Design of guided adaptive mathematical morphology for multimodal images

doi:10.11772/j.issn.1001-9081.2021122168

Journal of Computer Applications ›› 2023, Vol. 43 ›› Issue (2): 560-566.DOI: 10.11772/j.issn.1001-9081.2021122168

• Multimedia computing and computer simulation • Previous Articles

Design of guided adaptive mathematical morphology for multimodal images

Mengdi SUN, Zhonggui SUN(), Xu KONG, Hongyan HAN

College of Mathematical Sciences，Liaocheng University，Liaocheng Shandong 252000，China

Received:2021-12-29 Revised:2022-04-14 Accepted:2022-04-19 Online:2022-05-16 Published:2023-02-10
Contact: Zhonggui SUN
About author:SUN Mengdi， born in 1997， M. S. candidate. Her research interests include image processing， machine learning.
KONG Xu， born in 1981， Ph. D.， associate professor. His research interests include image processing， machine learning， applied mathematics.
HAN Hongyan， born in 1976， M. S.， associate professor. Her research interests include intelligent computation， image processing.
Supported by:
National Natural Science Foundation of China(11801249);Natural Science Foundation of Shandong Province(ZR2020MF040)

针对多模态图像的自适应引导形态学设计

孙梦迪, 孙忠贵(), 孔旭, 韩红燕

聊城大学数学科学学院，山东聊城 252000

通讯作者: 孙忠贵
作者简介:孙梦迪（1997—），女，山东滨州人，硕士研究生，主要研究方向：图像处理、机器学习
孔旭（1981—），男，山东曲阜人，副教授，博士，主要研究方向：图像处理、机器学习、应用数学
韩红燕（1976—），女，山东茌平人，副教授，硕士，主要研究方向：智能计算、图像处理。
基金资助:
国家自然科学基金资助项目(11801249);山东省自然科学基金资助项目(ZR2020MF040)

Abstract

Abstract:

Traditional Mathematical Morphology （TMM） is not well in structure-preserving， and the existing adaptive modified methods usually miss mathematical properties. To address the problems， a Guided Adaptive Mathematical Morphology （GAMM） for multimodal images was proposed. Firstly， the structure elements were constructed by considering the joint information of the input and the guidance images， so that the corresponding operators were more robust to the noise. Secondly， according to 3σ rule， the selected members of structure elements were able to be adapted to image contents. Finally， by using the Hadamard product of sparse matrices， the structure elements were imposed with a symmetry constraint. Both of the theoretical verification and simulation show that the corresponding operators of the proposed mathematical morphology can have important mathematical properties， such as order preservation and adjunction， at the same time. Denoising experimental results on multimodal images show that the Peak Signal-to-Noise Ratio （PSNR） of GAMM is 2 to 3 dB higher than those of TMM and Robust Adaptive Mathematical Morphology （RAMM）. Meanwhile， comparison of subjective visual effect shows that GAMM significantly outperforms TMM and RAMM in noise removal and structure preservation.

Key words: Guided Adaptive Mathematical Morphology (GAMM), multimodal image, mathematical property, robustness, sparse matrix

摘要：

针对传统数学形态学（TMM）细节保持能力较差，以及现有自适应改进方法数学性质丢失的问题，提出了一种针对多模态图像的自适应引导形态学（GAMM）。首先，通过考虑输入图像和引导图像的联合信息进行结构元素的构建，从而在一定程度上增强了相应算子对噪声的鲁棒性；其次，借助3σ原则，使结构元素成员的选取能够自适应于图像内容；最后，利用稀疏矩阵的哈达玛积对结构元素施加一个对称性约束。理论证明和仿真实验均表明所提形态学的相应算子能够同时具备保序性和附益性等重要数学性质。在多模态图像上进行去噪实验，结果表明GAMM比TMM以及近年所提出的鲁棒自适应形态学（RAMM）在峰值信噪比（PSNR）上高出约2~3 dB；同时，主观视觉效果对比表明了GAMM在噪声去除、结构保持方面明显优于TMM和RAMM。

关键词: 自适应引导形态学, 多模态图像, 数学性质, 鲁棒性, 稀疏矩阵

CLC Number:

TP391.4

Mengdi SUN, Zhonggui SUN, Xu KONG, Hongyan HAN. Design of guided adaptive mathematical morphology for multimodal images[J]. Journal of Computer Applications, 2023, 43(2): 560-566.

