BACKGROUND OF THE INVENTION TECHNICAL FIELD The invention relates to data compression. More particularly, the invention relates to a novel data compression technique referred to as zero-search, zero-memory vector quantization.

DESCRIPTION OF THE PRIOR ART The subject of this invention is a new technique for data compression, referred to herein as zero-search, zero-memory vector quantization. Vector quantization is a well-know and widely practiced method of lossy data compression. By"data compression"is meant that some body of data, typically a sequence of vectors of some fixed dimension, and requiring some amount of memory (for storage) or communication bandwidth (for transmission), is converted into a smaller amount of data. From this converted representation, usually referred to as the compressed form of the data, a reasonable facsimile of the original body of data may be reconstructed, via an appropriate decompression algorithm. Because data

reconstructed by this method may not exactly match the original, the scheme is said to be lossy. By contrast, a compression scheme with the property that the original data may always be exactly reconstructed from its compressed representation is said to be lossless.

Vector quantization operates by establishing a small set of vectors, called a codebook, which are representative of those that will be processed by the deployed system. When a vector is presented for compression, a computation determines the codebook entry that is closest to it, and the index of this entry within the codebook, rather than the vector itself, is transmitted (or stored) as a proxy of the input. Upon receipt (or readback) of index i, the input data are reconstructed by extracting the Ah entry from the codebook, and presenting this vector as a facsimile of the original. Though it can achieve very high rates of compression, two significant drawbacks of this method are: (1) The need to store the codebook (if the system is being used for data transmission, this must be done at both the sender and the receiver), and (2) The need for the sender to search the codebook, to find the closest match to the input vector.

SUMMARY OF THE INVENTION In this document we disclose a method for data compression, akin to but different from the existing vector quantization technique, that removes these drawbacks.

There is no explicit codebook and no search; the codebook memory and associated search computation are eliminated. Some memory and computation are still required, but these are dramatically reduced, compared to systems that do not exploit this method. The method has been validated in a demanding, real-

world data compression task, and found to yield over a 30-fold reduction in computation, and over a 300-fold reduction in memory (compared with a system that does not exploit the invention), while maintaining good fidelity of reconstruction. The invention teaches a variety of algorithms for compression; these are all exponentially more efficient, in both computation time and use of memory, than methods that do not exploit the invention.

Because there is no explicit codebook to be stored or searched, no such codebook need be generated either. This makes the method well suited to adaptive coding schemes, where the compression system adapts to the statistics of the data presented for processing. Both the complexity of the algorithm executed for adaptation, and the amount of data transmitted to synchronize the sender and receiver, are exponential more efficient than existing methods.

BRIEF DESCRIPTION OF THE DRAWINGS Figures 1 a and 1 b show data for compression, where Figure 1 a shows 200 two- dimensional vectors plotted, each marked by a dot, and seen to fall naturally into eight clusters, and Figure 1 b shows the same data plotted, along with nominal cluster centers, each marked by a star; Figure 2 shows the conventional K-Means algorithm, which is a widely used algorithm for determining vector quantization codebooks; Figure 3a is a Voronoi diagram which shows the cell boundaries, which are points in the plane equidistant from the code points of adjoining cells ; Figure 3b shows a Voronoi diagram with data points added;

Figures 4a and 4b show symmetry properties of adjacent code points and associated cell boundaries; Figure 5a shows a square and its Voronoi diagram; Figure 5b shows a variety of rectangles, each with (the same) perfect Voronoi diagram; Figure 6 show a perfect hypercube codebook, for D = 3; Figure 7a shows sampled symmetric data, where 1000 points are sampled from a uniform distribution on a disc of radius 2; Figure 7b shows a K-means solution, where a 4-point codebook is computed via 100 K-means iterations; Figures 8a and 8b show equivalent codebooks and Voronoi diagrams at two rotations; Figure 9 shows an optimal codebook with a perfect Voronoi diagram; Figures 10a and 10b show the intended effect of the symmetrizing transform, where Figure 10a shows data before applying the transform (15', 1000 points distributed within an ellipse) ; and Figure 10b shows data after applying the transform (T , same 1000 points, transformed by T) ; Figure 11 shows an algorithm to compress a vector v ;

Figure 12 shows an algorithm to decompress an index i ; Figures 13a and 13b show data with no particular symmetry, where Figure 13a shows an original data set # (1000 data points randomly distributed between two unaligned, overlapping ellipses) ; and Figure 13b shows a"symmetrized" data set, 7E, and depicting r- (a) ; Figure 14 shows a per-element distortion vs. hypercube rotation angle ; Figures 15a and 15b show rotation of a hypercube; equivalent rotation of data, where Figure 15a shows rotation of K, for 80, through an angle s* = 0.2827 (the optimal rotation); and Figure 15b shows equivalent rotation of data, through an angle-&num * ; Figure 16 shows a per-element distortion vs. hypercube rotation angle ; Figure 17 shows multiple hypercube compression; Figure 18 shows an algorithm to compress a vector v by a multiple hypercube method; Figure 19 shows an algorithm to decompress a hypercube, index pair Figures 20a and 20b show the effect of a conventional K-means algorithm, where Figure 20a shows transformed example data , and Figure 20B shows a result of the K-means algorithm, for K= 8; Figures 21 a and 21b show the effect of the orthant K-means algorithm, where Figure 21 a shows folded example data, = PM, and Figure 21b shows the

