Transmission errors recovery using fuzzy block similarity measures B. Solaiman, R. Pyndiah, O. Aitsab and Ch. Roux Télécomm Bretagne, B.P. 832, 29285 Brest Cedex, France Tel : 33 02 98 00 13 08, Fax : 33 02 98 00 10 98, e.mail :
[email protected] ABSTRACT In addition to the source coding artifacts, the encoded bit streams representing codeword indexes of a vector quantized image are vulnerable to transmission or media impairments. Impulse block noise in the received images is the main artifact due to transmission errors. In this paper, we use the fuzzy set theory to represent the vague concept of a block similarity with its spatial context. This approach is conducted in order to detect and to conceal the transmission errors. Error concealment by searching the best matching codeword is an attractive alternative to the commonly used interpolation approach. It reduces considerably the computational complexity at the receiver end. Simulation results show that the proposed method considerably improves the subjective quality of VQ images transmitted over noisy channels.
1. Introduction Vector Quantization (VQ) techniques have revealed a high potential for low bit rate image coding with relatively low complexity. The original image is first divided into small nxn non-overlapping blocks. VQ is then defined as a mapping Φ from ℜN (with N=n2) into a finite subset Ω of ℜN, where Ω={C1, ..,Ci, .. ,CM}, the VQ codebook, is a set of prototype vectors, Ci, called codewords. For each source block X, a codeword Ci in Ω is selected as the representation of X and only the index "i" of this codeword in the codebook is transmitted. In addition to the source coding VQ artifacts (block effect and staircase noise along the edges), the VQ indexes encoded bit stream is vulnerable to transmission errors resulting in a visually annoying impulse block noise in the reconstructed images. Several techniques have already been proposed when dealing with VQ indexes transmitted over noisy channels. Phamdo and Farvardin [1] have proposed the use of the correlation between successive indexes in a Markovian framework to correct the channel errors. Zeger and Gersho [2], developed a pseudo-gray coding to reassign the indexes of the codebook to reduce the distortion introduced by noisy channels. They rearrange the codebook so that similar indexes lie near each other in the input space. The ordered codebook design was also proposed using the Kohonen's Self Organization Feature Maps (SOFM) neural network[3-4]. Nevertheless, if the
transmission channel is a multipath Rayleigh channel, or, if the modulation scheme does not preserve the topological feature of the SOFM, then, the ordered nature of the codebook is highly degraded [4]. Therefore, the use of a post-processing algorithm in the spatial domain in order to locate and to recover erroneously received blocks is of great interest.
2. Similarity concept The "similarity" concept corresponds to a common form of information exchange among humans. The answer to the question whether a block is similar to its immediate neighboring is not a simple binary answer. For a given 8neighboring blocks surrounding the current block, all codewords are similar "to a certain degree" to this spatial context. In this study, the Fuzzy Set theory, provides a suitable mathematical tool to deal with problems where human reasoning and human perception are involved. Using the fuzzy sets notations, the similarity concept is defined as follows. Let the codebook set, Ω={C1, ..,CM}, define the universe of discourse (i.e; universal set) and Θ = {B1, B2, ..., B8} denotes the ordered set of the 8neighboring blocks of the current block B0. 8-neighboring blocks of block "B " 0 B B 1 2 B 4
B 3 B 5
B B 6 7
B 8
Figure 1, Block B0 and its 8- neighboring blocks Each block B is divided into four sub-blocks {SB1, SB2, SB3, SB4}. The different sub-blocks are not necessarily disjoint and may have some overlapping pixels. Each subblock is then characterized by the mean gray level value of its constituting pixels mSBi. The block similarity problem with its 8-neighboring blocks is then "distributed" to the constituting sub-blocks similarities with their own closest sub-blocks as illustrated in figure 2. Each sub-block SBi is surrounded by five sub-blocks belonging to three neighboring blocks {SBi (j), i=1, 2, 3, 4, and j = 1, 2,.., 5}. The sub-blocks similarity is thus defined as the spatial
mean gray level continuity between two adjacent subblocks. The sub-block SBi is said to be "close" to its neighboring if it is close enough to at least one of its neighboring sub-blocks SBi(j).