孙梦迪, 孙忠贵, 孔旭, 韩红燕. 针对多模态图像的自适应引导形态学设计[J]. 《计算机应用》唯一官方网站, 2023, 43(2): 560-566.

Figures/Tables 8

Fig. 1 Flowchart for GAMM

Fig. 2 Verification of properties

Fig. 3 RGB-depth image test set

Tab. 1 Results of PSNR in RGB-depth image denoising

图像	滤波窗口	输入			TMM			RAMM			GAMM
图像	滤波窗口	$δ = 10$	$δ = 30$	$δ = 50$	$δ = 10$	$δ = 30$	$δ = 50$	$δ = 10$	$δ = 30$	$δ = 50$	$δ = 10$	$δ = 30$	$δ = 50$
Art	$3 × 3$	28.08	18.66	14.55	30.59	28.14	24.80	29.38	27.63	24.65	32.70	28.23	24.81
	$5 × 5$				28.77	27.75	25.91	28.87	27.60	25.93	31.21	27.86	25.93
	$7 × 7$				27.37	27.07	25.51	27.57	27.12	25.68	30.33	27.57	25.73
Book	$3 × 3$	28.09	18.63	14.53	31.60	27.63	24.64	31.41	27.54	24.65	33.66	27.63	24.67
	$5 × 5$				30.53	27.70	26.20	30.47	27.65	26.03	33.66	27.75	26.29
	$7 × 7$				30.11	27.57	26.37	30.26	27.58	26.42	33.64	27.96	26.42
Doll	$3 × 3$	28.11	18.64	14.53	32.65	27.93	25.00	32.35	27.91	25.05	33.87	27.94	25.95
	$5 × 5$				31.98	28.68	26.78	32.06	28.64	26.67	34.19	28.69	26.79
	$7 × 7$				31.62	28.79	26.86	31.67	28.79	27.04	34.26	28.92	27.96
Laundry	$3 × 3$	28.09	18.73	14.48	33.90	28.95	25.35	33.21	28.83	25.31	34.07	29.01	25.41
	$5 × 5$				31.91	29.78	26.67	32.02	29.71	26.64	33.62	29.82	26.83
	$7 × 7$				31.41	29.33	26.04	31.42	29.46	26.21	32.59	29.84	26.36
Moebius	$3 × 3$	28.11	18.64	14.47	31.80	27.98	24.91	31.42	27.79	24.91	33.44	28.08	24.93
	$5 × 5$				30.40	28.32	26.32	30.48	28.20	26.24	32.91	28.82	26.55
	$7 × 7$				29.94	24.97	26.36	30.02	27.91	26.46	32.79	28.12	27.10
Reindeer	$3 × 3$	28.15	18.65	14.57	33.46	28.74	25.12	30.95	28.10	24.96	33.93	28.81	25.18
	$5 × 5$				29.98	29.24	26.98	30.26	28.99	26.75	33.57	30.01	27.86
	$7 × 7$				28.00	29.02	26.95	28.20	28.78	26.95	30.83	29.26	26.95