result of the orthant K-means algorithm, for K = 2 (the markers correspond to the a vectors of the desired Khypercubes) ; Figures 22a and 22b show a comparison of conventional and orthant K-Means, where Figure 22a shows code points from conventional K-means, and Figure 22b shows code points from orthant K-means ; and Figure 23 shows the orthant K-means algorithm.

DETAILED DESCRIPTION OF THE INVENTION In this document we disclose a method for implementing zero-search, zero- memory vector quantization. There is no explicit codebook and no search; the codebook memory and associated search computation are eliminated. Some memory and computation are still required, but these are dramatically reduced, compared to systems that do not exploit this method. The method has been validated in a demanding, real-world data compression task, and found to yield over a 30-fold reduction in computation, and over a 300-fold reduction in memory (compared with a system that does not exploit the invention), while maintaining good fidelity of reconstruction.

Because there is no explicit codebook to be stored or searched, no such codebook need be generated either. This makes the method well suited to adaptive coding schemes, where the compression system adapts to the statistics of the data presented for processing. Both the complexity of the algorithm executed for adaptation, and the amount of data transmitted to synchronize the sender and receiver, are exponentially more efficient than existing methods.

Vector quantization is described in detail in Allen Gersho and Robert M. Gray, Vector Quantization and Signal Compression, Kluwer Academic Publishers, 1992, and the reader is urged to consult that book for a thorough treatment of the subject. Nevertheless, for completeness this document provides enough background for the reader to understand the operation of our invention, its benefits, and how it differs from current methods. The reader is assumed to have a basic knowledge of linear algebra and multivariate statistics, notably the concepts of eigenvectors and covariance, at the levels of references Howard Anton, Elementary Linear Algebra, John Wiley and Sons, 1973 and Paul G.

Hoel, Introduction to Mathematical Statistics, John Wiley and Sons, New York, NY, 5t'edition, 1984, respectively. Understanding certain of the extensions to the basic method requires some familiarity with the Lie theory of continuous groups, at the level of Morton L. Curtis, Matrix Groups. Springer-Verlag, New York, NY, second edition, 1984.

The material presented here will be framed in terms of the compression of digitized human speech, which is transmitted through a communication channel, and then processed by an automatic speech recognition (ASR) system. This is the context in which the invention was developed, and in which it is presently being applied. However, there is nothing in the invention that limits its application exclusively to speech, or exclusively to data transmission. For instance, the invention may be applied to the transmission and storage of video, or any other digitized signal.

Review of Vector Quantization Let us suppose that we have a body of data V to be compressed, which is presented as a series of N vectors vo... VN1, each vector consisting of D elements, and each element comprising b bits. Absent application of any data

compression method, to transmit V from a sender to a receiver requires us to propagate some N x D x b bits of data through a communication channel.

Assuming the communication is error-free, this is lossless transmission: the receiver gets an exact copy of V.

Often in practice the communicating parties do not require the decompressed data to be an exact copy of the original. A reasonable facsimile may suffice, providing that it can be obtained at a suitably low price and/or high speed, and that the received copy is close enough to the genuine article. To put it another way, the sender and receiver may be willing to sacrifice some fidelity, in favor of a lower phone bill, and/or a lower transmission time.

Suppose this is so, and moreover that the vectors v.... of V are distributed so that they fall naturally into a relatively small number of clusters. Figure 1 exhibits such a distribution, where some 200 two-dimensional vectors are seen to fall into eight clusters.

Let us now choose a nominal center point of each cluster, not necessarily drawn from the points belonging to the cluster. Call these points C () through c,. Providing the spread of each cluster around its center point is not too large, we may compress the data as follows : For each vector v that we wish to transmit, we send the index i of the cluster center cj that is closest to v.

In general, if this scheme is applied with a total of K clusters, and hence with K associated cluster centers, then this entails transmitting no more than 10g2 K (bits for each vector sent. The transmission of the indices may itself be subject to an entropic compression scheme; see below.

If log2 K is small compared to D x b, which is the size in bits of one uncompressed vector, then this entails a substantial reduction in the amount of data transmitted. For instance, in the case of Figure 1, where D = 2, suppose each vector element is a 32-bit floating point number. Then, instead of transmitting D x 32 = 64 bits per vector, we can get by with sending only log2 K = log2 8 = 3 bits per vector; a more than 20-fold reduction in the required bandwidth. At the receiving end, when index i arrives, the receiver extracts entry c, from the codebook, and presents it as the (approximate) reconstruction of the original vector v.

