Fuzzy metrics and fuzzy logic for colour image filtering

El filtrado de imagen es una tarea fundamental para la mayoría de los sistemas de visión por computador cuando las imágenes se usan para análisis automático o, incluso, para inspección humana. De hecho, la presencia de ruido en una imagen puede ser un grave impedimento para las sucesivas tareas de procesamiento de imagen como, por ejemplo, la detección de bordes o el reconocimiento de patrones u objetos y, por lo tanto, el ruido debe ser reducido. En los últimos años el interés por utilizar imágenes en color se ha visto incrementado de forma significativa en una gran variedad de aplicaciones. Es por esto que el filtrado de imagen en color se ha convertido en un área de investigación interesante. Se ha observado ampliamente que las imágenes en color deben ser procesadas teniendo en cuenta la correlación existente entre los distintos canales de color de la imagen. En este sentido, la solución probablemente más conocida y estudiada es el enfoque vectorial. Las primeras soluciones de filtrado vectorial, como por ejemplo el filtro de mediana vectorial (VMF) o el filtro direccional vectorial (VDF), se basan en la teoría de la estadística robusta y, en consecuencia, son capaces de realizar un filtrado robusto. Desafortunadamente, estas técnicas no se adaptan a las características locales de la imagen, lo que implica que usualmente los bordes y detalles de las imágenes se emborronan y pierden calidad. A fin de solventar este problema, varios filtros vectoriales adaptativos se han propuesto recientemente. En la presente Tesis doctoral se han llevado a cabo dos tareas principales: (i) el estudio de la aplicabilidad de métricas difusas en tareas de procesamiento de imagen y (ii) el diseño de nuevos filtros para imagen en color que sacan provecho de las propiedades de las métricas difusas y la lógica difusa. Los resultados experimentales presentados en esta Tesis muestran que las métricas difusas y la lógica difusa son herramientas útiles para diseñar técnicas de filtrado,Morillas Gómez, S. (2007). Fuzzy metrics and fuzzy logic for colour image filtering [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/1879Palanci

Perceptual similarity between color images using fuzzy metrics

“NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Visual Communication and Image Representation. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Visual Communication and Image Representation, [Volume 34, January 2016, Pages 230–235] https://doi.org/10.1016/j.jvcir.2015.04.003In many applications of the computer vision field measuring the similarity between (color) images is of paramount importance. However, the commonly used pixelwise similarity measures such as Mean Absolute Error, Peak Signal to Noise Ratio, Mean Squared Error or Normalized Color Difference do not match well with perceptual similarity. Recently, it has been proposed a method for gray-scale image similarity that correlates quite well with the perceptual similarity and it has been extended to color images. In this paper we use the basic ideas in this recent work to propose an alternative method based on fuzzy metrics for perceptual color image similarity. Experimental results employing a survey of observations show that the global performance of our proposal is competitive with best state of the art methods and that it shows some advantages in performance for images with low correlation among some image channels. (C) 2015 Elsevier Inc. All rights reserved.Grecova, S.; Morillas Gómez, S. (2016). Perceptual similarity between color images using fuzzy metrics. Journal of Visual Communication and Image Representation. 34:230-235. doi:10.1016/j.jvcir.2015.04.003S2302353

On the importance of metrics in practical applications

[EN] Students motivation for learning mathematical concepts can be increased when showing the usefulness of these concepts in practical problems. One important mathematical concept is the concept of metric space and, more related to the applications, the concept of metric function. In this work we aim to illustrate how important is to appropriately choose the metric when dealing with a practical problem. In particular, we focus on the problem of detection of noisy pixels in colour images. In this context, it is very important to appropriately measure the distances and similarities between the image pixels, which is done by means of an appropriate metric. We study the performance of different metrics, including recent fuzzy metrics, within a specific filter to show that it is indeed a critical choice to appropriately solve the task.Camarena, J.; Morillas, S.; Cisneros, F. (2011). On the importance of metrics in practical applications. Modelling in Science Education and Learning. 4:119-128. doi:10.4995/msel.2011.3066SWORD119128

Robustifying Vector Median Filter

This paper describes two methods for impulse noise reduction in colour images that outperform the vector median filter from the noise reduction capability point of view. Both methods work by determining first the vector median in a given filtering window. Then, the use of complimentary information from componentwise analysis allows to build robust outputs from more reliable components. The correlation among the colour channels is taken into account in the processing and, as a result, a more robust filter able to process colour images without introducing colour artifacts is obtained. Experimental results show that the images filtered with the proposed method contain less noisy pixels than those obtained through the vector median filter. Objective measures demonstrate the goodness of the achieved improvement

