US 20010016080 A1 Abstract A multiple description (MD) joint source-channel (JSC) encoder in accordance with the invention encodes n components of an image signal for transmission over m channels of a communication medium. In an illustrative embodiment which uses statistical redundancy between the different descriptions of the image signal, the encoder forms vectors from transform coefficients of the image signal separated both in frequency and in space. The vectors may be formed such that the spatial separation between the transform coefficients is maximized. A correlating transform is then applied, followed by entropy coding, grouping as a function of frequency, and application of a cascade transform. In an illustrative embodiment which uses deterministic redundancy between the different descriptions of the image signal, the encoder may apply a linear transform, followed by quantization, to generate the multiple descriptions of the image signal. For example, vectors may be formed from transform coefficients of the image signal so as to include coefficients of like frequency separated in space. The vectors are expanded by multiplication with a frame operator, and then quantized using a step size which may be a function of frequency.
Claims(22) 1. A method of processing an image signal for transmission, comprising the steps of:
encoding a plurality of components of the image signal in a multiple description joint source-channel encoder for transmission over a plurality of channels, wherein the encoding step includes forming vectors from coefficients of the image signal such that the coefficients associated with a given one of the vectors are separated in at least one of frequency and space; and transmitting the encoded components of the image signal. 2. The method of claim 1 3. The method of claim 1 4. The method of claim 1 5. The method of claim 4 6. The method of claim 4 computing a transform of the image;
quantizing coefficients of the resulting transform;
forming vectors of transform coefficients separated in frequency and space;
applying correlating transforms to at least a subset of the vectors;
applying entropy coding to the transformed vectors;
grouping the coded vectors as a function of frequency; and
applying a cascade transform to at least a subset of the resulting groups.
7. The method of claim 1 8. The method of claim 1 9. The method of claim 8 computing a transform of the image signal;
forming vectors from coefficients of the resulting transform, wherein each vector includes coefficients of like frequency, separated in space;
expanding the vectors by multiplication with a frame operator; and
quantizing the expanded vectors using a quantization step size which is a function of frequency.
10. The method of claim 1 11. An apparatus for encoding an image signal for transmission, comprising:
a multiple description joint source-channel encoder for encoding a plurality of components of the image signal for transmission over a plurality of channels, wherein the encoder forms vectors from coefficients of the image signal such that the coefficients associated with a given one of the vectors are separated in at least one of frequency and space. 12. The apparatus of claim 11 13. The apparatus of claim 11 14. The apparatus of claim 11 15. The apparatus of claim 14 16. The apparatus of claim 14 17. The apparatus of claim 11 18. The apparatus of claim 11 19. The apparatus of claim 18 20. The apparatus of claim 11 21. The apparatus of claim 11 22. The apparatus of claim 21 Description [0001] The present application is a continuation-in-part of U.S. patent application Ser. No. 09/030,488 filed Feb. 25, 1998 in the name of inventors Vivek K. Goyal and Jelena Kovacevic and entitled “Multiple Description Transform Coding Using Optimal Transforms of Arbitrary Dimension.” [0002] The present invention relates generally to multiple description transform coding (MDTC) of signals for transmission over a network or other type of communication medium, and more particularly to MDTC of images. [0003] Multiple description transform coding (MDTC) is a type of joint source-channel coding (JSC) designed for transmission channels which are subject to failure or “erasure.” The objective of MDTC is to ensure that a decoder which receives an arbitrary subset of the channels can produce a useful reconstruction of the original signal. One type of MDTC introduces correlation between transmitted coefficients in a known, controlled manner so that lost coefficients can be statistically estimated from received coefficients. This correlation is used at the decoder at the coefficient level, as opposed to the bit level, so it is fundamentally different than techniques that use information about the transmitted data to produce likelihood information for the channel decoder. The latter is a common element in other types of JSC coding systems, as shown, for example, in P. G. Sherwood and K. Zeger, “Error Protection of Wavelet Coded Images Using Residual Source Redundancy,” Proc. of the 31st Asilomar Conference on Signals, Systems and Computers, November 1997. Other types of MDTC may be based on techniques such as frame expansions, as described in V. K. Goyal et al., “Multiple Description Transform Coding: Robustness to Erasures Using Tight Frame Expansions,” In Proc. IEEE Int. Symp. Inform. Theory, August 1998. [0004] A known MDTC technique for coding pairs of independent Gaussian random variables is described in M. T. Orchard et