This scheme is known as vector quantization. The set of K cluster centers t} is known as the codebook ; the cluster centers themselves are also called code words or code points. While vector quantization can yield a high rate of compression, this comes at a price.

First, the sender must devote memory to storing the codebook ; we will refer to this as the sender's codebook cost Second, for each vector vt that is to be transmitted, the sender must determine, via a computation, the index i of the cluster center ci that is closest to vt ; we will refer to this as the sender's search cost Third, the receiver must likewise store the codebook, so that upon receipt of the index/, the cluster center c may be selected from the codebook, and presented as a reasonable facsimile of the original vt we will refer to this as the receiver's codebook cost Fourth, the communicating parties have given up some fidelity. The receiver has obtained not V = #vO ... vN-1# but # = ##O ... #N-1#. The fine structure of each cluster about its center has been erased. We will refer to this as coding distortion.

Vector quantization can be expressed succinctly in terms of two functions, one for compression and one for decompression. Let K = {c0K ... cK-1K} be the set of K code points, let I = {0 ... K - 1} be the set of valid indices, and suppose v is drawn from the space of real, dimensional vectors. Then the compression algorithm is expressed by a function g, defined by fl"- (yj_) = argmin I iv-c7 11 iei (1) In words, is the index of the code word in K that is closest to v. The (not exactly) inverse decompression algorithm is likewise expressed by a function hK : I # RD, defined by M eE We call (1) and (2) above the compression and decompression equations respectively. It will be useful in what follows to have a succinct expression for the map from a vector v to its reconstructed facsimile 11.. Note that this is neatly provided by the composition of gK with hK, because tX. We will denote the composition hK # gK by ~K ; that is # = ~K(v) = (hK # gK)(v) = hK(gK(v)). (3

We will refer to this as the quantization equation, and call t the quantization function.

The Problem Solved by the Invention Our pedagogical example above has a small codebook, totaling 8 centers x 2 elements/center x 32 bits/element = 512 bits. Due to the small number of entries, and the short vector lengths, it is easy to search this codebook for the closest codeword.

However, this is not typical. To give some feeling for the characteristics of a real- world data compression task, we consider the application that motivated this invention: compression of human speech, represented as a sequence of twelve- dimensional mel-frequency cepstral coefficient (MFCC) vectors. Prior to the introduction of this invention, this application used a vector quantization scheme that involved two distinct codebooks, at both sender and receiver, each containing 4096 code words, with each vector element a 4-byte floating point number. The memory for each pair of codebooks was therefore 2 codebooks x 4096 vectors/codebook x 12 elements/vector x 32 bits/element = 3,145, 728 bits. Furthermore, the sender's search cost dominated, by a factor of 3 to 5, the cost of the MFCC computation that was generating the data to be compressed.

The compression achieved in this application was to compress a 384 bit vector into 24 bits of data.

Elimination of these costs is the central problem that the invention solves. In this document we disclose a method for data compression that reduces the sender and receiver codebook costs to zero, and the sender's search cost to zero. There is no explicit codebook to be searched for compression, and none used for reconstruction, so all of these costs are simply eliminated. Yet the method, when

it may be applied, does not significantly decrease the degree of compression achieved, or increase the coding distortion. In the real world example discussed just above, the invention when applied yielded identical compression, again reducing a 384 bit vector to 24 bits of compressed data. Yet the fidelity of the reconstructed data was good enough to achieve per-word recognition accuracy, by an automatic speech recognition system, that matches the results achieved by the prior, costly compression scheme.

In place of the codebook and search costs, our invention imposes a transform storage and compute cost, on both the sender and receiver. These costs are so low-on the order of 102 to 103 smaller than typical search and codebook costs- that they may be considered negligible.

Metrics, Codebooks and Voronoi Diagrams Some fine points of vector quantization must be noted here, which will figure in the discussion to follow.

First, we have so far sidestepped the issue of just what constitutes a cluster of data points, or indeed what measure should be used to decide which code point is closest to a given vector. Figures 1a and 1b show data for compression, where Figure 1 a shows 200 two-dimensional vectors plotted, each marked by a dot, and seen to fall naturally into eight clusters, and Figure 1 b shows the same data plotted, along with nominal cluster centers, each marked by a star. By presenting the concept with the aid of Figures 1a and 1b, we have implicitly endorsed the Euclidean norm as the appropriate metric. However, unless the encoded data arose from some real-world geodetic measurements-for instance, the distribution of uncollected golf balls on a driving range-there is really no inherent reason to use this measure. Indeed, in an abstract setting, such as the

coding of MFCC vectors, there is substantial reason to prefer the Mahalanobis norm. We discuss this topic further below. For now we note that the metric that appears in the compression equation (1) is understood to be arbitrary.

Second, we have made no mention of a host of practical details concerning the codebook, notably how the sender and receiver establish a common codebook, how closely a codebook is tied to a particular data set intended for compression, how clusters are found, and how cluster center points (that is, code words) are chosen. For concreteness of this exposition we will make some simplifying assumptions on these topics; we will explore variations on these assumptions after presenting the fundamental idea.