On completable fuzzy metric spaces

In this paper we construct a non-completable fuzzy metric space in the sense of George and Veeramani which allows to answer an open question related to continuity on the real parameter t. In addition, the constructed space is not strong (non-Archimedean).Juan Jose Minana acknowledges the support of Conselleria de Educacion, Formacion y Empleo (Programa Vali+d para investigadores en formacion) of Generalitat Valenciana, Spain and the support of Universitat Politecnica de Valencia under Grant PAID-06-12 SP20120471.Gregori Gregori, V.; Miñana, J.; Morillas, S. (2015). On completable fuzzy metric spaces. Fuzzy Sets and Systems. 267:133-139. https://doi.org/10.1016/j.fss.2014.07.009S13313926

Colour image denoising by eigenvector analysis of neighbourhood colour samples

[EN] Colour image smoothing is a challenging task because it is necessary to appropriately distinguish between noise and original structures, and to smooth noise conveniently. In addition, this processing must take into account the correlation among the image colour channels. In this paper, we introduce a novel colour image denoising method where each image pixel is processed according to an eigenvector analysis of a data matrix built from the pixel neighbourhood colour values. The aim of this eigenvector analysis is threefold: (i) to manage the local correlation among the colour image channels, (ii) to distinguish between flat and edge/textured regions and (iii) to determine the amount of needed smoothing. Comparisons with classical and recent methods show that the proposed approach is competitive and able to provide significative improvements.Latorre-Carmona, P.; Miñana, J.; Morillas, S. (2020). Colour image denoising by eigenvector analysis of neighbourhood colour samples. Signal Image and Video Processing. 14(3):483-490. https://doi.org/10.1007/s11760-019-01575-5S483490143Plataniotis, K.N., Venetsanopoulos, A.N.: Color Image Processing and Applications. Springer, Berlin (2000)Lukac, R., Smolka, B., Martin, K., Plataniotis, K.N., Venetsanopoulos, A.N.: Vector Filtering for Color Imaging. IEEE Signal Processing Magazine, Special Issue on Color Image Processing 22, 74–86 (2005)Lukac, R., Plataniotis, K.N.: A taxonomy of color image filtering and enhancement solutions. In: Hawkes, P.W. (ed.) Advances in Imaging and Electron Physics, vol. 140, pp. 187–264. Elsevier Acedemic Press, Amsterdam (2006)Buades, A., Coll, B., Morel, J.M.: Nonlocal image and movie denoising. Int. J. Comput. Vis. 76, 123–139 (2008)Tomasi, C., Manduchi, R.: Bilateral filter for gray and color images. In: Proceedings of IEEE International Conference Computer Vision, pp. 839–846 (1998)Elad, M.: On the origin of bilateral filter and ways to improve it. IEEE Trans. Image Process. 11, 1141–1151 (2002)Kao, W.C., Chen, Y.J.: Multistage bilateral noise filtering and edge detection for color image enhancement. IEEE Trans. Consum. Electron. 51, 1346–1351 (2005)Garnett, R., Huegerich, T., Chui, C., He, W.: A universal noise removal algorithm with an impulse detector. IEEE Trans. Image Process. 14, 1747–1754 (2005)Morillas, S., Gregori, V., Sapena, A.: Fuzzy Bilateral Filtering for color images. Lecture Notes Comput. Sci. 4141, 138–145 (2006)Zhang, B., Allenbach, J.P.: Adaptive bilateral filter for sharpness enhancement and noise removal. IEEE Trans. Image Process. 17, 664–678 (2008)Kenney, C., Deng, Y., Manjunath, B.S., Hewer, G.: Peer group image enhancement. IEEE Trans. Image Process. 10, 326–334 (2001)Morillas, S., Gregori, V., Hervás, A.: Fuzzy peer groups for reducing mixed Gaussian-impulse noise from color images. IEEE Trans. Image Process. 18, 1452–1466 (2009)Plataniotis, K.N., Androutsos, D., Venetsanopoulos, A.N.: Adaptive fuzzy systems for multichannel signal processing. Proc. IEEE 87, 1601–1622 (1999)Schulte, S., De Witte, V., Kerre, E.E.: A fuzzy noise reduction method for colour images. IEEE Trans. Image Process. 16, 1425–1436 (2007)Shen, Y., Barner, K.: Fuzzy vector median-based surface smoothing. IEEE Trans. Vis. Comput. Graph. 