{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T05:22:31Z","timestamp":1768454551308,"version":"3.49.0"},"reference-count":57,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,6]],"date-time":"2021-01-06T00:00:00Z","timestamp":1609891200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100016394","name":"ANR","doi-asserted-by":"publisher","award":["ANR-17-CE38-0009"],"award-info":[{"award-number":["ANR-17-CE38-0009"]}],"id":[{"id":"10.13039\/501100016394","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The estimation of image quality and noise perception still remains an important issue in various image processing applications. It has also become a hot topic in the field of photo-realistic computer graphics where noise is inherent in the calculation process. Unlike natural-scene images, however, a reference image is not available for computer-generated images. Thus, classic methods to assess noise quantity and stopping criterion during the rendering process are not usable. This is particularly important in the case of global illumination methods based on stochastic techniques: They provide photo-realistic images which are, however, corrupted by stochastic noise. This noise can be reduced by increasing the number of paths, as proved by Monte Carlo theory, but the problem of finding the right number of paths that are required in order to ensure that human observers cannot perceive any noise is still open. Until now, the features taking part in the human evaluation of image quality and the remaining perceived noise are not precisely known. Synthetic image generation tends to be very expensive and the produced datasets are high-dimensional datasets. In that case, finding a stopping criterion using a learning framework is a challenging task. In this paper, a new method for characterizing computational noise for computer generated images is presented. The noise is represented by the entropy of the singular value decomposition of each block composing an image. These Singular Value Decomposition (SVD)-entropy values are then used as input to a recurrent neural network architecture model in order to extract image noise and in predicting a visual convergence threshold of different parts of any image. Thus a new no-reference image quality assessment is proposed using the relation between SVD-Entropy and perceptual quality, based on a sequence of distorted images. Experiments show that the proposed method, compared with experimental psycho-visual scores, demonstrates a good consistency between these scores and stopping criterion measures that we obtain.<\/jats:p>","DOI":"10.3390\/e23010075","type":"journal-article","created":{"date-parts":[[2021,1,6]],"date-time":"2021-01-06T09:15:16Z","timestamp":1609924516000},"page":"75","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Stopping Criterion during Rendering of Computer-Generated Images Based on SVD-Entropy"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6071-744X","authenticated-orcid":false,"given":"J\u00e9r\u00f4me","family":"Buisine","sequence":"first","affiliation":[{"name":"University of Littoral C\u00f4te d\u2019Opale (ULCO), LISIC, BP 719, 62228 Calais CEDEX, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3165-5363","authenticated-orcid":false,"given":"Andr\u00e9","family":"Bigand","sequence":"additional","affiliation":[{"name":"University of Littoral C\u00f4te d\u2019Opale (ULCO), LISIC, BP 719, 62228 Calais CEDEX, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4907-8813","authenticated-orcid":false,"given":"R\u00e9mi","family":"Synave","sequence":"additional","affiliation":[{"name":"University of Littoral C\u00f4te d\u2019Opale (ULCO), LISIC, BP 719, 62228 Calais CEDEX, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Samuel","family":"Delepoulle","sequence":"additional","affiliation":[{"name":"University of Littoral C\u00f4te d\u2019Opale (ULCO), LISIC, BP 719, 62228 Calais CEDEX, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christophe","family":"Renaud","sequence":"additional","affiliation":[{"name":"University of Littoral C\u00f4te d\u2019Opale (ULCO), LISIC, BP 719, 62228 Calais CEDEX, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kajiya, J.T. (1986, January 18\u201322). The rendering equation. Proceedings of the 13th Annual Conference on Computer Graphics and Interactive Techniques, (SIGGRAPH\u201986), Dallas, TX, USA.","DOI":"10.1145\/15922.15902"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/226150.226151","article-title":"Monte Carlo Techniques for Direct Lighting Calcultations","volume":"15","author":"Shirley","year":"1996","journal-title":"ACM Trans. Graph."