Specifically, we will assume that the designer of the system supplies a fixed codebook to the sender and receiver when the system is deployed. This codebook is developed by examination of a collection of examples containing E vectors, each of dimension D, comprising data typical of the kind that will be presented to the deployed system, once in operation. That is, the properties of the data processed by the deployed system are assumed to match closely with those of S, at least with respect to the characteristics that are important for vector quantization. is assumed to be very large ; in particular E >> D, the dimension of the vectors. We will refer to the elements of g variously as example points or training data.

Regarding the method for creating the codebook, we will now briefly describe the widely used iterative K-means algorithm. This method is so named because it iteratively establishes a sequence of codebooks by repeatedly partitioning the training data into a collection of Kclusters, and then determining the code words of a new codebook as the arithmetic means of the cluster members. The details of the algorithm are found in Figure 2.

In the following discussion we will make use of the Voronoi diagram, a notion that is related to the partitioning step of the K-means algorithm. Figure 2 shows the conventional K-means algorithm. This is a widely used algorithm for determining vector quantization codebooks. As discussed in Figure 2, at step 6a of each iteration, the K-means algorithm partitions the elements of S with respect to a set of code points, thereby decomposing into a collection of disjoint subsets. In much the same way, the space of all possible data vectors, from which the elements of E are drawn, may itself be partitioned with respect to a given codebook into non-overlapping regions, called Voronoi cells. There is one cell per code point; each cell consists of those points in the space that are closer to the given code point than any other. This partition depends as well upon the particular metric in use; in this discussion we will use the Euclidean metric. A diagram that exhibits the boundaries between these cells-the boundaries consist of the points that are equidistant from two or more code points-is known as a Voronoi diagram.

Figure 3a is a Voronoi Diagram. The collection of code points shown implicitly divides the plane into non-overlapping cells, one per code point. The cell for code point c, consists of all the points in the plane that are closer to ci than to any other entry in the codebook. The Voronoi diagram shows the cell boundaries, which are points in the plane equidistant from the code points of adjoining cells.

Figure 3b shows a Voronoi diagram with data points added. This illustrates how the Voronoi diagram implicitly represents the compression function. Figure 3a is the Voronoi diagram for the code points of Figure 1. Note that from symmetry considerations, the boundary between any two adjoining cells is always the perpendicular bisector of the line segment that connects the code points associated with each cell.

For graphic clarity, the example presented in the figure treats the plane, However the notion of the Voronoi diagram is well-defined in any normed, finite dimensional vector space. In the case of R-0, the Voronoi cells are D- dimensional polytopes, bounded by D-1-dimensional hyperplanes. Moreover, we have this generalization of the observation at the end of the preceding paragraph: in the case of the Euclidean metric, the boundary between adjoining cells is comprised of (a portion of) the hyperplane that bisects, and is orthogonal to, the line connecting the two associated code points.

Voronoi diagrams and vector quantization are intimately related, as follows. The Voronoi diagram for a given codebook K, and the association between code points and Voronoi cells, amounts to a graphical presentation of the information that is encapsulated in the compression function gK. For if we know that a given vector v lies in a specific Voronoi cell, then the cell's associated code point ci is the closest in Kto v, and thus gr" (u) This idea is exhibited in Figure 3b.

Figures 4a and 4b show symmetry properties of adjacent code points and associated cell boundaries. As these two examples, illustrate, the boundaries between adjacent Voronoi cells are perpendicular bisectors of lines connecting the code points of the respective cells.

The Fundamental Idea The fundamental idea of the invention is the following observation : If in a plane, a set of four code points are chosen to lie at the vertices of a square centered at the origin, with edges orthogonal to the coordinate axes, say t + + +t5-t

then the associated Voronoi cells are the four quadrants of a conventional Cartesian coordinate system, and the cell boundaries are the coordinate axes themselves.

Figures 5a and 5b show perfect Voronoi diagrams. Figure 5a shows a square and its Voronoi diagram. Figure 5b shows a variety of rectangles, each with (the same) perfect Voronoi diagram. Figure 5a also shows the meaning of the radius vector, for the rectangle with vertices marked by *. Figure 5a exhibits this codebook, and its associated Voronoi diagram, which we will refer to as a perfect Voronoi diagram. We say the Voronoi diagram is perfect because the cell boundaries correspond exactly with the coordinate axes of the plot.

In a slight abuse of terminology, we will say that the square of Figure 5a has radius 1 because it encloses a circle that has radius 1, in the conventional sense.

Some generalization of this claim is possible. As a consequence of the symmetry considerations discussed in the preceding section, the same perfect Voronoi diagram obtains, no matter what the square's dimensions. Indeed, the figure does not even have to be a square. It can be a rectangle, of arbitrary edge lengths, providing that the rectangle is centered at the origin, and that the sides of the rectangle are orthogonal to the coordinate axes. The associated Voronoi diagram will always be perfect; this is exhibited in Figure 5, right panel.