10, 252–265 (2004)Lukac, R., Plataniotis, K.N., Smolka, B., Venetsanopoulos, A.N.: cDNA microarray image processing using fuzzy vector filtering framework. Fuzzy Sets Syst. 152, 17–35 (2005)Smolka, B.: On the new robust algorithm of noise reduction in color images. Comput. Graph. 27, 503–513 (2003)Van de Ville, D., Nachtegael, M., Van der Weken, D., Philips, W., Lemahieu, I., Kerre, E.E.: Noise reduction by fuzzy image filtering. IEEE Trans. Fuzzy Syst. 11, 429–436 (2003)Schulte, S., De Witte, V., Nachtegael, M., Van der Weken, D., Kerre, E.E.: Histogram-based fuzzy colour filter for image restoration. Image Vis. Comput. 25, 1377–1390 (2007)Nachtegael, M., Schulte, S., Van der Weken, D., De Witte, V., Kerre, E.E.: Gaussian noise reduction in grayscale images. Int. J. Intell. Syst. Technol. Appl. 1, 211–233 (2006)Schulte, S., De Witte, V., Nachtegael, M., Mélange, T., Kerre, E.E.: A new fuzzy additive noise reduction method. Lecture Notes Comput. Sci. 4633, 12–23 (2007)Morillas, S., Schulte, S., Mélange, T., Kerre, E.E., Gregori, V.: A soft-switching approach to improve visual quality of colour image smoothing filters. In: Proceedings of Advanced Concepts for Intelligent Vision Systems ACIVS07, Lecture Notes in Computer Science, vol. 4678, pp. 254–261 (2007)Lucchese, L., Mitra, S.K.: A new class of chromatic filters for color image processing: theory and applications. IEEE Trans. Image Process. 13, 534–548 (2004)Lee, J.A., Geets, X., Grégoire, V., Bol, A.: Edge-preserving filtering of images with low photon counts. IEEE Trans. Pattern Anal. Mach. Intell. 30, 1014–1027 (2008)Russo, F.: Technique for image denoising based on adaptive piecewise linear filters and automatic parameter tuning. IEEE Trans. Instrum. Meas. 55, 1362–1367 (2006)Shao, M., Barner, K.E.: Optimization of partition-based weighted sum filters and their application to image denoising. IEEE Trans. Image Process. 15, 1900–1915 (2006)Ma, Z., Wu, H.R., Feng, D.: Partition based vector filtering technique for suppression of noise in digital color images. IEEE Trans. Image Process. 15, 2324–2342 (2006)Ma, Z., Wu, H.R., Feng, D.: Fuzzy vector partition filtering technique for color image restoration. Comput. Vis. Image Underst. 107, 26–37 (2007)Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)Sroubek, F., Flusser, J.: Multichannel blind iterative image restoration. IEEE Trans. Image Process. 12, 1094–1106 (2003)Hu, J., Wang, Y., Shen, Y.: Noise reduction and edge detection via kernel anisotropic diffusion. Pattern Recognit. Lett. 29, 1496–1503 (2008)Li, X.: On modeling interchannel dependency for color image denoising. Int. J. Imaging Syst. Technol., Special issue on applied color image processing 17, 163–173 (2007)Keren, D., Gotlib, A.: Denoising color images using regularization and correlation terms. J. Vis. Commun. Image Represent. 9, 352–365 (1998)Lezoray, O., Elmoataz, A., Bougleux, S.: Graph regularization for color image processing. Comput. Vis. Image Underst. 107, 38–55 (2007)Elmoataz, A., Lezoray, O., Bougleux, S.: Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing. IEEE Trans. Image Process. 17, 1047–1060 (2008)Blomgren, P., Chan, T.: Color TV: total variation methods for restoration of vector-valued images. IEEE Trans. Image Process. 7, 304–309 (1998)Tschumperlé, D., Deriche, R.: Vector-valued image regularization with PDEs: a common framework from different applications. IEEE Trans. Pattern Anal. Mach. Intell. 27, 506–517 (2005)Plonka, G., Ma, J.: Nonlinear regularized reaction-diffusion filters for denoising of images with textures. IEEE Trans. Image Process. 17, 1283–1294 (2007)Melange, T., Zlokolica, V., Schulte, S., De Witte, V., Nachtegael, M., Pizurca, A., Kerre, E.E., Philips, W.: A new fuzzy motion and detail adaptive video filter. Lecture Notes Comput. Sci. 4678, 640–651 (2007)De Backer, S., Pizurica, A., Huysmans, B., Philips, W., Scheunders, P.: Denoising of multicomponent images using wavelet least-squares estimators. Image Vis. Comput. 