Coding B-Frames of Color Videos with Fuzzy Transforms

Abstract

We use a new method based on discrete fuzzy transforms for coding/decoding frames of color videos in which we determine dynamically the GOP sequences. Frames can be differentiated into intraframes, predictive frames, and bidirectional frames, and we consider particular frames, called Δ-frames (resp., R-frames), for coding P-frames (resp., B-frames) by using two similarity measures based on Lukasiewicz 𝑡-norm; moreover, a preprocessing phase is proposed to determine similarity thresholds for classifying the above types of frame. The proposed method provides acceptable results in terms of quality of the reconstructed videos to a certain extent if compared with classical-based F-transforms method and the standard MPEG-4.

1. Introduction

A video can be considered as a sequence of frames of sizes 𝑁×𝑀; a frame is an image that can be compressed by using a lossy compression method. We can classify each frame as intraframe (for short, I-frame), predictive frame (for short, P-frame), and bidirectional frame (for short, B-frame) which is more compressible than I-frame. A B-frame can be predicted or interpolated from an earlier and/or later frame. In order to avoid a growing propagation error, a B-frame is not used as a reference to make further predictions in most encoding standards except in AVC [1]. A frame can be considered as a P-frame if it is “similar” to the previous I-frame in the frame sequence; otherwise, it must be considered as a new I-frame. This similarity relation between a P-frame and the previous I-frame is fundamental in video-compression processes because a P-frame has values in its pixels very close to the pixels of the previous I-frame. This suggests to define a frame containing differences between a P-frame and the previous I-frame, called Δ-frame which has a low quantity of information and hence it can be coded with a low compression rate. A P-frame is decoded via the previous I-frame and the Δ-frame. In the MPEG-4 method [2, 3], that adopts the JPEG technique [4] for coding/decoding frames, the I-frames, P-frames, and B-frames are arranged in a Group of Picture (for short, GOP) sequence. A B-frame is reconstructed by using either the previous or successive I-frame. Here the results of [5] are improved by using a technique based on F-transforms for coding B-frames. For convenience, we assume that the first frame of a video is an I-frame. We assign an ID number to each frame of the video. Then we can say that the 𝑘th frame is a B-frame or a P-frame if it is “very similar” to the previous 𝑖th I-frame in the sense that its similarity Sim(𝑖,𝑘) a parameter defined on the Lukasiewicz 𝑡-norm (see formula (12)) is greater than a threshold value Sim𝑃 [5]; otherwise the 𝑘th frame is assumed to be a new I-frame as the first frame of the successive GOP sequence.

The first algorithm is used for determining the GOP sequences; the second algorithm is used for determining the type of P-frame or B-frame. The first frame of the GOP sequence is always an I-frame and the last frame is a P-frame. The function “analyze GOP sequence (ID1, ID2)” reported in Algorithm 1 describes this process, where ID1 is the ID of the first I-frame and ID2 is the ID of the last P-frame in the GOP sequence. This function is used for determining if the 𝑘th frame in the GOP sequence, where ID1<𝑘<ID2, is a B-frame or a P-frame. We define a threshold similarity Sim𝐵, and we compare it with the frame whose ID is formed from the integer [𝑀𝑠] contained in the mean 𝑀𝑠 of the previous I-frame or P-frame and the 𝑘th frame by obtaining a similarity value Sim(𝑘,[𝑀𝑠]). In the array element NP[𝑘] we insert the ID number of the last frame after the 𝑘th frame for which Sim(𝑘,[𝑀𝑠])<Sim𝐵 holds. The variable 𝑖 contains the ID number of the previous I-frame or P-frame; it is initially called ID1; the variable 𝑤 points to the last frame in the GOP sequence; it is called ID2.

Algorithm 1 (analyze GOP sequence (ID1, ID2)). Pseudo-code for determining a GOP sequence(1)𝑖=ID of the first I-frame //𝑖 is the ID of first frame of the video(2)𝑤= number of frames //𝑤 is the ID of the last P-frame of the video(3)𝑘=𝑖+1(4)IF 𝑘<𝑤(5)Calculate the similarity Sim(𝑖,𝑘) between the 𝑘th frame and the 𝑖th frame(6)If Sim(𝑖,𝑘)<Sim𝑃, (a)the 𝑘th frame is a B-frame or a P-frame and is inserted in the GOP sequence(b)𝑘=𝑘+1(7)Else (a)analyse GOP sequence (𝑖,𝑘−1)(b)𝑖=𝑘(c)go to (3)(8)End.

