Low-complexity rate-distortion optimization of sampling rate and bit-depth for compressed sensing of images. Entropy, 22.","DOI":"10.3390\/e22010125"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/s11265-015-1087-0","article-title":"The Rate-Distortion Optimized Compressive Sensing for Image Coding","volume":"86","author":"Jiang","year":"2017","journal-title":"J. Signal Process. Syst."},{"key":"ref_23","first-page":"1549","article-title":"Joint sampling rate and bit-depth optimization in compressive video sampling","volume":"16","author":"Liu","year":"2014","journal-title":"IEEE Trans. Multimed."},{"key":"ref_24","unstructured":"Huang, J., and Mumford, D. (1999, January 23\u201325). Statistics of natural images and models. Proceedings of the 1999 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Fort Collins, CO, USA."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1661","DOI":"10.1109\/83.869177","article-title":"A mathematical analysis of the DCT coefficient distributions for images","volume":"9","author":"Lam","year":"2000","journal-title":"IEEE Trans. Image Process."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Pudi, V., Chattopadhyay, A., and Lam, K.Y. (2018, January 27\u201330). Efficient and Lightweight Quantized Compressive Sensing using \u03bc-Law. Proceedings of the IEEE International Symposium on Circuits and Systems, Florence, Italy.","DOI":"10.1109\/ISCAS.2018.8351505"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"600","DOI":"10.1109\/TIP.2003.819861","article-title":"Image quality assessment: From error visibility to structural similarity","volume":"13","author":"Wang","year":"2004","journal-title":"IEEE Trans. Image Process."},{"key":"ref_28","first-page":"172","article-title":"Rate Control in DCT Video Coding for Low-Delay Communications","volume":"9","year":"1997","journal-title":"IEEE Trans. Circuits Syst. Video Technol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"656","DOI":"10.1109\/83.568923","article-title":"Lossless compression of continuous-tone images via context selection, quantization, and modeling","volume":"6","author":"Wu","year":"1997","journal-title":"IEEE Trans. Image Process."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Qian, C., Zheng, B., and Lin, B. (2015, January 15\u201317). Nonuniform quantization for block-based compressed sensing of images in differential pulse-code modulation framework. Proceedings of the 2014 2nd International Conference on Systems and Informatics, Shanghai, China.","DOI":"10.1109\/ICSAI.2014.7009392"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1109\/97.763145","article-title":"Estimation based on entropy matching for generalized Gaussian PDF modeling","volume":"6","author":"Aiazzi","year":"1999","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_32","unstructured":"Kennedy, J., and Eberhart, R. (December, January 27). Particle swarm optimization. Proceedings of the IEEE International Conference on Neural Networks, Perth, Australia."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s00500-016-2474-6","article-title":"Particle swarm optimization algorithm: An overview","volume":"22","author":"Wang","year":"2018","journal-title":"Soft Comput."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/S0169-7439(97)00061-0","article-title":"Introduction to multi-layer feed-forward neural networks","volume":"39","author":"Svozil","year":"1997","journal-title":"Chemom. Intell. Lab. Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1109\/72.557662","article-title":"Capabilities of a four-layered feedforward neural network: Four layers versus three","volume":"8","author":"Tamura","year":"1997","journal-title":"IEEE Trans. Neural Networks"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1016\/S0924-0136(00)00498-2","article-title":"BP-neural network predictor model for plastic injection molding process","volume":"103","author":"Sadeghi","year":"2000","journal-title":"J. Mater. Process. Technol."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1186\/s13640-017-0184-3","article-title":"Lossy image compression based on prediction error and vector quantisation","volume":"2017","author":"Ayoobkhan","year":"2017","journal-title":"Eurasip J. Image Video Process."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"898","DOI":"10.1109\/TPAMI.2010.161","article-title":"Contour detection and hierarchical image segmentation","volume":"33","author":"Maire","year":"2011","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1007\/s11263-008-0197-6","article-title":"Fields of experts","volume":"82","author":"Roth","year":"2009","journal-title":"Int. J. Comput. Vis."},{"key":"ref_40","unstructured":"Skretting, K. (2020, October 13). Huffman Coding and Arithmetic Coding. Available online: https:\/\/www.mathworks.com\/matlabcentral\/fileexchange\/2818-huffman-coding-and-arithmetic-coding."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Mun, S., and Fowler, J.E. (2009, January 7\u201310). Block compressed sensing of images using directional transforms. Proceedings of the 16th IEEE International Conference on Image Processing, Cairo, Egypt.","DOI":"10.1109\/DCC.2010.90"},{"key":"ref_42","first-page":"59","article-title":"Thirteen ways to look at the correlation coefficient","volume":"42","year":"1988","journal-title":"Am. Stat."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/10\/1354\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:16:16Z","timestamp":1760166976000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/10\/1354"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,16]]},"references-count":42,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2021,10]]}},"alternative-id":["e23101354"],"URL":"https:\/\/doi.org\/10.3390\/e23101354","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2021,10,16]]}}}