Tab. 1 Results of PSNR in RGB-depth image denoising

图像	滤波窗口	输入			TMM			RAMM			GAMM
图像	滤波窗口	$δ = 10$	$δ = 30$	$δ = 50$	$δ = 10$	$δ = 30$	$δ = 50$	$δ = 10$	$δ = 30$	$δ = 50$	$δ = 10$	$δ = 30$	$δ = 50$
Art	$3 × 3$	28.08	18.66	14.55	30.59	28.14	24.80	29.38	27.63	24.65	32.70	28.23	24.81
	$5 × 5$				28.77	27.75	25.91	28.87	27.60	25.93	31.21	27.86	25.93
	$7 × 7$				27.37	27.07	25.51	27.57	27.12	25.68	30.33	27.57	25.73
Book	$3 × 3$	28.09	18.63	14.53	31.60	27.63	24.64	31.41	27.54	24.65	33.66	27.63	24.67
	$5 × 5$				30.53	27.70	26.20	30.47	27.65	26.03	33.66	27.75	26.29
	$7 × 7$				30.11	27.57	26.37	30.26	27.58	26.42	33.64	27.96	26.42
Doll	$3 × 3$	28.11	18.64	14.53	32.65	27.93	25.00	32.35	27.91	25.05	33.87	27.94	25.95
	$5 × 5$				31.98	28.68	26.78	32.06	28.64	26.67	34.19	28.69	26.79
	$7 × 7$				31.62	28.79	26.86	31.67	28.79	27.04	34.26	28.92	27.96
Laundry	$3 × 3$	28.09	18.73	14.48	33.90	28.95	25.35	33.21	28.83	25.31	34.07	29.01	25.41
	$5 × 5$				31.91	29.78	26.67	32.02	29.71	26.64	33.62	29.82	26.83
	$7 × 7$				31.41	29.33	26.04	31.42	29.46	26.21	32.59	29.84	26.36
Moebius	$3 × 3$	28.11	18.64	14.47	31.80	27.98	24.91	31.42	27.79	24.91	33.44	28.08	24.93
	$5 × 5$				30.40	28.32	26.32	30.48	28.20	26.24	32.91	28.82	26.55
	$7 × 7$				29.94	24.97	26.36	30.02	27.91	26.46	32.79	28.12	27.10
Reindeer	$3 × 3$	28.15	18.65	14.57	33.46	28.74	25.12	30.95	28.10	24.96	33.93	28.81	25.18
	$5 × 5$				29.98	29.24	26.98	30.26	28.99	26.75	33.57	30.01	27.86
	$7 × 7$				28.00	29.02	26.95	28.20	28.78	26.95	30.83	29.26	26.95

Tab. 2 Results of SSIM in RGB-depth image denoising

图像	滤波窗口	输入			TMM			RAMM			GAMM
图像	滤波窗口	$δ = 10$	$δ = 30$	$δ = 50$	$δ = 10$	$δ = 30$	$δ = 50$	$δ = 10$	$δ = 30$	$δ = 50$	$δ = 10$	$δ = 30$	$δ = 50$
Art	$3 × 3$	0.576	0.155	0.080	0.891	0.678	0.473	0.871	0.664	0.468	0.896	0.678	0.481
	$5 × 5$				0.871	0.801	0.678	0.872	0.795	0.665	0.873	0.857	0.679
	$7 × 7$				0.845	0.821	0.651	0.844	0.817	0.639	0.846	0.821	0.691
Book	$3 × 3$	0.468	0.103	0.050	0.906	0.656	0.439	0.900	0.649	0.432	0.934	0.657	0.439
	$5 × 5$				0.924	0.818	0.700	0.923	0.807	0.676	0.973	0.872	0.754
	$7 × 7$				0.924	0.871	0.805	0.926	0.865	0.786	0.951	0.874	0.836
Doll	$3 × 3$	0.484	0.106	0.048	0.885	0.650	0.445	0.878	0.644	0.442	0.895	0.655	0.446
	$5 × 5$				0.893	0.806	0.686	0.900	0.797	0.666	0.900	0.815	0.692
	$7 × 7$				0.887	0.854	0.789	0.888	0.848	0.775	0.895	0.891	0.797
Laundry	$3 × 3$	0.480	0.107	0.050	0.911	0.662	0.461	0.902	0.658	0.454	0.946	0.662	0.471
	$5 × 5$				0.923	0.825	0.697	0.920	0.815	0.678	0.972	0.869	0.751
	$7 × 7$				0.918	0.865	0.776	0.916	0.862	0.767	0.949	0.890	0.775
Moebius	$3 × 3$	0.482	0.106	0.050	0.898	0.659	0.449	0.891	0.652	0.443	0.943	0.659	0.453
	$5 × 5$				0.909	0.815	0.688	0.910	0.806	0.666	0.969	0.865	0.696
	$7 × 7$				0.900	0.858	0.777	0.901	0.853	0.766	0.949	0.887	0.782
Reindeer	$3 × 3$	0.497	0.113	0.057	0.918	0.669	0.458	0.905	0.661	0.450	0.951	0.675	0.459
	$5 × 5$				0.920	0.826	0.704	0.922	0.814	0.685	0.981	0.882	0.767
	$7 × 7$				0.900	0.870	0.807	0.903	0.863	0.790	0.953	0.874	0.842