Algorithm 2. Pseudo-code for determining type of frames(1)𝑖=ID1//𝑖 is the ID of the first frame of the GOP sequence(2)𝑤=ID2//𝑤 is the ID of the last P-frame of the GOP sequence (3) For each 𝑘 in [𝑖+1,𝑤−1] NP[𝑘]=𝑘//InitializeNP[𝑘](4)𝑠=𝑘+1(5) Create the [𝑀𝑠]th frame as a new frame whose normalized pixels are obtained as the mean between the normalized pixels of the 𝑖th and 𝑠th frames (6) Calculate the similarity Sim(𝑘,[𝑀𝑠]) between the 𝑘th and [𝑀𝑠]th frames. If Sim(𝑘,[𝑀𝑠])<Sim𝐵, (a)NP[𝑘]=𝑠−1(b)Else 𝑠=𝑠+1(c)go to step (6)(7) next for(8)NPMin=min(NP[𝑘])(9) The frames between the 𝑖th and NPMin-th frames are labelled as B-frames(10) The NPMin-th frame is labelled as a P-frame(11) If NPmin<𝑤 then(a)𝑖=NPMin, (b)go to step (2)(12) End.

In our approach we determine a GOP sequence at each step. The frame after the last P-frame is the I-frame of the new GOP sequence. After determining the GOP sequences of the color video, we use the F-transforms [5, 7–10] for compressing the frames. The F-transform method has been developed in [5]. In this paper each frame is converted in the 𝑌𝑈𝑉 space. Indeed, since the human eye perceives an image mostly in the 𝑌 band (brightness) with respect to the 𝑈 and 𝑉 bands (chrominance), we can use a stronger compression rate for coding the image in 𝑈 and 𝑉 bands with respect to that one used for coding the image in the 𝑌 band, without loss of information in the reconstructed image. In [5] the authors show that the quality of the reconstructed images is better than the one obtained using the F-transform method directly in the 𝑅𝐺𝐵 space (see also [11, 12]). The proposed method is widely discussed in Section 4. In Sections 2 and 3 the theory of F-transforms and its application are recalled for image compression, respectively. In Section 5 the results are deduced on a large color videos dataset.

2. Fuzzy Transforms

We recall from [9] some essential definitions. Let 𝑛≥3 and 𝑥₁,𝑥₂,…,𝑥_𝑛 be points (nodes) of [𝑎,𝑏] such that 𝑥₁=𝑎<𝑥₂<⋯<𝑥_𝑛=𝑏. The fuzzy sets 𝐴₁,…,𝐴_𝑛∶[𝑎,𝑏]→[0,1] form a fuzzy partition of [𝑎,𝑏] if

(1)𝐴_𝑖(𝑥_𝑖)=1 for any 𝑖=1,2,…,𝑛;

(2)𝐴_𝑖(𝑥)=0 if 𝑥∉(𝑥_𝑖−1,𝑥_𝑖+1), where 𝑖=1,2,…,𝑛 and 𝑥₀=𝑥₁=𝑎, 𝑥_𝑛+1=𝑥_𝑛=𝑏;

(3)𝐴_𝑖(𝑥) is a continuous function on [𝑎,𝑏];

(4)𝐴_𝑖(𝑥) is strictly increasing on the interval [𝑥_𝑖−1,𝑥_𝑖] for 𝑖=2,…,𝑛 and is strictly decreasing on the interval [𝑥_𝑖,𝑥_𝑖+1] for 𝑖=1,…,𝑛−1;

(5) for any 𝑥∈[𝑎,𝑏], ∑^𝑛_𝑖=1𝐴_𝑖(𝑥)=1.