The significance of these observations is that because the Voronoi diagram is perfect, the compression function is trivial to compute. By inspecting only the signs of the elements of v, it is possible to determine in which quadrant-and hence in which Voronoi cell-the vector lies, and thereby determine the index of the associated code point. No search per se is involved, and no codebook is ever consulted. Yet the nearest code point is known exactly.

All these observations have generalizations from 2 dimensions to D dimensions, as follows. The associated figure, the vertices of which constitute the codebook, is a dimensional hypercube, or D-hypercube for short. The vertices of a D- hypercube of radius 1, centered at the origin, with faces that are orthogonal to the coordinate axes, are determined as the D-fold Cartesian product K = {+1, -1} # {+1, -1} # ### # {+1, -1}, (5) where the doubleton {+1,-1} appears D times in the product. This figure has 2° vertices, corresponding to a codebook with 2° entries. (The notation 2D denotes the integer 2 raised to the D th power, a function that grows exponentially with D.) Note that the advantage afforded by this construction becomes ever greater as D increases. For the size of the notional codebook, and hence the workload of the associated search, both grow exponentially with D. Yet both codebook and search may be dispensed with: finding the code point of 'nearest to a given vector v reduces to inspecting the sign of its D elements.

Figure 6 shows a perfect hypercube codebook, for D = 3. The radius vector for this codebook is We will refer to as constituted in (5) as a radius 1 hypercube codebook. If the quantity 1 in (5) is replaced by an arbitrary positive real number, we will say that ~ is a radius a hypercube codebook.

In the two-dimensional case, both squares and rectangles had perfect Voronoi diagrams. There is a suitable generalization for D dimensions as well. In D dimensions, a Voronoi diagram is perfect if and only if each cell boundary is comprised of D-1 dimesional hyperplanes that are the span of D-1 coordinate

axes. Now let # = #α1, α2,..., αD# # RD be a vector with strictly positive elements. Then again by symmetry considerations, the codebook defined by K = {+α1, -α1} # {+α2, -α2} # ### # {+αD, -αD} (8) also has a perfect Voronoi diagram. This is shown, along with the meaning of the elements of a, in Figure 5a. In a slight abuse of language, we will say that this is a radius Cl hypercube codebook, even though in fact it is more properly described as a hyperrectangle ; we will also refer to S as the hypercube's radius vector.

Note that by virtue of (6), the element vector CE defines a codebook, hereafter written comprising 2 D elements. We will refer to as an implicit codebook and each of the vertices of (as an implicit code point This representation affords a great economy. Suppose each coordinate of CE is recorded with a precision of b bits. Then the implicit codebook (is specified in full by b x D bits. Whereas recording all the code points of explicitly, that is, as elements of, requires some b x D x 2° bits. It is in this sense that the implicit representation of a hypercube codebook is exponentially smaller than the explicit version.

There is no guarantee that any hypercube codebook will provide suitably low distortion for a given compression task. Surprisingly, suitable hypercube codebooks may be found for many data compression tasks, via a second element of the invention. We now explain the idea behind this second element.

To develop the necessary intuition, we focus our attention now on the two- dimensional case, and suppose that the data to be compressed are uniformly distributed in a disc centered at the origin. An example of such a data set,

consisting of 1000 points sampled from a uniform distribution within a circle of radius 2, appears in Figure 7a, which shows sampled symmetric data, where 1000 points are sampled from a uniform distribution on a disc of radius 2.

In this special case, it can be shown that the best possible arrangement of four code points, from the standpoint of minimizing coding distortion, is obtained when they are located at the vertices of a square, likewise centered at the origin. An experimental verification is provided in the Figure 7b, which shows the results of 100 K-means iterations, executed to determine a codebook of four elements.

The locations of the code points approximate the vertices of a square, centered at the origin. Indeed, it is possible to show analytically that the best possible codebook is a square with an edge length of 16/37r =1. 6977, which is close to the dimensions in the figure.

Now comes a key observation. We continue to consider the hypothetical case of a symmetric disc of data points, with the codebook formed from a square centered at the origin. The distortion obtained with this codebook is independent of the orientation of the square. This follows from a symmetry argument: because the data to be compressed are, by assumption, uniformly distributed within the disc, there can be no preferred direction for orienting the square that defines the codebook. The idea is illustrated in Figures 8a and 8b.

This being the case, we can choose the orientation as we wish, without paying any penalty, or conversely, realizing any improvement, in the coding distortion. In particular, we may orient the square so that its Voronoi diagram is perfect. If we do so, and if the square has the optimal edge length mentioned above, we obtain a vector quantization scheme that requires neither a search nor an explicit codebook, and also achieves the lowest possible distortion, for any four-element codebook. This choice is exhibited in Figure 9.