neighboring. In this case, the threshold must be selected close to unity. On the contrary, for higher σ values (corresponding to edge blocks), the threshold must be selected small enough in order to avoid false alarms. Therefore, the use of a threshold as a function of σ seems adequate. The following exponential function relating the threshold ψ to the standard deviation σ, was found to realize a good tradeoff (where "a" is a positive real constant): ψ =
e -aσ
(4)
3. Errors recovery : considered sub-block
: sub-blocks of interest
Figure 2, SB similarity with their neighboring Therefore, if Ai/Θ denotes the fuzzy set over the universal set of codewords and defining the sub-block SBi similarity with Θ, then, the membership strength to Ai/Θ is given by the following fuzzy If-Then rule If {(SBi is close to SBi (1)) .OR.... .OR. (SBi is close to SBi (5)) } Then {(SBi is Ai/Θ)} (1) Hence, the overall strength µAi/Θ(SBi) is given by µAi/Θ(SBi)=Max (µClose(SBi,SBi(j))) ,j=1,.,5 (2) where, µClose(SBi, SBi(j)) is a triangular fuzzy membership function defined on the parameter mSBimSBi(j). Finally, and conditionally to the set Θ, the vague statement "Block B0 is similar to its spatial context" defines a fuzzy set "A/Θ" over the universal set Ω. The similarity strength of the current block B0 with its 8neighboring blocks, µA/Θ(B0), is maximized when all the elementary sub-blocks, constituting the block B0, assume the spatial mean gray level continuity: µA/Θ(B0) = Min ( µAj/Θ(SBj) ), j=1, ..,4 (3) An important issue remains: for which similarity strengths can the block be considered (in the binary sense) as similar or not to its surrounding blocks?. The simplest way is to use a fixed threshold ψ for which, if µA/Θ(B0) < ψ, then the corresponding transmitted index of the block B0 is considered as being affected by transmission errors. If a small threshold value is selected, then, a very small amount of erroneously received blocks can be detected. On the contrary, larger threshold values will result in an important amount of "false alarms" (i.e.; correctly received blocks for which the similarity strength is less than the fixed threshold). Most of these false alarms will generally occur on contour blocks. In order to realize a tradeoff between the number of correctly detected erroneous blocks and false alarms, the use of an adaptive threshold is a key issue of the proposed approach. In this study, the adaptive threshold is determined as a function of the standard deviation "σ" of the set of pixels contained in the 8-neighboring blocks Θ. Small values of σ means that the considered block is surrounded by a homogeneous
In order to enhance the reconstructed images transmitted over the noisy channels, the common Automatic Retransmission Request technique based on the retransmission of erroneous blocks is often considered. This technique is not very efficient for real time video or point to multipoint transmission systems. Recently, postprocessing techniques have been proposed for solving this problem. In these techniques, erroneous blocks are reconstructed by using the available data in the neighboring blocks through the use of an interpolation method. Linear interpolation [5], multi-directional edgebased interpolation [6] and the Bayesian interpolation [7] are some of the methods currently used for this task. In the particular case of VQ images, this interpolation problem can be avoided. In fact, given that the basic "alphabet" used in reconstructing the images is the codebook, then, the search for the best matching codeword (in terms of the most similar) constitutes a good alternative to the interpolation approach. Given the codebook Ω ={C1, ..,Ci, .. ,CM}, and assume that a given block B0 is "judged" as an erroneously received block. In this case, and conditionally to the set of its 8-neighboring blocks Θ = {B1, B2, ..., B8}, the error concealment problem is formulated as follows: Let Si , i=1, 2, .., M, denote the similarity strength between the codeword Ci (considered instead of the erroneously received codeword, B0 ) and its 8-neighboring set of blocks Θ. The best matching codeword Ci0 is then, the one maximizing Si Ci0 iff. Si0 = Max (Si) , i= 1, ..,M (5) Therefore, the previous methods defining the similarity between a given block and its 8-neighboring blocks, for detecting the erroneous blocks can be extended for the error concealment process.