We say that {𝐴₁,𝐴₂,…,𝐴_𝑛} constitute a symmetric fuzzy partition if the following hold:

(6) equidistance of the nodes, that is, 𝑥_𝑖=𝑎+ℎ⋅(𝑖−1) for 𝑖=1,2,…,𝑛, where ℎ=(𝑏−𝑎)/(𝑛−1);

(7)𝐴_𝑖(𝑥_𝑖−𝑥)=𝐴_𝑖(𝑥_𝑖+𝑥) for any 𝑥∈[0,ℎ] and 𝑖=2,…,𝑛−1;

(8)  𝐴_𝑖+1(𝑥)=𝐴_𝑖(𝑥−ℎ) for any 𝑥∈[𝑥_𝑖,𝑥_𝑖+1] and 𝑖=1,2,…,𝑛−1.

Considering functions 𝑓 taking values on a finite set 𝑃={𝑝₁,…,𝑝_𝑚}⊆[𝑎,𝑏], 𝑓∶𝑃→[0,1], we suppose that 𝑃 is sufficiently dense with respect to a fuzzy partition {𝐴₁,𝐴₂,…,𝐴_𝑛} of [𝑎,𝑏], that is, if 𝑚>𝑛 and for each 𝑖=1,…,𝑛 there exists an index 𝑗∈{1,…,𝑚} such that 𝐴_𝑖(𝑝_𝑗)>0. Now let 𝑛,𝑚≥3, 𝑦₁,𝑦₂,…,𝑦_𝑚∈[𝑐,𝑑] be other 𝑚 assigned nodes such that 𝑦₁=𝑐<⋯<𝑦_𝑚=𝑑. Let 𝐶₁,…,𝐶_𝑚∶[𝑐,𝑑]→[0,1] be another fuzzy partitions of [𝑐,𝑑]. Let 𝑓∶𝑃×𝑄→[0,1] be a function defined on the finite set 𝑃×𝑄={𝑝₁,…,𝑝_𝑁}×{𝑞₁,…,𝑞_𝑀}⊆[𝑎,𝑏]×[𝑐,𝑑], with 𝑁>𝑛 and 𝑀>𝑚, where 𝑃 (resp., 𝑄) is sufficiently dense with respect to some fuzzy partition {𝐴₁,𝐴₂,…,𝐴_𝑛} of [𝑎,𝑏] (resp., {𝐶₁,…,𝐶_𝑚} of [𝑐,𝑑]). Then [𝐹_𝑘𝑙],𝐹_𝑘𝑙∈[0,1], 𝑘=1,…,𝑛 and 𝑙=1,…,𝑚, is the fuzzy matrix which is defined as discrete F-transform of 𝑓 with respect to {𝐴₁,𝐴₂,…,𝐴_𝑛} and {𝐶₁,…,𝐶_𝑚} if the following holds: 𝐹_𝑘𝑙=∑^𝑀_𝑗=1∑^𝑁_𝑖=1𝑓𝑝_𝑖,𝑞_𝑗𝐴_𝑘𝑝_𝑖𝐶_𝑙𝑞_𝑗∑^𝑀_𝑗=1∑^𝑁_𝑖=1𝐴_𝑘𝑝_𝑖𝐶_𝑙𝑞_𝑗. (1) Afterwards we define 𝑓^𝐹_𝑛𝑚∶𝑃×𝑄→[0,1] to be the inverse F-transform of 𝑓 with respect to {𝐴₁,𝐴₂,…,𝐴_𝑛} and {𝐶₁,…,𝐶_𝑚} as 𝑓^𝐹_𝑛𝑚𝑝_𝑖,𝑞_𝑗=^𝑛𝑚_𝑘=1𝑙=1𝐹_𝑘𝑙𝐴_𝑘𝑝_𝑖𝐶_𝑙𝑞_𝑗. (2) The following theorem holds.