al., “Redundancy Rate-Distortion Analysis of Multiple Description Coding Using Pairwise Correlating Transforms,” Proc. IEEE Int. Conf. Image Proc., Santa Barbara, Calif., October 1997. This MDTC technique provides optimal 2×2 transforms for coding pairs of signals for transmission over two channels. However, this technique as well as other conventional techniques fail to provide optimal generalized n×m transforms for coding any n signal components for transmission over any m channels. In addition, conventional transforms such as those in the M. T. Orchard et al. reference fail to provide a sufficient number of degrees of freedom, and are therefore unduly limited in terms of design flexibility. Moreover, the optimality of the 2×2 transforms in the M. T. Orchard et al. reference requires that the channel failures be independent and have equal probabilities. The conventional techniques thus generally do not provide optimal transforms for applications in which, for example, channel failures either are dependent or have unequal probabilities, or both. These and other drawbacks of conventional MDTC prevent its effective implementation in many important applications. [0005] The invention provides MDTC techniques which can be used to implement optimal or near-optimal n×m transforms for coding any number n of signal components for transmission over any number m of channels. A multiple description (MD) joint source-channel (JSC) encoder in accordance with an illustrative embodiment of the invention encodes n components of an image signal for transmission over m channels of a communication medium, in applications in which at least one of n and m may be greater than two, and in which the failure probabilities of the m channels may be non-independent and non-equivalent. [0006] In accordance with one aspect of the invention, the MD JSC encoder may be configured to provide statistical redundancy between different descriptions of the image signal. For example, the encoder may form vectors from discrete cosine transform (DCT) coefficients of the image signal separated both in frequency and in space. The vectors may be formed such that the spatial separation between the DCT coefficients is maximized. A correlating transform is applied to the resulting vectors, followed by entropy coding, grouping of the coded vectors as a function of frequency, and application of a cascade transform to each of the groups, in order to generate the multiple descriptions of the image signal. [0007] In accordance with another aspect of the invention, the MD JSC encoder may be configured to provide deterministic redundancy between different descriptions of the image signal. For example, the encoder may form vectors from DCT coefficients of the image signal so as to include coefficients of like frequency separated in space. The vectors are expanded by multiplication with a frame operator, and then quantized using a step size which may be a function of frequency, in order to generate the multiple descriptions of the image signal. In both the statistical redundancy and deterministic redundancy embodiments noted above, other types of linear transforms may be used in place of the DCT. [0008] An MD JSC encoder in accordance with the invention may include a series combination of N “macro” MD encoders followed by an entropy coder, and each of the N macro MD encoders includes a parallel arrangement of M “micro” MD encoders. Each of the M micro MD encoders implements one of: (i) a quantizer block followed by a transform block, (ii) a transform block followed by a quantizer block, (iii) a quantizer block with no transform block, and (iv) an identity function. In addition, a given n×m transform implemented by the MD JSC encoder may be in the form of a cascade structure of several transforms each having dimension less than n×m. This general MD JSC encoder structure allows the encoder to implement any desired n×m transform while also minimizing design complexity. [0009] The MDTC techniques of the invention do not require independent or equivalent channel failure probabilities. As a result, the invention allows MDTC to be implemented effectively in a much wider range of applications than has heretofore been possible using conventional techniques. The MDTC techniques of the invention are suitable for use in conjunction with signal transmission over many different types of channels, including, for example, lossy packet networks such as the Internet, wireless networks, and broadband ATM networks. [0010]FIG. 1 shows an exemplary communication system in accordance with the invention. [0011]FIG. 2 shows a multiple description (MD) joint source-channel (JSC) encoder in accordance with the invention. [0012]FIG. 3 shows an exemplary macro MD encoder for use in the MD JSC encoder of FIG. 2. [0013]FIG. 4 shows an entropy encoder for use in the MD JSC encoder of FIG. 2. [0014]FIGS. 5A through 5D show exemplary micro MD encoders for use in the macro MD encoder of FIG. 3. [0015]FIGS. 6A, 6B and [0016]FIG. 7 illustrates an exemplary 4×4 cascade structure which may be used in an MD JSC encoder in accordance with the invention. [0017]FIGS. 8 and 9 are flow diagrams illustrating exemplary image encoding processes in accordance with the invention. [0018] The invention will be illustrated below in conjunction with exemplary MDTC systems. The techniques described may be applied to transmission of a wide variety of different types of signals, including data signals, speech