26, 1038–1051 (2008)Dengwen, Z., Wengang, C.: Image denoising with an optimal threshold and neighboring window. Pattern Recognit. Lett. 29, 1694–1697 (2008)Schulte, S., Huysmans, B., Pizurica, A., Kerre, E.E., Philips, W.: A new fuzzy-based wavelet shrinkage image denoising technique. In: Proceedings of Advanced Concepts for Intelligent Vision Systems ACIVS06, Lecture Notes in Computer Science, vol. 4179, pp. 12–23 (2006)Pizurica, A., Philips, W.: Estimating the probability of the presence of a signal of interest in multiresolution single and multiband image denoising. IEEE Trans. Image Process. 15, 654–665 (2006)Scheunders, P.: Wavelet thresholding of multivalued images. IEEE Trans. Image Process. 13, 475–483 (2004)Sendur, L., Selesnick, I.W.: Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency. IEEE Trans. Signal Process. 50, 2744–2756 (2002)Balster, E.J., Zheng, Y.F., Ewing, R.L.: Feature-based wavelet shrinkage algorithm for image denoising. IEEE Trans. Image Process. 14, 2024–2039 (2005)Miller, M., Kingsbury, N.: Image denoising using derotated complex wavelet coefficients. IEEE Trans. Image Process. 17, 1500–1511 (2008)Zhang, B., Fadili, J.M., Starck, J.L.: Wavelets, ridgelets, and curvelets for poisson noise removal. IEEE Trans. Image Process. 17, 1093–1108 (2008)Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3D transform-domain collaborative filtering. IEEE Trans. Image Process. 16, 2080–2095 (2007)Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Color image denoising via sparse 3D collaborative filtering with grouping constraint in luminance-chrominance space. In: Proceedings of the IEEE International Conference on Image Processing ICIP2007 , pp. 313–316 (2007)Hao, B.B., Li, M., Feng, X.C.: Wavelet iterative regularization for image restoration with varying scale parameter. Signal Process. Image Commun. 23, 433–441 (2008)Zhao, W., Pope, A.: Image restoration under significat additive noise. IEEE Signal Process. Lett. 14, 401–404 (2007)Gijbels, I., Lambert, A., Qiu, P.: Edge-preserving image denoising and estimation of discontinuous surfaces. IEEE Trans. Pattern Anal. Mach. Intell. 28, 1075–1087 (2006)Liu, C., Szeliski, R., Kang, S.B., Zitnik, C.L., Freeman, W.T.: Automatic estimation and removal of noise from a single image. IEEE Trans. Pattern Anal. Mach. Intell. 30, 299–314 (2008)Oja, E.: Principal components, minor components, and linear neural networks. Neural Netw. 5, 927–935 (1992)Takahashi, T.: Kurita, T.: Robust de-noising by kernel PCA. In: Proceedings of ICANN2002, Lecture Notes in Computer Science, vol. 2145, pp. 739–744 (2002)Park, H., Moon, Y.S.: Automatic denoising of 2D color face images using recursive PCA reconstruction. In: Proceedings of Advanced Concepts for Intelligent Vision Systems ACIVS06, Lecture Notes in Computer Science, vol. 4179, pp. 799–809 (2006)Teixeira, A.R., Tomé, A.M., Stadlthanner, K., Lang, E.W.: KPCA denoising and the pre-image problem revisited. Digital Signal Process. 18, 568–580 (2008)Astola, J., Haavisto, P., Neuvo, Y.: Vector median filters. Proc. IEEE 78, 678–689 (1990)Morillas, S., Gregori, V., Sapena, A.: Adaptive marginal median filter for colour images. Sensors 11, 3205–3213 (2011)Morillas, S., Gregori, V.: Robustifying vector median filter. Sensors 11, 8115–8126 (2011)Dillon, W.R., Goldstein, M.: Multivariate Analysis: Methods and Applications. Wiley, Hoboken (1984)Jackson, J.E.: A User’s Guide to Principal Components. Wiley, Hoboken (2003)Camacho, J., Picó, J.: Multi-phase principal component analysis for batch processes modelling. Chemom. Intell. Lab. Syst. 81, 127–136 (2006)Nomikos, P., MacGregor, J.: Multivariate SPC charts for monitoring batch processes. Technometrics 37, 41–59 (1995)Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)Grecova, Svetlana, Morillas, Samuel: Perceptual similarity between color images using fuzzy metrics. J. Vis. Commun. Image Represent. 34, 230–235 (2016)Fairchild, M.D., Johnson, G.M.: iCAM framework for image appearance differences and quality. J. Electron. Imaging 13(1), 126–138 (2004)Immerkaer, J.: Fast noise variance estimation. Comput. Vis. Image Underst. 64, 300–302 (1996