Theorem 3. Let 𝑓∶𝑃×𝑄→[0,1] be a function assigned on 𝑃×𝑄={𝑝₁,…,𝑝_𝑁}×{𝑞₁,…,𝑞_𝑀}⊆[𝑎,𝑏]×[𝑐,𝑑]. Then for every 𝜀>0, there exist two integers 𝑛(𝜀), 𝑚(𝜀) with 𝑛(𝜀)<𝑁, 𝑚(𝜀)<𝑀 and some fuzzy partitions {𝐴₁,𝐴₂,…,𝐴_𝑛(𝜀)} of [𝑎,𝑏] and {𝐶₁,𝐶₂,…,𝐶_𝑚(𝜀)} of [𝑐,𝑑] for which 𝑃 and 𝑄 are sufficiently dense with respect to these partitions, respectively, and such that the following inequality holds for every 𝑖=1,…,𝑁, 𝑗=1,…,𝑀: 𝑓𝑝_𝑖,𝑞_𝑗−𝑓^𝐹_{𝑛(𝜀)𝑚(𝜀)}𝑝_𝑖,𝑞_𝑗<𝜀. (3)

3. The Coding/Decoding Process

Let 𝑅 be an image of sizes 𝑁×𝑀, considered as a fuzzy relation 𝑅∶(𝑖,𝑗)∈{1,…,𝑁}×{1,…,𝑀}→[0,1]; that is, 𝑅(𝑖,𝑗)=𝑃(𝑖,𝑗)/𝐿_𝑡, with 𝑃(𝑖,𝑗) being the normalized value of the pixel with respect to the length 𝐿𝑡 of the scale used. For simplicity, let 𝑝_𝑖=𝑖, 𝑞_𝑗=𝑗, 𝑎=𝑐=1, 𝑏=𝑁, and 𝑑=𝑀. Let the fuzzy sets 𝐴₁,…,𝐴_𝑛∶[1,𝑁]→[0,1] and 𝐶₁,…,𝐶_𝑚∶[1,𝑀]→[0,1], with 𝑛<𝑁 and 𝑚<𝑀, form a fuzzy partition of [1,𝑁] and [1,𝑀], respectively. Following [8], 𝑅 is subdivided in submatrices 𝑅_𝐵 of sizes 𝑁(𝑅_𝐵)×𝑀(𝑅_𝐵), 𝑅_𝐵∶(𝑖,𝑗)∈{1,…,𝑁(𝑅_𝐵)}×{1,…,𝑀(𝑅_𝐵)}→[0,1], called blocks, coded to matrices of sizes 𝑛(𝑅_𝐵)×𝑚(𝑅_𝐵), (𝑛(𝑅_𝐵)<𝑁(𝑅_𝐵), 𝑚(𝑅_𝐵)<𝑀(𝑅_𝐵)) via the following discrete F-transforms [𝐹^𝐵_𝑘𝑙] for every (𝑘,𝑙)∈{1,…,𝑛(𝑅_𝐵)}×{1,…,𝑚(𝑅_𝐵)} as 𝐹^𝐵_𝑘𝑙=∑^{𝑀(𝑅𝐵)}_𝑗=1∑^{𝑁(𝑅𝐵)}_𝑖=1𝑅_𝐵𝑖,𝑗𝐴_𝑘(𝑖)𝐶_𝑙𝑗∑^{𝑀(𝑅𝐵)}_𝑗=1∑^{𝑁(𝑅𝐵)}_𝑖=1𝐴_𝑘(𝑖)𝐶_𝑙𝑗, (4) and decode [𝐹^𝐵_𝑘𝑙] via 𝑅^𝐹_{𝑛(𝑅𝐵)𝑚(𝑅𝐵)}∶(𝑖,𝑗)∈{1,…,𝑁(𝑅_𝐵)}×{1,…,𝑀(𝑅_𝐵)}→[0,1] defined as 