This all depends upon the presumption that the data are uniformly distributed within a circle centered at the origin, a condition that seems highly artificial and unlikely. But this then immediately suggests the following general scheme: given a D-dimensional vector quantization problem, let us seek a simple, invertible function that may be applied to its associated example data set zu such that the transformed data set will approximate a uniform distribution within a D- dimensional sphere. We may then compress transformed vectors arising from the data source, by quantizing them with respect to a suitable D-hypercube codebook. We will refer to the desired function as a symmetrizing transform.

This proposal is notably incomplete, in that it does not provide any hint of what form such a transform might take, much less a prescription for how to find it. Nor does it supply the radius vector for the codebook. These however are the next two elements of the invention that we will explain : an algorithm for finding the desired transform, and one for determining the radius vector of the associated codebook.

Two important caveats must be mentioned.

First, in the general case of data distributed in RD, as opposed to the particular case of R explored here, there is no guarantee that a D-hypercube codebook, of any radius vector, will achieve the minimal distortion for a codebook of 2° elements. This is so even if the vectors to be compressed are indeed distributed uniformly within a D-sphere. Nevertheless, in practice the method of the invention has proven to be highly effective.

Second, even if a given codebook is optimal for its size, in the sense of achieving minimum distortion for a fixed number of code words, there is no

guarantee that the reconstructed data will exhibit sufficiently high fidelity to the original.

In the next two sections we explain how to compute the symmetrizing transform, and the optimal D-hypercube codebook, respectively.

Finding a Symmetrizing Transform Here we explain how to find the required symmetrizing transform. The technique we will develop is not the only possible approach to the problem. Independent of this particular method, we claim priority on the general idea of efficient vector quantization via D-hypercube codebooks, with a symmetrizing transform determined by this or any other technique, or without any transform step at all.

Finding the Transform By assumption we are supplied with a large example data set # # RD, which exhibits the same statistics as the data that will be encountered in deployment.

Without loss of generality we may further assume that # has zero mean; that is #v## v = 0@, and by extension the same is true of the source data itself. If this is not the case then we compute a mean µ = #v## v/E, form a new example collection, and operate thereafter on, incorporating the subtraction of, u as the first step of the transform we seek. Here, we have written E for the number of vectors in.

Now observe that if a zero-mean data set s is spherically symmetric about the origin, its covariance matrix, which we will write as Z (D), is a scalar times the identity matrix. Although the converse is decidedly not true-there are zero mean data sets, not spherically symmetric, for which the covariance matrix is a scaled identity matrix-we will nevertheless adopt the following plan: we seek a linear <BR> <BR> Z(T#) = I<BR> transform, such that. We will then use T as the symmetrizing transform. Here I denotes the D x D identity matrix, and the <BR> <BR> <BR> <BR> notation T# means (Tv # v # #), which is a copy of # with T applied to each vector. The intended effect of T is illustrated in Figures 10a and 1 Ob. Figure 10a shows data before transform (#, 1000 points distributed within an ellipse); Figure 10b shows data after transform (T#, same 1000 points, transformed by 7).

We proceed to develop an algorithm for finding T, consistent with the symmetry criterion just adopted. It is worth noting here that other criteria may be used, which will likely lead to different transforms. The method is a standard manipulation in linear algebra, yielding a so-called whitening transform ; one novel aspect of the invention lies in the way that we apply this technique.

Recall the definition of the covariance matrix of a zero-mean data set,

By assumption each element v of # is a D-element column vector; vu denotes its row-vector transpose. The symbol @ denotes outer product. Note that b y virtue of this definition, (g) is a symmetric, positive-semidefinite matrix.

Hence by virtue of the spectral theorem (see Howard Anton, Elementary Linear Algebra, John Wiley and Sons, 1973; Theorems 6.7 and 6. 8), there exists an orthonormal collection of D eigenvectors, (tK ZD), with associated real, non- negative eigenvalues (11K » D). By definition these are column vectors, and satisfy

Z(#)zi = #izi. (8) The eigenvectors and associated eigenvalues may be found by any chosen method of numerical linear algebra ; consult reference William H. Press, Saul A.

Teukolsky, William T. Vetterling, and Brian P. Flannery, Numerical Recipes n C++: The Art of Scientific Computing, Cambridge University Press, second edition, 2002, for details.

Now form a matrix S, the rows of which are transposed eigenvectors, thus r I i L- S = zD or eq tl. Z rY I The lines are intended to suggest the direction in which the vectors extend within the matrix.

We proceed to compute the matrix Z(#) St; by the definition of matrix multiplication we have where the second equality follows from (8).

The eigenvectors are orthonormal, that is 4 i = 1 if i= j, and 0 otherwise.

Hence

In other words, the result is a matrix with the associated eigenvalues appearing in order on the diagonal, and zeros everywhere else. Let us write A for the rightmost matrix in the preceding equation, and likewise #-1/2 for dagF K XD/). Both matrices are symmetric and diagonal ; it is evident that #--1/2 #(#-1/2)t = @, the D x D identity matrix.