Tab. 2 Results of SSIM in RGB-depth image denoising

图像	滤波窗口	输入			TMM			RAMM			GAMM
图像	滤波窗口	$δ = 10$	$δ = 30$	$δ = 50$	$δ = 10$	$δ = 30$	$δ = 50$	$δ = 10$	$δ = 30$	$δ = 50$	$δ = 10$	$δ = 30$	$δ = 50$
Art	$3 × 3$	0.576	0.155	0.080	0.891	0.678	0.473	0.871	0.664	0.468	0.896	0.678	0.481
	$5 × 5$				0.871	0.801	0.678	0.872	0.795	0.665	0.873	0.857	0.679
	$7 × 7$				0.845	0.821	0.651	0.844	0.817	0.639	0.846	0.821	0.691
Book	$3 × 3$	0.468	0.103	0.050	0.906	0.656	0.439	0.900	0.649	0.432	0.934	0.657	0.439
	$5 × 5$				0.924	0.818	0.700	0.923	0.807	0.676	0.973	0.872	0.754
	$7 × 7$				0.924	0.871	0.805	0.926	0.865	0.786	0.951	0.874	0.836
Doll	$3 × 3$	0.484	0.106	0.048	0.885	0.650	0.445	0.878	0.644	0.442	0.895	0.655	0.446
	$5 × 5$				0.893	0.806	0.686	0.900	0.797	0.666	0.900	0.815	0.692
	$7 × 7$				0.887	0.854	0.789	0.888	0.848	0.775	0.895	0.891	0.797
Laundry	$3 × 3$	0.480	0.107	0.050	0.911	0.662	0.461	0.902	0.658	0.454	0.946	0.662	0.471
	$5 × 5$				0.923	0.825	0.697	0.920	0.815	0.678	0.972	0.869	0.751
	$7 × 7$				0.918	0.865	0.776	0.916	0.862	0.767	0.949	0.890	0.775
Moebius	$3 × 3$	0.482	0.106	0.050	0.898	0.659	0.449	0.891	0.652	0.443	0.943	0.659	0.453
	$5 × 5$				0.909	0.815	0.688	0.910	0.806	0.666	0.969	0.865	0.696
	$7 × 7$				0.900	0.858	0.777	0.901	0.853	0.766	0.949	0.887	0.782
Reindeer	$3 × 3$	0.497	0.113	0.057	0.918	0.669	0.458	0.905	0.661	0.450	0.951	0.675	0.459
	$5 × 5$				0.920	0.826	0.704	0.922	0.814	0.685	0.981	0.882	0.767
	$7 × 7$				0.900	0.870	0.807	0.903	0.863	0.790	0.953	0.874	0.842