𝑅^𝐹_{𝑛(𝑅𝐵)𝑚(𝑅𝐵)}=^{𝑀(𝑅𝐵)}_𝑗=1^{𝑁(𝑅𝐵)}_𝑖=1𝐹^𝐵_𝑘𝑙𝐴_𝑘(𝑖)𝐶_𝑙𝑗 (5) which approximates 𝑅_𝐵 in the sense of Theorem 3; that is, there exist, for every 𝜀>0, two integers 𝑛(𝑅_𝐵,𝜀), 𝑚(𝑅_𝐵,𝜀) such that the following holds for every (𝑖,𝑗)∈{1,…,𝑁(𝑅_𝐵)}×{1,…,𝑀(𝑅_𝐵)}: 𝑅_𝐵𝑖,𝑗−𝑅^𝐹_{𝑛(𝑅𝐵,𝜀)𝑚(𝑅𝐵,𝜀)}𝑖,𝑗<𝜀. (6) Unfortunately the previous theorem does not suggest a method for finding such integers, and then we try to assign values to 𝑛(𝑅_𝐵)=𝑛(𝑅_𝐵,𝜀) and 𝑚(𝑅_𝐵)=𝑚(𝑅_𝐵,𝜀) for getting compression rates given by 𝜌𝑅_𝐵=𝑛𝑅_𝐵⋅𝑚𝑅_𝐵𝑁𝑅_𝐵⋅𝑀𝑅_𝐵 (7) which are useful to code any original block 𝑅_𝐵. The recomposition of the blocks 𝑅^𝐹_{𝑛(𝑅𝐵)𝑚(𝑅𝐵)} gives the image 𝑅^𝐹 whose PSNR with respect to the original image 𝑅 is calculated via the following well-known formula: PSNR𝑅,𝑅^𝐹=20log₁₀𝐿𝑡∑^𝑁_𝑖=1∑^𝑀_𝑗=1𝑅𝑖,𝑗−𝑅^𝐹𝑖,𝑗²./𝑁×𝑀 (8) In accordance with [8], in the proposed experiments the best results are deduced with the symmetric fuzzy partitions 𝐴₁,…,𝐴_{𝑛(𝑅𝐵)}∶[1,𝑁(𝑅_𝐵)]→[0,1] and 𝐶₁,…,𝐶_{𝑚(𝑅𝐵)}∶[1,𝑀(𝑅_𝐵)]→[0,1] defined as 𝐴₁(𝑖)=𝜋0.5cosℎ(𝑖−1)+1if1≤𝑖≤𝑥₂,𝐴0else,_𝑘(𝑖)=𝜋0.5cosℎ𝑖−𝑥_𝑘+1if𝑥_𝑘≤𝑖≤𝑥_𝑘+1,𝐴0else,_{𝑛(𝑅𝐵)}(𝑖)=𝜋0.5cosℎ𝑖−𝑥_{𝑛(𝑅𝐵−1)}+1if𝑥_{𝑛(𝐵)−1}≤𝑖≤𝑁𝑅_𝐵,0else, (9) where 𝑘=2,…,𝑛(𝑅_𝐵)−1, ℎ=(𝑁(𝑅_𝐵)−1)/(𝑛(𝑅_𝐵)−1), 𝑥_𝑘=1+ℎ⋅(𝑘−1), and 𝐶₁𝑗=𝜋0.5cos𝑠𝑗−1+1if1≤𝑗≤𝑦₂,𝐶0else,_𝑡𝑗=𝜋0.5cos𝑠𝑗−𝑦_𝑡+1if𝑦_𝑡−1≤𝑗≤𝑥_𝑡+1,𝐶0else,_{𝑚(𝑅𝐵)}𝑗=𝜋0.5cos𝑠𝑗−𝑦_{𝑚(𝑅𝐵)−1}+1if𝑦_{𝑚(𝑅𝐵)−1}≤𝑗≤𝑀𝑅_𝐵,0else, (10) where 𝑡=2,…,𝑚(𝑅_𝐵)−1, 𝑠=(𝑀(𝑅_𝐵)−1)/(𝑚(𝑅_𝐵)−1), and 𝑦_𝑡=1+𝑠⋅(𝑡−1).