This leads us directly to the transform that we desire, as follows. Multiplying both the left most and right most expressions in (11) on the left by #-1/2 and on the right by (#-1/2)t, we have (Here the final equality follows by simply working out the product of the three indicated matrices. ) Substituting the definition of from (7) in this yields

where the last equality follows from the identity (AB)t = BtAt, for any matrices conformable for multiplication. (See Howard Anton, Elementary Linear Algebra, John Wiley and Sons, 1973; page 68 property iv.) Now set T = A-1/2S, (17) and let M=M as at the start of this section. Note that M and £ have the same number of elements ; we write y = E Moreover, fatso has zero mean: because Tis a linear transformation, and s has zero mean. Picking up from (16), it is evident that

because the left most sum combines outer products formed from transformed elements of £, and the right most combines these same outer products, formed from the same transformed elements. Comparing the right most expression in (19) with (7), we see that it is g (, the covariance of the zero-mean data series . Thus, applying T to each element of £ yields a data set that has/as its covariance matrix, and so Tis the transform we seek.

Degenerate Cases One will have noticed that while we are assured, because Z is positive- semidefinite, that each #i # 0, these inequalities are not guaranteed to be strict.

That is, the possibility exists that Ai is zero, for one or more values of i. This would apparently foil the method discussed here because it would be impossible to form A-1.

In fact, the exact opposite is true. For if some- = 0, this means that every vector in #, and by extension every vector encountered in deployment, because # is assumed to be examplary, is orthogonal to the corresponding eigenvector z ;. Hence, the data to be compressed all lie in a D-1-dimensional hyperplane. We may therefore project all the data into an appropriate D-1- dimensional space (specifically, the hyperplane orthogonal to z ;), and proceed to apply our method to the resulting data set. This yields even higher compression because only D-1 bits are transmitted for each vector processed. If more than one eigenvalue is zero, this procedure can be iterated until only non-zero eigenvalues remain.

Efficient Joint Representation of T and T-'

The matrix T-'is needed to perform the decompression step; it is also needed for so-called closed-loop compression, a technique that is commonly used in conjunction with lossy data compression methods. Storing both T and T-1 explicitly requires 2 # D2 # F bits, where F is the number of bits per matrix entry.

While this is vastly smaller than a typical vector quantization codebook, it could still consume a significant amount of memory.

For this reason, we now describe a method that exploits the algebraic form of T and T-'to represent both matrices, but which yields a memory reduction, by approximately a factor of two, over storing the two matrices explicitly. There is a small increase in program size and computation associated with this method.

However, it is an effective way of trading memory cost for compute cost.

To begin we establish the form of T-1. From equation (17) above we have 2. it follows (see Howard Anton, Elementary Linear Algebra, John Wiley and Sons, 1973; Theorem 1.6) that T-1 = S-1 (#-1/2)-1. A simple calculation shows that S St = I, because S is matrix of orthonormal eigenvectors, and so S-1 = St. Moreover, if we define #1/2 = diag(#11/2,...,#D1/2), it is clear that A, #1/2 = I, and so (#-1/2)-1 = #1/2. Hence Thus, from equations (17) and (20), the actions of both T and T-'on any vector may be computed from the entries of S, #-1/2 and #1/2 alone.

This means that it suffices to store only these latter three matrices, at a cost of (D2 +2D) # F bits of memory. This follows because each diagonal matrix can be represented by only its D non-zero elements. Moreover, if necessary either one

of the diagonal matrices may be dispensed with, because the corresponding diagonal elements are just reciprocals of one another; this further reduces the cost to (D2 + D) # F bits.

Finding the Optimal Radius Vector In this section we explain how to find the radius vector # for the optimal D- hypercube codebook. This is determined by analysis of u = T#, the collection of transformed example data. The radius vector oc we obtain is optimal in the sense that it yields minimal mean square coding distortion, among all D- hypercube codebooks, for this data. Note that 0'is a D-dimensional vector, defined by its elements α1, K, αD. We refer to # as the optimal radius vector.

It will be useful to define the following helper function (6 {» 1} xi L C {t1 if + 1 1 otherwise.

In words, #(x) is 1 if x is positive or 0, and -1 if x is negative. Two properties of #, which are both ture for all x # R, are <BR> <BR> <BR> <BR> x#(x) = #x# (22)<BR> <BR> <BR> #(x)#(x) = 1. (23) Now suppose is a D-hypercube codebook, with radius vector 0. For any given u # u, the closest code point in r is # = ~K(@)(u) = #α1 #(u1), α2 #(u2), ..., αD #(ud)#. (24)

We seek to adjust t to minimize the average coding distortion # of the example set U, defined as Note that the norm used in equation (26) is the standard Euclidean metric, computed in the transformed space. We will have more to say about this point below. Differentiating this quantity with respect to each element 9 of a, we have But 0 is 0 unless i = j, in which case it is 1. Hence the inner sum collapses to the j= iterm only, and we are left with which we have equated to 0 as the condition for optimality. Multiplying by U/2 to clear the constants, and transposing subtracted terms to the right hand side, we have equivalently

where we have exploited the fact that ai is a constant with respect to the left hand summation, and can be moved outside it. But as previously noted, C = 1 and ut (. Hence we have But the left hand sum is the result of adding together U copies of 1, and hence itself equals U. Dividing through by U, this leaves us with for each/= 1,..., D. This defines the optimal D-hypercube codebook.