signals, audio signals, image signals, and video signals, in either compressed or uncompressed formats. The term “channel” as used herein refers generally to any type of communication medium for conveying a portion of an encoded signal, and is intended to include a packet or a group of packets. The term “packet” is intended to include any portion of an encoded signal suitable for transmission as a unit over a network or other type of communication medium. The term “linear transform” should be understood to include a discrete cosine transform (DCT) as well as any other type of linear transform. The term “vector” as used herein is intended to include any grouping of coefficients or other elements representative of at least a portion of a signal. [0019]FIG. 1 shows a communication system [0020]FIG. 2 illustrates the MD JSC encoder [0021]FIG. 4 indicates that the entropy coder [0022]FIGS. 5A through 5D illustrate a number of possible embodiments for each of the micro MD [0023]FIGS. 6A through 6C illustrate the manner in which the MD JSC encoder [0024] A general model for analyzing MDTC techniques in accordance with the invention will now be described. Assume that a source sequence {x [0025] An MDTC coding structure for implementation in the MD JSC encoder [0026] 1. The source vector x is quantized using a uniform scalar quantizer with stepsize Δ:x [0027] 2. The vector x [0028] 3. The components of y are independently entropy coded. [0029] 4. If m>n, the components of y are grouped to be sent over the m channels. [0030] When all of the components of y are received, the reconstruction process is to exactly invert the transform {circumflex over (T)} to get {circumflex over (x)}=x [0031] Starting with a linear transform T with a determinant of one, the first step in deriving a discrete version {circumflex over (T)} is to factor T into “lifting” steps. This means that T is factored into a product of lower and upper triangular matrices with unit diagonals T=T {circumflex over (T)}(x [0032] The lifting structure ensures that the inverse of {circumflex over (T)} can be implemented by reversing the calculations in (1): {circumflex over (T)} [0033] The factorization of T is not unique. Different factorizations yield different discrete transforms, except in the limit as Δ approaches zero. The above-described coding structure is a generalization of a 2×2 structure described in the above-cited M. T. Orchard et al. reference. As previously noted, this reference considered only a subset of the possible 2×2 transforms; namely, those implementable in two lifting steps. [0034] It is important to note that the illustrative embodiment of the invention described above first quantizes and then applies a discrete transform. If one were to instead apply a continuous transform first and then quantize, the use of a nonorthogonal transform could lead to non-cubic partition cells, which are inherently suboptimal among the class of partition cells obtainable with scalar quantization. See, for example, A. Gersho and R. M. Gray, “Vector Quantization and Signal Compression,” Kluwer Acad. Pub., Boston, Mass., 1992. The above embodiment permits the use of discrete transforms derived from nonorthogonal linear transforms, resulting in improved performance. [0035] An analysis of an exemplary MDTC system in accordance with the invention will now be described. This analysis is based on a number of fine quantization approximations which are generally valid for small Δ. First, it is assumed that the scalar entropy of y={circumflex over (T)}([x] [0036] The rate can be estimated as follows. Since the quantization is fine, y [0037] where k [0038] The minimum rate occurs when the product from i=1 to n of σ [0039] The distortion will now be estimated, considering first the average distortion due only to quantization. Since the quantization noise is approximately uniform, the distortion is Δ [0040] and is independent of T. [0041] The case when l>0 components are lost will now be considered. It first must be determined how the reconstruction will proceed. By renumbering the components if necessary, assume that y {circumflex over (x)}=E[x|y [0042] [0043] If the correlation matrix of y is partitioned in a way compatible with the partition ofy as: then it can be shown that the conditional signal y [0044] correlation matrix A Δ R [0045] such that ∥x−{circumflex over (x)}∥ is given by:
[0046] where U is the last l columns of T [0047] The distortion with l erasures is denoted by D [0048] For a case in which each channel has a failure probability of p and the channel failures are independent, the weighting
[0049] makes the weighted sum {overscore (D)} the overall expected MSE. Other choices of weighting could be used in alternative embodiments. Consider an image coding example in which an image is split over ten packets. One might want acceptable image quality as long as eight or more packets are received. In this case, one could set α [0050] The above expressions may be used to determine optimal transforms which minimize the weighted sum {overscore (D)} for a given rate R. Analytical solutions to this minimization problem are possible in many applications. For example, an analytical solution is possible for the general case in which n=2 components are sent over m=2 channels, where the channel failures have unequal probabilities and may be dependent. Assume that the channel failure probabilities in this general case are as given in the following table.