4. Our Proposal

The proposed process includes the following steps:(1)each color frame, seen as a fuzzy relation, is converted from the space 𝑅𝐺𝐵 to the space 𝑌𝑈𝑉;(2)a classification of the frames is made via the previous algorithms;(3)the compression rate 𝜌_𝐼=𝜌_𝐼(𝑅_𝐵) of the I-frames is the mean of three (possibly different) compression rates used in the three bands, that is, if any block 𝑅_𝐵 of an I-frame has sizes (say) 𝑁_𝐼𝑌(𝑅_𝐵)×𝑀_𝐼𝑌(𝑅_𝐵) in the band 𝑌 and is coded to a block of sizes (say) 𝑛_𝐼𝑌(𝑅_𝐵)×𝑚_𝐼𝑌(𝑅_𝐵) for which the related compression rate is given by 𝜌_𝐼𝑌=𝜌_𝐼𝑌(𝑅_𝐵)=(𝑛_𝐼𝑌(𝑅_𝐵)⋅𝑚_𝐼𝑌(𝑅_𝐵))⋅(𝑁_𝐼𝑌(𝑅_𝐵)⋅𝑀_𝐼𝑌(𝑅_𝐵))⁻ and the analogous meaning has the symbols 𝜌_𝐼𝑈, 𝜌_𝐼𝑉. Of course we have 𝜌_𝐼=(𝜌_𝐼𝑌+𝜌_𝐼𝑈+𝜌_𝐼𝑉)/3. A similar meaning can be given to 𝜌_Δ=𝜌_Δ(𝑅_𝐵) (resp., 𝜌_𝑅=𝜌_𝑅(𝑅_𝐵)) for Δ-frames (resp., R-frames).

A color image in the 𝑅𝐺𝐵 space with pixels normalized in [0,1] is converted to 𝑌𝑈𝑉 space via the formula [5] 𝑌𝑈𝑉=𝑅𝐺𝐵+0.2990.5870.1140−0.169−0.3320.5000.500−0.419−0.08130.50.5. (11) Since no misunderstanding can arise, a frame is denoted by a capital letter instead of its ID number in a sequence of a video. In step (2), the similarity measure adopted in [5] is used for classifying the type of frame. It is based on the Lukasiewicz 𝑡-norm between two frames 𝐹 and 𝐺, with 𝐹,𝐺∶(𝑖,𝑗)∈{1,2,…,𝑁}×{1,2,…,𝑀}→[0,1], defined as Sim(𝐹,𝐺)=^𝑁𝑀_𝑖=1𝑗=1𝐹𝐹×()1−max𝑖,𝑗,𝐺𝑖,𝑗+min𝑖,𝑗,𝐺𝑖,𝑗𝑁×𝑀⁻¹. (12) In the 𝜇th band (𝜇∈{𝑌,𝑈,𝑉}) we will use the symbol Sim_𝜇(𝐹,𝐺). The authors [5] have shown that Lukasiewicz 𝑡-norm provides the best results with respect to other 𝑡-norms as the classical Min and the arithmetical product. For convenience, we assume that the first frame of a video is an I-frame. For determining a GOP sequence in a single band, it can be verified if the successive frame 𝐺 is a B-frame or a P-frame, that is, if it is “very similar” to the preceding I-frame 𝐹 in the sense that Sim(𝐹,𝐺)<Sim𝐵, with Sim𝐵∈[0,1] being a prefixed threshold value; otherwise 𝐺 is assumed to be a new I-frame. We determine a GOP sequence in an assigned band using (12) with the following process:(1)we consider the first frame 𝐹 as an I-frame;(2)we compare 𝐹 with the successive frame 𝐺; (3)if Sim(𝐹,𝐺)<Sim𝑃, the frame 𝐺 is a B-frame or a P-frame and is enclosed in the GOP sequence. Then we consider the successive frame 𝐺 and go to step (2); otherwise 𝐺 is a new I-frame. The previous frame is a P-frame and represents the last frame of the GOP sequence.

