Fuzzy relation equations for coding/decoding processes of images and videos

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Abstract

We adopt fuzzy relation equations with continuous triangular norms for compression/decompression processes of grey images, colour images in the RGB space and frames of videos, by comparing the results of the reconstructed images with standard methods like JPEG and MPEG-4. Any image is subdivided in blocks and each block is coded/decoded using arbitrary fuzzy sets as coders in the fuzzy equations. We evaluate the peak signal to noise ratio (PSNR) on the decompressed images for several values of the compression rates as measure of the quality of images and frames reconstructed. The original frames of a video are classified in Intra-frames and Predictive frames by using a similarity measure based on the well known Lukasiewicz t-norm.

Introduction

This work follows other papers ([1], [2], [4], [7], [9]) where various types of fuzzy relation equations with continuous triangular norms [6] are used for compression/decompression of images.

Any monochromatic image is interpreted as a fuzzy relation (or fuzzy matrix) R in which the entries are the normalized values of the pixels. This method (see, e.g., [11]) is based essentially on the fact that the reconstructed image is obtained as the greatest or the smallest solution of a system of fuzzy equations having also R as a solution. In the sequel we use a continuous triangular norm (for short, t-norm) t:[0,1]2→[0,1] and, as usually, we put xty=t(x,y) for all x,y∈[0,1]. The hypothesis of continuity makes possible to define uniquely the residuum “→t ” of “t” as (x→ty)=sup{z∈[0,1]:xtz⩽y} for all x,y∈[0,1] (see, e.g., [6]). In fuzzy logic the most used t-norms with the related residua are the following:Gödelt-norm:xGy=min(x,y),(x→Gy)=1ifx⩽y,(x→Gy)=yifx>yGoguent-norm:xPy=xy,(x→Py)=1ifx⩽y,(x→Py)=y/xifx>yLukasiewiczt-norm:xLy=max(0,x+y−1),(x→Ly)=min(1,1−x+y)

We also shall consider the concept of bi-residuum “↔t” associated to the t-norm “t” defined as (xty)=min{(xty),(ytx)} for all x,y∈[0,1]. If t=L, it is easily seen that (xLy)=1−max{x,y}+min{x,y} for all x,y∈[0,1] and this formula is useful to define a similarity measure between two frames of a motion extracted from a well known database [17]. Other applications in fuzzy setting concerning this similarity measure can be found in [15]. We are motivated from our previous paper [1] where the best results were obtained using the Lukasiewicz t-norm. In [11] the authors show processes for coding/decoding colour images in the RGB and YUV spaces by using fuzzy relation equations of max-t type, where “t” is the Yager t-norm [6]. In [2] the authors propose the same thematics of [7] by using adjoint fuzzy relation equations and they divide the original image in submatrices, called blocks.

Here we apply the same ideas to processes of compression/decompression of grey images, colour images in the RGB space and frames of videos. Indeed we divide the original image in blocks and we compress each block R with a fuzzy relation equation of max-t type and an adjoint fuzzy relation equation by obtaining a block with smaller dimensions. The greatest solution S of the fuzzy relation equation of max-t type and the smallest solution D of the adjoint fuzzy relation equation are successively considered and hence suitably and slightly modified in the fuzzy relations Š and Ď, respectively, for the decompression process. Hence we assume the fuzzy relation Ř=(Ď+Š)/2 as reconstructed image of R.

We evaluate the peak signal to noise ratio (PSNR) on the decompressed images for several values of the compression rates as measure of the quality of images and frames reconstructed. By continuing our previous papers ([2], [7]), here we consider in our experiments only the t-norms L and P. The basic procedure is described in Section 2 and in Section 3 we show our results on grey images extracted also from the database [17], by calculating the related PSNR for the values 0.25 and 0.44 of the compression rates. Section 4 contains the results of our experiments made on some colour images of 256 × 256 pixels (extracted always from [17]), by evaluating the PSNR in the RGB space for the same values of the compression rates. Section 5 deals with the compression/reconstruction of Intra-frames and Predictive frames which compose the motion considered. The related experiments are presented in Section 6 and final comments are contained in the Section 7.

Section snippets

Theoretical preliminaries

Here we recall some definitions and concepts of the theory of fuzzy relation equations [3] used in the sequel. Let h, k, m, n be positive integers such that km and hn, further let Is={1,…,s} be the set of the first s positive integers and be assigned the fuzzy sets A1,…,Ak:Im→[0,1] and B1,…,Bh:In→[0,1]. For brevity of notation, we put Ap(i)=Api for all pIk, iIm and Bq(j)=Bqj for all qIh, jIn. A monochromatic image R of sizes m×n (pixels) is considered as a fuzzy relation R:(i,j)∈Im×In

By coding/decoding grey images

In order to give a precise idea of the above method and to consider a wide variety of details in the images, from [17] we have extracted the famous Lena of Fig. 1, Camera of Fig. 2, Parrot of Fig. 3, each considered as a fuzzy relation of sizes 256 × 256.

We have made two processes of compression/decompression: in the first (resp., second) one, we have divided these images in square blocks of sizes mb×nb=3×3 (resp., 4 × 4) compressed to blocks of sizes kb×hb=2×2 with the formula (8) under the above t

By coding/decoding colour images

In this Section we show how our method works on colour images in the RGB space, by considering well known images extracted from [17]. The procedure, described in Section 2, is applied to the three primary components of the original image obtained in the bands R, G and B involved (see, e.g., Fig. 10, Fig. 11, Fig. 12, Fig. 13).

We have also examined the following well known images: Lena in Fig. 14 and Peppers in Fig. 15, each considered as a fuzzy relation of sizes 256 × 256 in each of the three

By coding/decoding frames of videos

Here we adopt the procedure of Section 2 to code a motion, defined as a sequence of N colour images or frames, each considered obviously as fuzzy relation of sizes m×n in each band of the RGB space. From now on, any consideration is made in each of three bands R, G and B.

Experimental results on videos

We have considered from [17] the two motions called “Tennis” and “Sflowg”, each composed of 40 frames of sizes 240 × 360. The first frame of both motions is assumed as I-frame and we have classified the 40 frames of both videos in I-frames and P-frames by fixing the following values of the threshold Θ (see Table 6).

By choosing Θ=Θ1 (resp. Θ=Θ2), we have that an I-frame is followed from two (resp., four) P-frames. For brevity, we present the results on the sequence of the first four frames, by

Concluding comments

In continuation of our previous papers [1], [2], [7], here we have proposed known fuzzy relation equations of max-t type and related adjoint fuzzy equations, being “t” a continuous triangular norm, for coding/decoding processes of grey images, colour images in the RGB spaces and frames of videos. Our method has been also compared about the compression times and the compression rates with the standard methods like JPEG and MPEG-4. Further improvements are naturally to be proposed, the most

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