[0051] If the transform T is given by:
[0052] minimizing (2) over transforms with a determinant of one gives a minimum possible rate of: [0053] The difference ρ=R−R* is referred to as the redundancy, i.e., the price that is paid to reduce the distortion in the presence of erasures. Applying the above expressions for rate and distortion to this example, and assuming that σ [0054] The optimal value of bc is then given by:
[0055] The value of (bc) [0056] If p [0057] Using a transform from this set gives:
[0058] For values of σ [0059] Although the conventional 2×2 transforms described in the above-cited M. T. Orchard et al. reference can be shown to fall within the optimal set of transforms described herein when channel failures are independent and equally likely, the conventional transforms fail to provide the above-noted extra degree of freedom, and are therefore unduly limited in terms of design flexibility. Moreover, the conventional transforms in the M. T. Orchard et al. reference do not provide channels with equal rate (or, equivalently, equal power). The extra degree of freedom in the above example can be used to ensure that the channels have equal rate, i.e., that R [0060] As previously noted, the invention may be applied to any number of components and any number of channels. For example, the above-described analysis of rate and distortion may be applied to transmission of n=3 components over m=3 channels. Although it becomes more complicated to obtain a closed form solution, various simplifications can be made in order to obtain a near-optimal solution. If it is assumed in this example that σ [0061] Optimal or near-optimal transforms can be generated in a similar manner for any desired number of components and number of channels. [0062]FIG. 7 illustrates one possible way in which the MDTC techniques described above can be extended to an arbitrary number of channels, while maintaining reasonable ease of transform design. This 4×4 transform embodiment utilizes a cascade structure of 2×2 transforms, which simplifies the transform design, as well as the encoding and decoding processes (both with and without erasures), when compared to use of a general 4×4 transform. In this embodiment, a 2×2 transform T [0063] Illustrative embodiments of the invention more particularly directed to transmission of images will be described below with reference to the flow diagrams of FIGS. [0064] The flow diagram of FIG. 8 illustrates an example of an MDTC process particularly well suited for use with still images. In this example, the process codes four channels using a technique which operates on source vectors with uncorrelated components. In accordance with the invention, a suitable approximation of this condition can be obtained by forming vectors from discrete cosine transform (DCT) coefficients separated both in frequency and in space. It should be noted that the use of the DCT in the embodiments of FIGS. [0065] After the above steps [0066] In the embodiment of FIG. 8, the importance of the DC coefficient may dictate allocating most of the redundancy to the group containing the DC coefficient. In an alternative embodiment, it may be assumed that the quantized DC coefficient is communicated reliably through some other means, e.g., a separate channel. The remaining coefficients are then separated, e.g, into those that are placed in groups of four and those that are sent by one of the four channels only. Because the optimal allocation of redundancy between the groups is often difficult to determine, it may instead be desirable to allocate approximately the same redundancy to each group. The AC coefficients for each block are then sent over one of the four channels. It can be shown that such an embodiment provides a higher quality reconstructed image when one of four packets is lost, at the expense of worse rate-distortion performance when there are no packet losses. In addition, the expected number of bits for each channel is approximately equal, which facilitates packetization. This is in contrast to certain conventional techniques in which one must multiplex channel bit streams in order to produce packets of approximately the same size. [0067] It should be noted that effects of factors such as coarse quantization, dead zone, divergence from Gaussian, run length coding and Huffinan coding are not addressed in the above examples, but could be addressed through, e.g., an expansive numerical optimization. The encoding process could be further improved by, e.g., using a perceptually tuned quantization matrix as suggested by the JPEG standard, rather than the uniform quantization used for simplicity in the above examples. Using perceptually tuned quantization, one can design a system which, e.g., performs as well as conventional systems when two or four of four packets arrive, but which performs better when one or three packets arrive. [0068] In the embodiment of FIG. 8, the redundancy in the source representation is statistical, i.e., the distribution of one part of the representation is reduced in variance by conditioning on another part. Another possible technique for implementing MDTC of images in accordance with the invention, illustrated in the flow diagram of FIG. 9, uses a deterministic redundancy between descriptions. Consider a conventional discrete block code which represents k input symbols through a set of n output symbols such that any k of the n can be used to recover the original k. One possible example is a systematic (n, k) Reed-Solomon code over GF(2 [0069] An alternative to the above-described discrete block coding involves using a linear transform from R [0070] Assume that we have a tight frame Φ={φ [0071] The flow diagram of FIG. 9 is an example of the above-described deterministic redundancy approach, using a frame alternative to a (10, 8) block code. For the 10×8 frame operator F we use a matrix corresponding to a length −10 real Discrete Fourier Transform (DFT) of a length-8 sequence. This matrix can be constructed as F=[F [0072] In order to obtain the benefit of perceptual tuning, we apply this technique to DCT coefficients and use quantization step sizes as in a typical JPEG decoder. FIG. 9 illustrates the encoding process. In step [0073] The reconstruction for the above-described frame-based process may follow a least-squares strategy. It can be shown that the frame-based process of FIG. 9 provides better performance than a corresponding systematic block code when less than eight packets are received, and the performance degrades gracefully as the number of lost packets increases. It should be noted, however, that the process of FIG. 9 may not provide better performance than the corresponding block code when all ten packets are received. [0074] The above-described embodiments of the invention are intended to be illustrative only. For example, image characteristics, e.g., resolution, block size, etc., coding parameters, e.g., quantization, frame type, etc., and other aspects of the examples of FIGS. Referenced by
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