After determining the GOP sequence, we check if each frame of the sequence is a B-frame or a P-frame by using the previous algorithms. In step (3) we finally compress the frames. In order to reduce the mean compression rate for a P-frame, in [5] and references therein, the authors introduce a “difference” frame D, called Δ-frame, between a P-Frame 𝐺 and I-frame 𝐹 by defining 𝐷∶(𝑖,𝑗)∈{1,2,…,𝑁}×{1,2,…,𝑀}→[0,1] as 𝐷=𝐹𝑖,𝑗𝑖,𝑗−𝐺𝑖,𝑗+12. (13) The usage of the Δ-frame has the advantage of using a stronger compression rate for the P-frames with respect to the I-frames; indeed a P-frame 𝐺 has values in its pixels very close to the pixels of the previous I-frame. Hence the Δ-frame 𝐷 in (13) has a low quantity of information and it can be coded with a low compression rate. Then, if 𝐷^′ and 𝐹^′ are the frames obtained after coding/decoding 𝐹 and 𝐷, the frame 𝐺^′ (reconstruction of the frame 𝐺), with 𝐷^′, 𝐹^′, 𝐺^′∶(𝑖,𝑗)∈{1,2,…,𝑁}×{1,2,…,𝑀}→[0,1], is deduced from the membership values of 𝐹^′ and 𝐷^′ via the following formula: 𝐺^′𝑖,𝑗=max0,𝐹^′𝑖,𝑗−2𝐷^′𝑖,𝑗+1max1,𝐹^′𝑖,𝑗−2𝐷^′.𝑖,𝑗+1 (14) Now we present a new schema for coding/decoding a B-frame which is inserted in a GOP between an I-frame 𝐹 and a P-frame 𝐺. Then we consider a frame 𝑅 given by 𝑅=𝐹𝑖,𝑗𝑖,𝑗+𝐺𝑖,𝑗/2−𝐵𝑖,𝑗+12 (15) and we code it. Let 𝑅^′ be the frame obtained after decoding 𝑅, with 𝑅^′∶(𝑖,𝑗)∈{1,2,…,𝑁}×{1,2,…,𝑀}→[0,1]. All the coding/decoding processes are realized via the F-transforms with the symmetric fuzzy partition given in Section 3. We reconstruct the B-frame, say 𝐵^′, by combining the membership values of 𝐹^′, 𝐺^′, and 𝑅^′ via the following formula: 𝐵^′=𝐹𝑖,𝑗max0,^′𝑖,𝑗+𝐺^′𝑖,𝑗/2−2𝑅^′𝑖,𝑗+1max1,𝐹^′𝑖,𝑗+𝐺^′𝑖,𝑗/2−2𝑅^′.𝑖,𝑗+1 (16) We use the formulas (14) and (16) for reconstructing the P-frames and the B-frames in the videos, respectively. In accordance with [5], we convert each image in the 𝑅𝐺𝐵 space by using the formula 𝑅𝐺𝐵=𝑌𝑈𝑉+101.40751−0.3455−0.716911.779000.50.50.5. (17) For simplicity of presentation, in our tests here we adopt 𝑀(𝑅_𝐵)=𝑁(𝑅_𝐵), 𝑚(𝑅_𝐵)=𝑛(𝑅_𝐵). In [5] a preprocessing phase is adopted for determining the threshold Sim𝑃 calculated with the following steps: (1)if the initial frame 𝐹 is considered as an I-frame, we compress 𝐹 in the 𝜇th band (𝜇∈{𝑌,𝑈,𝑉}) with compression rate 𝜌_𝐼𝜇; each successive frame is a P-frame 𝐺 and we archive the similarity value Sim_𝜇(𝐹,𝐺) calculated with formula (12); we compress the Δ-frame 𝐷 in the 𝜇th band with compression rate equal to 𝜌_𝑃𝜇 (less than of 𝜌_𝐼𝜇) and if 𝐷^′ is the related decompressed frame, we derive the P-frame 𝐺^′ via (14);(2)each P-frame 𝐺 is also coded in the 𝜇th band with compression rate 𝜌_𝑃𝜇 and let 𝐺^′′ be the decoded P-frame by using directly the F-transforms; then we determine the difference diff(PSNR)=|PSNR(𝐺^′′,𝐺)−PSNR(𝐺^′,𝐺)|;(3)the trend of diff(PSNR) is plotted with respect to the similarity Sim_𝜇(𝐹,𝐺) in each band of the image. As similarity threshold, we assume that value of Sim_𝜇(𝐹,𝐺) such that diff(PSNR) does not exceed a prefixed limit is equal to 3 (cf. [5] for details);(4)then the threshold Sim𝑃 is given by Sim𝑃=max_𝐺∈GOPmaxSim_𝜇(𝐹,𝐺)∶𝜇∈{𝑌,𝑈,𝑉} (18)with 𝐹 being the first I-frame of the GOP sequence. In our tests, in addition we put Sim𝐵=Sim𝑃 in the preprocessing phase.