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.ins.2013.12.029","article-title":"Feature selection with SVD entropy: Some modification and extension","volume":"264","author":"Banerjee","year":"2014","journal-title":"Inf. Sci."},{"key":"ref_4","unstructured":"Lafortune, E.P., and Willems, Y.D. (1993, January 5\u201310). Bi-Directional Path Tracing. Proceedings of the CompuGraphics, Alvor, Portugal."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Veach, E., and Guibas, L.J. (1997, January 3\u20138). Metropolis light transport. Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH\u201997), Los Angeles, CA, USA.","DOI":"10.1145\/258734.258775"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/2366145.2366214","article-title":"Adaptive rendering with non-local means filtering","volume":"31","author":"Rousselle","year":"2012","journal-title":"ACM Trans. Graph. (TOG)"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/2366145.2366213","article-title":"SURE-based optimization for adaptive sampling and reconstruction","volume":"31","author":"Li","year":"2012","journal-title":"ACM Trans. Graph. (TOG)"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/2641762","article-title":"Adaptive rendering based on weighted local regression","volume":"33","author":"Moon","year":"2014","journal-title":"ACM Trans. Graph. (TOG)"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Vorba, J., Hanika, J., Herholz, S., M\u00fcller, T., K\u0159iv\u00e1nek, J., and Keller, A. (August, January 28). Path guiding in production. Proceedings of the ACM SIGGRAPH 2019 Courses, Los Angeles, CA, USA.","DOI":"10.1145\/3305366.3328091"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1109\/LSP.2010.2043888","article-title":"A two-step framework for constructing blind image quality indices","volume":"17","author":"Moorthy","year":"2010","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3350","DOI":"10.1109\/TIP.2011.2147325","article-title":"Blind image quality assessment: From natural scene statistics to perceptual quality","volume":"20","author":"Moorthy","year":"2011","journal-title":"IEEE Trans. Image Process."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Venkatanath, N., Praneeth, D., Bh, M.C., Channappayya, S.S., and Medasani, S.S. (March, January 27). Blind image quality evaluation using perception based features. Proceedings of the 2015 Twenty First National Conference on Communications (NCC), IIT Bombay Mumbai, Maharashtra, India.","DOI":"10.1109\/NCC.2015.7084843"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ye, P., Kumar, J., Kang, L., and Doermann, D. (2013, January 23\u201328). Real-time no-reference image quality assessment based on filter learning. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Portland, OR, USA.","DOI":"10.1109\/CVPR.2013.132"},{"key":"ref_14","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_15","first-page":"1","article-title":"New Measurements Reveal Weaknesses of Image Quality Metrics in Evaluating Graphics Artifacts","volume":"Volume 31","author":"Herzog","year":"2012","journal-title":"ACM Transactions on Graphics (Proc. of SIGGRAPH Asia)"},{"key":"ref_16","unstructured":"Sheikh, H. (2021, January 02). LIVE Image Quality Assessment Database Release 2. Available online: http:\/\/live.ece.utexas.edu\/research\/quality."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.image.2014.10.009","article-title":"Image database TID2013: Peculiarities, results and perspectives","volume":"30","author":"Ponomarenko","year":"2015","journal-title":"Signal Process. Image Commun."},{"key":"ref_18","unstructured":"LIVE (2021, January 02). Liver Immersive Images. Available online: http:\/\/live.ece.utexas.edu\/research\/quality\/immersive_images\/."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1137\/040616024","article-title":"A review of image denoising algorithms, with a new one","volume":"4","author":"Buades","year":"2005","journal-title":"Multiscale Model. Simul."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"208","DOI":"10.5201\/ipol.2011.bcm_nlm","article-title":"Non-local means denoising","volume":"1","author":"Buades","year":"2011","journal-title":"Image Process. Line"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2024","DOI":"10.1109\/TIP.2005.859385","article-title":"Feature-based wavelet shrinkage algorithm for image denoising","volume":"14","author":"Balster","year":"2005","journal-title":"IEEE Trans. Image Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"99","DOI":"10.3233\/ICA-2005-12108","article-title":"Image denoising using neighbouring wavelet coefficients","volume":"12","author":"Chen","year":"2005","journal-title":"Integr. Comput.