Colour image smoothing through a soft-switching mechanism using a graph model

[EN] In this study, the authors propose a soft-switching ¿lter to improve the performance of recent colour image smoothing ¿lters when processing homogeneous image regions. The authors use a recent ¿lter mixed with the classical arithmetic mean ¿lter (AMF). The recent method is used to process image pixels close to edges, texture and details and the AMF is only used to process homogeneous regions. To this end, the authors propose a method based on the graph theory to distinguish image details and homogeneous regions and to perform a soft switching between the two ¿lters. Experimental results show that the proposed method provides improved results which supports the appropriateness of the graph theory-based method and suggests that the same structure can be used to improve the performance of other non-linear colour image smoothing methods.The authors acknowledge the support of Spanish Ministry of Science and Innovation under grant MTM2009-12872-C02-01, Spanish Ministry of Science and Technology under grant MTM2010-18539 and DGCYT under grant MTM2009-08933Jordan Lluch, C.; Morillas, S.; Sanabria Codesal, E. (2012). Colour image smoothing through a soft-switching mechanism using a graph model. IET Image Processing. 6(9):1293-1298. doi:10.1049/IET-IPR.2011.0164S129312986

A note on local bases and convergence in fuzzy metric spaces

In the context of fuzzy metrics in the sense of George and Veeramani, we study when certain families of open balls centered at a point are local bases at this point. This question is related to p-convergence and s-convergence. © 2013 Elsevier B.V. All rights reserved.Samuel Morillas acknowledges the support of Universitat Politenica de Valencia under Grant PAID-05-12 SP20120696.Gregori Gregori, V.; Miñana Prats, JJ.; Morillas Gómez, S. (2014). A note on local bases and convergence in fuzzy metric spaces. Topology and its Applications. 163:142-148. https://doi.org/10.1016/j.topol.2013.10.013S14214816

Adaptive Marginal Median Filter for Colour Images

This paper describes a new filter for impulse noise reduction in colour images which is aimed at improving the noise reduction capability of the classical vector median filter. The filter is inspired by the application of a vector marginal median filtering process over a selected group of pixels in each filtering window. This selection, which is based on the vector median, along with the application of the marginal median operation constitutes an adaptive process that leads to a more robust filter design. Also, the proposed method is able to process colour images without introducing colour artifacts. Experimental results show that the images filtered with the proposed method contain less noisy pixels than those obtained through the vector median filter

Using suprathreshold color-difference ellipsoids to estimate any perceptual color-difference

[EN] Relating instrumentally measured to visually perceived colour-differences is one of the challenges of advanced colorimetry. Lately, the use of color difference formulas is becoming more important in the computer vision field as it is a key tool in advancing towards perceptual image processing and understanding. In the last decades, the study of contours of equal color-differences around certain color centers has been of special interest. In particular, the contour of threshold level difference that determines the just noticeable differences (JND) has been deeply studied and, as a result, a set of 19 different ellipsoids of suprathreshold color-difference is available in the literature. In this paper we study whether this set of ellipsoids could be used to compute any color difference in any region of the color space. To do so, we develop a fuzzy multi-ellipsoid model using the ellipsoids information along with two different metrics. We see that the performance of the two metrics vary significantly for very small, small, medium and large color differences. Therefore, we also study how to adapt two metric parameters to optimize performance. The obtained results outperform the currently CIE-recommended colordifference formula CIEDE2000.S. Morillas acknowledges the support of grants PRX16/00050 and PRX17/00384 (Ministerio de Educacion, Cultura y Deporte) and MTM2015-64373-13 (MINECO/FEDER, UE). The authors thank Dr. Manuel Melgosa, Dr. Luis Gomez-Robledo, Dr. Esther Sanabria-Codesal, Dr. Francisco Montserrat and Mr. Fu Jiang for providing useful materials, information and suggestions.Morillas, S.; Fairchild, MD. (2018). Using suprathreshold color-difference ellipsoids to estimate any perceptual color-difference. Journal of Visual Communication and Image Representation. 55:142-148. https://doi.org/10.1016/j.jvcir.2018.05.022S1421485