5. The Results

For brevity of discussion, we show the results obtained for the color video “tennis2” [6]. We present all the results by assuming 𝜌_𝐼≈0.262 for the I-frames, 𝜌_Δ≈0.027 for the Δ-frames, and 𝜌_𝑅≈0.020 for the R-frames. Figures 1(a)–1(d) show the first frame of the video and the corresponding single-band images in the 𝑌𝑈𝑉 space, respectively. As example of Diff(PSNR), Figure 2 contains the plots of Diff(PSNR) ≤ 3 for the similarity values obtained in 𝑌, 𝑈, and 𝑉 bands for which we choose Sim_𝑌(𝐹,𝐺)>0.948=Sim𝑃 (as average). As examples we show some Δ-frames and R-frames in each band.

(a)

(b)

(c)

(d)

Figure 1 

(a) Frame 1 of “tennis2” [6], (b) Frame 1 in 𝑌 band, (c) Frame 1 in 𝑈 band, and (d) Frame 1 in 𝑉 band.

Figure 2 

Diff(PSNR) with the similarity in 𝑌, 𝑈, and 𝑉 bands.

(i) 𝑌  Band. The first P-frame is given by the fourth frame. Figure 3(a) contains the Δ-frame obtained by using (13) from the fourth frame and the first frame (an I-frame). The second and the third frames are B-frames. Figure 3(b) (resp., Figure 3(c)) shows the R-frame obtained by using (15) from the second (resp., third) frame, the first frame (an I-frame), and the fourth frame (a P-frame).

(a)

(b)

(c)

Figure 3 

(a) Δ-frame from Frame 4 in 𝑌 band, (b) R-frame from Frame 2 in 𝑌 band, and (c) R-frame from Frame 3 in 𝑌 band.

(ii) 𝑈  Band. The first P-frame is given by the sixth frame. Figure 4(a) contains the Δ-frame obtained by using (13) from the sixth frame and the first frame (an I-frame). The frames 2, 3, and 4 are B-frames. Figures 4(b)–4(d) show the R-frames obtained by using (15) from the first frame (an I-frame), the B-frames 2, 3, and 4, and the sixth frame (a P-frame), respectively.

(a)

(b)

(c)

(d)

Figure 4 

(a) Δ-frame from Frame 6 in 𝑈 band, (b) R-frame from Frame 2 in 𝑈 band, (c) R-frame from Frame 3 in 𝑈 band, and (d) R-frame from Frame 4 in 𝑈 band.

(iii)𝑉  Band. The first P-frame is given by the fifth frame. Figure 5(a) contains the Δ-frame obtained by using (13) from the sixth frame and the first frame (an I-frame). The frames 2, 3, and 4 are B-frames. Figures 5(b)–5(d) show the R-frames obtained by using (15) from the first frame (an I-frame), the B-frames 2, 3, and 4, and the fifth frame (a P-frame), respectively.

(a)

(b)

(c)

(d)

Figure 5 

(a) Δ-frame from Frame 5 in 𝑉 band, (b) R-frame from Frame 2 in 𝑉 band, (c) R-frame from Frame 3 in 𝑉 band, and (d) R-frame from Frame 4 in 𝑉 band.

All the results obtained for the video “tennis2” are synthetized in Table 1.

Table 1 

Results for “tennis2” [6] in the proposed method.

Figures 6(a)–6(c) contain Frame 2 decoded with the proposed method, classical F-transforms, and MPEG-4, respectively.

(a)

(b)

(c)

Figure 6 

(a) Frame 2 in the proposed method, (b) Frame 2 in F-transforms, and (c) Frame 2 in MPEG-4.

In Table 2 we report the final PSNR index in the three methods.

Table 2 

Comparison with other methods for “tennis2” [6].

6. Conclusions

We present a new method for coding/decoding color videos, in which we classify a frame in I-frame, P-frame, and B-frame using similarity measures for determining the GOP sequences and the type of frames. Our method seems to be fully comparable with classical F-transforms and MPEG-4 for similar mean compression rates to a certain extent.

Acknowledgments

The authors thank the referees and the editor whose suggestions have greatly improved the contents of this paper.