Fig. 4 Visual effect of RGB-depth image denoising

Fig. 5 Visual effect of flash-flash-free image restoration

Fig. 6 Visual effect of NIR-RGB image denoising

References 30

1	SERRA J. Image Analysis and Mathematical Morphology［M］. London： Academic Press， 1982：536. 10.2307/2531038
2	LEI T， JIA X H， LIU T L， et al. Adaptive morphological reconstruction for seeded image segmentation［J］. IEEE Transactions on Image Processing， 2019， 28（11）：5510-5523. 10.1109/tip.2019.2920514
3	HUANG W L， LIU J X. Robust seismic image interpolation with mathematical morphological constraint［J］. IEEE Transactions on Image Processing， 2020， 29：819-829. 10.1109/tip.2019.2936744
4	GONZALEZ R C， WOODS R E， EDDINS S L. Digital Image Processing Using Matlab［M］. 2nd ed. Knoxville， TN： Gatesmark Publishing， 2009：486-534.
5	SOILLE P. Morphological Image Analysis： Principles and Applications［M］. 2nd. Berlin： Springer， 2004. 10.1007/978-3-662-05088-0
6	ROERDINK J B T M. Adaptivity and group invariance in mathematical morphology［C］// Proceedings of the 16th IEEE International Conference on Image Processing. Piscataway： IEEE， 2009：2253-2256. 10.1109/icip.2009.5413983
7	VELASCO-FORERO S， ANGUIO J. On nonlocal mathematical morphology［C］// Proceedings of the 2013 International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing， LNCS 7883. Berlin： Springer， 2013：219-230.
8	ĆURIĆ V， LANDSTRÖM A， THURLEY M J， et al. Adaptive mathematical morphology - a survey of the field［J］. Pattern Recognition Letters， 2014， 47：18-28. 10.1016/j.patrec.2014.02.022
9	DEBAYLE J， PINOLI J C. Spatially adaptive morphological image filtering using intrinsic structuring elements［J］. Image Analysis and Stereology， 2005， 24（3）：145-158.
10	LERALLUT R， DECENCIÈRE É， MEYER F. Image filtering using morphological amoebas［J］. Image and Vision Computing， 2007， 25（4）：395-404. 10.1016/j.imavis.2006.04.018
11	CUISENAIRE O. Locally adaptable mathematical morphology using distance transformations［J］. Pattern Recognition， 2006， 39（3）：405-416. 10.1016/j.patcog.2005.07.009
12	TREECE G. The bitonic filter： linear filtering in an edge-preserving morphological framework［J］. IEEE Transactions on Image Processing， 2016， 25（11）： 5199-5211. 10.1109/tip.2016.2605302
13	MILANFAR P. A tour of modern image filtering： new insights and methods， both practical and theoretical［J］. IEEE Signal Processing Magazine， 2013， 30（1）：106-128. 10.1109/msp.2011.2179329
14	PETSCHNIGG G， SZELISKI R， AGRAWALA M， et al. Digital photography with flash and no-flash image pairs［J］. ACM Transactions on Graphics， 2004， 23（3）：664-672. 10.1145/1015706.1015777
15	HE K M， SUN J， TANG X O. Guided image filtering［J］. IEEE Transactions on Pattern Analysis and Machine Intelligence， 2013， 35（6）： 1397-1409. 10.1109/tpami.2012.213
16	LI Z G， ZHENG J H， ZHU Z J， et al. Weighted guided image filtering［J］. IEEE Transactions on Image Processing， 2015， 24（1）：120-129. 10.1109/tip.2014.2371234
17	SUN Z G， HAN B， LI J， et al. Weighted guided image filtering with steering kernel［J］. IEEE Transactions on Image Processing， 2020， 29：500-508. 10.1109/tip.2019.2928631
18	韩博，孙忠贵. 基于结构相似性感知的引导图像滤波［J］. 聊城大学学报（自然科学版）， 2020，33（5）：9-15.
	HAN B， SUN Z G. Guided image filtering based on structure similarity aware［J］. Journal of Liaocheng University （Natural Science Edition）， 2020， 33（5）：9-15.
19	PUKELSHEIM F. The three sigma rule［J］. American Statistician， 1994， 48（2）：88-91. 10.1080/00031305.1994.10476030
20	SUSSNER P. Lattice fuzzy transforms from the perspective of mathematical morphology［J］. Fuzzy Sets and Systems， 2016， 288：115-128. 10.1016/j.fss.2015.09.018
21	SUN Z G， GAO X B， ZHANG D M. A nonlocal operator model for morphological image processing［C］// Proceedings of the 2017 IEEE International Conference on Image Processing. Piscataway： IEEE， 2017：230-234. 10.1109/icip.2017.8296277
22	孙忠贵，高新波，张冬梅，等. 一个具有附益性及保序性算子的形态学非局部拓展［J］. 电子学报， 2018， 46（8）：1969-1975. 10.3969/j.issn.0372-2112.2018.08.023
	SUN Z G， GAO X B， ZHANG D M， et al. A nonlocal extension of morphology with adjunction and order-preservation operators［J］. Acta Electronica Sinica， 2018， 46（8）： 1969-1975. 10.3969/j.issn.0372-2112.2018.08.023
23	王朋伟，牛瑞卿. 基于灰度形态学与小波相位滤波的高分辨率遥感影像边缘检测［J］. 计算机应用， 2011， 31（9）： 2481-2484. 10.3724/SP.J.1087.2011.02481
	WANG P W， NIU R Q. Edge detection of high resolution remote sensing images based on morphology and wavelet phase filtering［J］. Journal of Computer Applications， 2011， 31（9）： 2481-2484. 10.3724/SP.J.1087.2011.02481
24	SHUI P L， WANG F P. Anti-impulse-noise edge detection via anisotropic morphological directional derivatives［J］. IEEE Transactions on Image Processing， 2017， 26（10）：4962-4977. 10.1109/tip.2017.2726190
25	SHUANG X C， ZHANG K. An edge detection algorithm based on adaptive weight multi-structure element mathematical morphology［J］. Energy Procedia， 2011， 13：9983-9991.
26	RONSE C. Why mathematical morphology needs complete lattices［J］. Signal Processing， 1990， 21（2）：129-154. 10.1016/0165-1684(90)90046-2
27	VERD-MONEDERO R， ANGULO J， SERRA J. Anisotropic morphological filters with spatially-variant structuring elements based on image-dependent gradient fields［J］. IEEE Transactions on Image Processing， 2011， 20（1）：200-212. 10.1109/tip.2010.2056377
28	APTOULA E， LEFÈVRE S. A comparative study on multivariate mathematical morphology［J］. Pattern Recognition， 2007， 40（11）：2914-2929. 10.1016/j.patcog.2007.02.004
29	SHEN X Y， ZHOU C， XU L， et al. Mutual-structure for joint filtering［J］. International Journal of Computer Vision， 2017， 125（1/2/3）：19-33. 10.1007/s11263-017-1021-y
30	YAN Q， SHEN X Y， XU L， et al. Cross-field joint image restoration via scale map［C］// Proceedings of the 2013 IEEE International Conference on Computer Vision. Piscataway： IEEE， 2013：1537-1544. 10.1109/iccv.2013.194