-Aided Eng."},{"key":"ref_23","first-page":"63","article-title":"Image restoration via Wiener filtering in the frequency domain","volume":"5","author":"Furuya","year":"2009","journal-title":"WSEAS Trans. Signal Process."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"26327","DOI":"10.1007\/s11042-020-09158-0","article-title":"Image noise reduction based on block matching in wavelet frame domain","volume":"79","author":"Muhammad","year":"2020","journal-title":"Multimed. Tools Appl."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Lefkimmiatis, S. (2017, January 21\u201326). Non-local color image denoising with convolutional neural networks. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Honolulu, HI, USA.","DOI":"10.1109\/CVPR.2017.623"},{"key":"ref_26","first-page":"341","article-title":"Image denoising and inpainting with deep neural networks","volume":"25","author":"Xie","year":"2012","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1109\/TBC.2011.2104671","article-title":"Objective Video quality Assessment Methods: A Classification, Review, and Performance Comparison","volume":"57","author":"Chikerur","year":"2011","journal-title":"IEEE Trans. Broadcast."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2823","DOI":"10.1007\/s11042-017-4454-y","article-title":"Monitoring of audio visual quality by key indicators","volume":"77","author":"Fernadez","year":"2018","journal-title":"Multimed. Tools Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3072959.3073601","article-title":"Interactive reconstruction of Monte Carlo image sequences using a recurrent denoising autoencoder","volume":"36","author":"Chaitanya","year":"2017","journal-title":"ACM Trans. Graph. (TOG)"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"97-1","DOI":"10.1145\/3072959.3073708","article-title":"Kernel-predicting convolutional networks for denoising Monte Carlo renderings","volume":"36","author":"Bako","year":"2017","journal-title":"ACM Trans. Graph."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1111\/cgf.13473","article-title":"Deep Adaptive Sampling for Low Sample Count Rendering","volume":"Volume 37","author":"Kuznetsov","year":"2018","journal-title":"Computer Graphics Forum"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1111\/cgf.13533","article-title":"Denoising Deep Monte Carlo Renderings","volume":"Volume 38","author":"Vicini","year":"2019","journal-title":"Computer Graphics Forum"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1123","DOI":"10.1007\/s11390-019-1964-2","article-title":"DEMC: A Deep Dual-Encoder Network for Denoising Monte Carlo Rendering","volume":"34","author":"Yang","year":"2019","journal-title":"J. Comput. Sci. Technol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1109\/83.557359","article-title":"Noise estimation and filtering using block-based singular value decomposition","volume":"6","author":"Konstantinides","year":"1997","journal-title":"IEEE Trans. Image Process."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Sae-Bae, N., and Udomhunsakul, S. (2007, January 18\u201320). Noise suppression using block-based singular value decomposition filtering. Proceedings of the 2007 Asia-Pacific Conference on Communications, Bangkok, Thailand.","DOI":"10.1109\/CGIV.2007.15"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Wang, S., Deng, C., Lin, W., Zhao, B., and Chen, J. (2013, January 15\u201318). A novel SVD-based image quality assessment metric. Proceedings of the 2013 IEEE International Conference on Image Processing, Melbourne, Australia.","DOI":"10.1109\/ICIP.2013.6738087"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"872","DOI":"10.1109\/TIP.2012.2219544","article-title":"Additive white Gaussian noise level estimation in SVD domain for images","volume":"22","author":"Liu","year":"2012","journal-title":"IEEE Trans. Image Process."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Esmaeilpour, M., Mansouri, A., and Mahmoudi-Aznaveh, A. (2013, January 10\u201312). A new SVD-based image quality assessment. Proceedings of the 2013 8th Iranian Conference on Machine Vision and Image Processing (MVIP), Teheran, Iran.","DOI":"10.1109\/IranianMVIP.2013.6780013"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Liu, W. (2014, January 8\u20139). Additive white Gaussian noise level estimation based on block SVD. Proceedings of the 2014 IEEE Workshop on Electronics, Computer and Applications, Ottawa, ON, Canada.","DOI":"10.1109\/IWECA.2014.6845781"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.neucom.2014.10.090","article-title":"Image noise detection in global illumination methods based on