4. Simulation results The transmission channel considered in the simulated system is first considered as an additive noise channel where the noise is assumed to be white and Gaussian. 3x3
blocks are considered in this study. Codebook design is performed using a three dimensional SOFM, 8x8x8 (512 codewords). This network is trained using source blocks issued from the Boat and the Bridge standard images. The VQ of the Lena and the F16 standard images corrupted by transmission errors and their enhanced versions are depicted in figure 3. The Symbol Error Rate, SER, for the received images is close to 6% corresponding to 1852 erroneously received blocks. Note that the topology preserving property of the SOFMs obtained through the ordered codebook coupled with a 64-QAM modulation [4] (having constellation points distributed similarly to the codewords in the SOFM) reduce considerably the visual impulse block noise artifact. "Residual" visible erroneous blocks correspond to those for which the erroneous indexes are quite "far" (on the SOFM) from the transmitted codewords. The first part of the conducted simulations concerns the evaluation of the proposed approach for detecting erroneous received blocks. The performances of the error detection approaches are evaluated using the two following indicators: 1) the number of Correctly Detected erroneous Blocks, CDB, and 2) the number of False Alarm Blocks, FAB, (i.e.; correct blocks detected as erroneous blocks). The obtained results are resumed in Table I. The use of a Gaussian channel model associated with the ordered codebook is thus shown to be a robust approach to transmission errors. In this case, and statistically speaking, the erroneous blocks are mainly "located" in the immediate vicinity of the originally transmitted codeword (on the SOFM). Thus, the proposed approach finds its interest in detecting erroneous blocks that are located far from the originally transmitted codewords on the SOFM. This situation is often encountered when the channel is a multipath Rayleigh channel or when the modulation is not adapted to the topology preserving feature of the Kohonen's SOFM. Using a Rayleigh channel model, obtained results concerning the image Lena are depicted in figure 4. In this case, a high SER (28%) is simulated. Visual inspection of the enhanced image shows that the impulse block noise is considerably reduced. Therefore, the proposed approach using an ordered SOFM codebook coupled with the fuzzy similarity measure (permitting to locate and to recover the erroneously received blocks in the spatial domain) constitutes a robust approach in VQ image transmission over noisy channels.
5. Conclusions
In this paper, the vague concept used in describing the block similarity with its immediate neighboring is formulated using the fuzzy set theory. The proposed approach seems promising in errors detection and recovery. Another attractive feature is that the proposed approach is parameter free. The only parameter used is the slant "a" applied in the adaptive threshold selection. This parameter is determined once as a function of the global number of potential number of erroneous blocks that the real time constraint of the proposed system is appealed to deal with. Actual studies are conducted in order to extend the proposed approach to higher block dimensions (≥ 4x4) and to the VQ video images enhancement. Acknowledgement This work has been supported by the Centre National d'Etudes des Télécommunications (CNET) and the Centre Commun d'Etudes de Télédiffusion et Télécommunication (CCETT). The authors would like to thank P.Siohan for his support and for the valuable discussions.
References [1]N.Phamdo et. al,"Optimal detection of discrete Markov sources over discrete memoryless channels-Application to combined source-channel coding," IEEE Trans. Inform. Theory, vol.40, no.1, pp.186-193, Jan1994. [2]K.Zeger and A.Gersho,"Pseudo-gray coding", IEEETrans.Com.v38, n.12, pp. 2147-2158, Dec. 90. [3]P.Knagenhjelm,"A recursive design method for robust vector quantization," Intern.Conf. Signal Proc. Applic. Technol., Boston, MA, pp.948-954, Nov. 92. [4]O.Aitsab, R. Pyndiah and B. Solaiman, "Optimization of multi-dimensional SOFM codebooks with QAM modulations for vector quantized images transmission," 3d Int. Workshop on Image and Signal Proc., 4-7 November, 1996, Manchester, United Kingdom. [5]Y. Wang and Q. Zhu, "Signal Loss Recovery in DCT-based image and video codecs," SPIE Conf. on Visual Commun. and Image Processing, vol. 1605, pp. 667-678, Boston, Nov. 1991. [6]W.Kwok & H.Sun, "Multidirectional Interpolation for spatial error concealment," IEEE Trans. Cons.r Electronics, vol. 39, no. 3, pp. 455-460, Aug. 1993. [7]P. Salama, N. Sharoff and E. J. Delp, "Error Concealment Techniques for encoded video streams, " ICIP'96, EPFL Switzerland, Sep. 1996.
(a)
(b)
(c) (d) Figure 3; Received (over a Gaussian channel) and the enhanced images: Lena (a),(b) and F16 (c), (d) PSNR
PSNR
SER
CDB
FAB
PSNR
(VQ image)
(received image)
(enhanced image)
Lena
30.74 dB
29.63 dB
6%
261
49
30.07 dB
F16
28.81 dB
26.32 dB
5.4 %
881
476
28.09 dB
Table. I, Performance indicators using the standard Lena and F16 images
(a) (b) Fig 4; Received as well as the enhanced Lena image over a Rayleigh transmission channel