[1]	GAO Gong, YANG Hongyu, LIU Hong. 3D point cloud face recognition based on deep learning [J]. Journal of Computer Applications, 2021, 41(9): 2736-2740.
[2]	LI Xingfeng, HUANG Yuqing, REN Zhenwen, LI Yihong. Robust multi-view clustering algorithm based on adaptive neighborhood [J]. Journal of Computer Applications, 2021, 41(4): 1093-1099.
[3]	Yunpeng GONG, Zhiyong ZENG, Feng YE. Person re-identification method based on grayscale feature enhancement [J]. Journal of Computer Applications, 2021, 41(12): 3590-3595.
[4]	Hua LI, Guifu LU, Qinru YU. Manifold regularized nonnegative matrix factorization based on clean data [J]. Journal of Computer Applications, 2021, 41(12): 3492-3498.
[5]	PEI Yiyao, GUO Huiming, ZHANG Danpu, CHEN Wenbo. Robust 3D object detection method based on localization uncertainty [J]. Journal of Computer Applications, 2021, 41(10): 2979-2984.
[6]	XING Zhiwei, QIAO Di, LIU Hong’en, GAO Zhiwei, LUO Xiao, LUO Qian. Optimization method of airport gate assignment based on relaxation algorithm [J]. Journal of Computer Applications, 2020, 40(6): 1850-1855.
[7]	WANG Benjie, NONG Liping, ZHANG Wenhui, LIN Jiming, WANG Junyi. 3D point cloud classification and segmentation network based on Spider convolution [J]. Journal of Computer Applications, 2020, 40(6): 1607-1612.
[8]	HUANG Guangqiu, XIE Rong. Cascading failure model of carbon emission spatial correlation system considering load overload [J]. Journal of Computer Applications, 2019, 39(6): 1829-1835.
[9]	LAI Jie, WANG Xiaodan, LI Rui, ZHAO Zhenchong. Denoising autoencoder based extreme learning machine [J]. Journal of Computer Applications, 2019, 39(6): 1619-1625.
[10]	XU Guangxian, WANG Dong. Secure network coding scheme based on chaotic encryption against wiretapping [J]. Journal of Computer Applications, 2019, 39(5): 1374-1377.
[11]	YU Jianglan, LI Xiangli, ZHAO Pengfei. Sparse non-negative matrix factorization based on kernel and hypergraph regularization [J]. Journal of Computer Applications, 2019, 39(3): 742-749.
[12]	LIU Weibo, ZENG Qingning, BU Yuting, ZHENG Zhanheng. Speech recognition method based on dual micro-array and convolutional neural network [J]. Journal of Computer Applications, 2019, 39(11): 3268-3273.
[13]	WANG Yan, LYU Meng, MENG Xiangfu, LI Yuhao. Image matching method with illumination robustness [J]. Journal of Computer Applications, 2019, 39(1): 262-266.
[14]	CHEN Ping, GONG Xun. Face alignment method based on scale self-adaption and incremental learning [J]. Journal of Computer Applications, 2018, 38(7): 2064-2069.
[15]	CAO Jingjing, XU Jieping, SHAO Shengqi. Hierarchical speech recognition model in multi-noise environment [J]. Journal of Computer Applications, 2018, 38(6): 1790-1794.

Design of guided adaptive mathematical morphology for multimodal images

针对多模态图像的自适应引导形态学设计

RichHTML

PDF

Knowledge

Abstract

Cite this article

share this article

Figures/Tables 8

References 30

Related Articles 15

Recommended Articles

Metrics