FRVM","volume":"164","author":"Constantin","year":"2015","journal-title":"Neurocomputing"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Constantin, J., Constantin, I., Bigand, A., and Hamad, D. (2016, January 24\u201329). Perception of noise in global illumination based on inductive learning. Proceedings of the 2016 International Joint Conference on Neural Networks (IJCNN), Vancouver, BC, Canada.","DOI":"10.1109\/IJCNN.2016.7727861"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.cag.2017.09.008","article-title":"Perception of noise and global illumination: Toward an automatic stopping criterion based on SVM","volume":"69","author":"Takouachet","year":"2017","journal-title":"Comput. Graph."},{"key":"ref_43","unstructured":"Bitterli, B. (2021, January 02). Rendering Resources. Available online: https:\/\/benedikt-bitterli.me\/resources\/."},{"key":"ref_44","unstructured":"Pharr, M., Jakob, W., and Humphreys, G. (2016). Physically Based Rendering: From Theory to Implementation, Morgan Kaufmann."},{"key":"ref_45","unstructured":"Golub, G.H., and Loan, C. (1983). Matrix Computations, The Johns Hopkins University Press."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"10101","DOI":"10.1073\/pnas.97.18.10101","article-title":"Singular value decomposition for genome-wide expression data processing and modeling","volume":"97","author":"Alter","year":"2000","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1109\/TASSP.1976.1162766","article-title":"Singular vvalue decompositions and digital image processing","volume":"24","author":"Andrews","year":"1976","journal-title":"IEEE Trans. Accoustics Speech Signal Process."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.dsp.2013.09.008","article-title":"Lossy image compression using singular value decomposition and wavelet difference reduction","volume":"24","author":"Rufai","year":"2014","journal-title":"Digit. Signal. Process."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Mikolov, T., Kombrink, S., Burget, L., \u010cernock\u1ef3, J., and Khudanpur, S. (2011, January 22\u201327). Extensions of recurrent neural network language model. Proceedings of the 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Prague, Czech Republic.","DOI":"10.1109\/ICASSP.2011.5947611"},{"key":"ref_50","unstructured":"Zaremba, W., Sutskever, I., and Vinyals, O. (2014). Recurrent neural network regularization. arXiv."},{"key":"ref_51","unstructured":"Gregor, K., Danihelka, I., Graves, A., Rezende, D.J., and Wierstra, D. (2015). Draw: A recurrent neural network for image generation. arXiv."},{"key":"ref_52","unstructured":"Liu, P., Qiu, X., and Huang, X. (2016). Recurrent neural network for text classification with multi-task learning. arXiv."},{"key":"ref_53","unstructured":"Sutskever, I., Martens, J., and Hinton, G.E. (July, January 28). Generating Text with Recurrent Neural Networks. Proceedings of the ICML\u201911 28th International Conference on Machine Learning, Bellevue, WA, USA."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Baccouche, M., Mamalet, F., Wolf, C., Garcia, C., and Baskurt, A. (2010, January 15\u201318). Action classification in soccer videos with long short-term memory recurrent neural networks. Proceedings of the International Conference on Artificial Neural Networks, Thessaloniki, Greece.","DOI":"10.1007\/978-3-642-15822-3_20"},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Ebrahimi Kahou, S., Michalski, V., Konda, K., Memisevic, R., and Pal, C. (2015, January 9\u201313). Recurrent neural networks for emotion recognition in video. Proceedings of the 2015 ACM on International Conference on Multimodal Interaction, Seattle, WA, USA.","DOI":"10.1145\/2818346.2830596"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1735","DOI":"10.1162\/neco.1997.9.8.1735","article-title":"Long short-term memory","volume":"9","author":"Hochreiter","year":"1997","journal-title":"Neural Comput."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1145","DOI":"10.1016\/S0031-3203(96)00142-2","article-title":"The use of the area under the ROC curve in the evaluation of machine learning algorithms","volume":"30","author":"Bradley","year":"1997","journal-title":"Pattern Recognit."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/1\/75\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:07:44Z","timestamp":1760159264000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/1\/75"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,6]]},"references-count":57,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["e23010075"],"URL":"https:\/\/doi.org\/10.3390\/e23010075","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,6]]}}}