The asymptotic cost of this algorithm is determined as follows. We assume that the symmetrizing transform T and the transformed example set 11 = TE have already been determined. Thus Z is a set of g vectors ; each vector comprising D dimensions.

There are D elements of the optimal radius vector s ; each element determined by an appropriate instance of equation (32).

A single instance of equation (32) requires U applications of the absolute value function, U-1 additions, and a single division; thus its asypmtotic cost is O (U) + O (U-1) + O (1) = O (U) operations. Since s is completely determined by D applications of (32), the asymptotic cost of the algorithm is O (D U).

It is worth noting that this cost is dominated by the cost of determining U. For each vector u # U is obtained from some

corresponding v in g by a matrix multiplication, u = Tv. The cost of this multiplication is O (D2), since each element of u is determined by D multiplications and D-1 additions, comprising O(D) arithmetic operations, and there are D elements of u to compute. Therefore the asymptotic cost of forming U is O(D2 U) operations.

The cost of finding the symmetrizing transform T depends upon the algorithm used. A typical algorithm for finding Tis the method of Jacobi rotations, described in William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P.

Flannery, Numerical Recipes in C++ : The Art of Scientific Computing, Cambridge University Press, second edition, 2002, Section 11.1. The asymptotic cost of this method is O (D3), though because it is an iterative algorithm, its exact running time is data dependent. Thus, even with the generation of M and T figured in, the D-optimal hypercube algorithm costs no more than O (D (U + D) ) operations, which is polynomial in both D and U.

Note that the procedure discussed here determines a full D-hypercube codebook, which implicitly defines 2° code points. By comparison, the K-means algorithm of Figure 2 has an asymptotic cost of at least O (U D 2 operations to create a codebook of equal size (that is, with K=2°). This is because the K- means algorithm must explicitly compute and record each of the 2D code points, and compare each element of # (which contains U vectors, since ### = #U#) with each candidate code point.

We can compare the efficiency of the two approaches by examining the ratio of their asymptotic costs when computing equal-sized codebooks; that is, (cost of K-means algorithm)/ (cost of optimal radius vector algorithm).

This ratio is (U D 2/ (D2 (U + D) ). Since this function grows exponentially with D, the optimal radius vector algorithm is exponentially more efficient than the K- means algorithm, when creating codebooks of equal size.

Compression and Decompression

With the necessary machinery now all in place, we can present complete compression and decompression algorithms.

Basic Algorithms We assume that, following the procedures given in the preceding two sections, we have analyzed the example set 6 to determine T and T-1, and found an associated D-hypercube codebook, with radius vector.

To clarify the discussion, we define a few"helper"functions as follows. The first is the"mark-negative"function m(x): R # {0, 1}, defined as if x # 0 otherwise In other words, m attains the value 1 if its argument is negative, and 0 if its argument is zero or positive.

We also define a family of bit-extraction-and-selection functions bitn P), where n is a non-negative integer, and P is an arbitrary real number, by if the jth binary digit of n is 0 ( 34)<BR> <BR> <BR> otherwise Here bits are numbered with bit 0 as the least significant bit of n.

The algorithms for compression and decompression are given in Figures 11 and 12 respectively.

Figure 11 shows an algorithm to compress a vector v. T is the symmetrizing transform. The function m (4 is defined herein. This algorithm implicitly defines the compression function 3 (), though the codebook is nowhere explicitly constructed or searched.

Figure 12 shows an algorithm to decompress an index i. The functions 6 P) are defined herein. The matrix T-1 is the inverse of the symmetrizing transform, given by equation (20). This algorithm implicitly defines the decompression function although the codebook is nowhere explicitly constructed or consulted.

Alternately, it can be reduced to a table lookup operation, if the receiver codebook cost is not a concern.

Metric Considerations We return now to the question of the metric in the compression equation. It should be apparent from the discussion to this point that the method operates in the transformed space T (RD). Because T is by assumption of full rank, T (R0) is RD itself. Our intention is to underscore that when compressing a given vector v, we will apply the compression function a (> to u = Tv. This function, as implicitly defined by the discussion elsewhere herein, selects the closest code point in with respect to the Euclidean norm in the transformed space. As we now show, this corresponds to using the Mahalanobis norm, computed with respect to the covariance matrix C), in the untransformed space. A discussion of the Mahalanobis norm, which is the natural one for abstract vector data, that is, vectors not derived from coordinate measurements in a Euclidean space, may be found in Richard O. Duda, Peter E. Hart, and David G. Stork, Pattern Classification, John Wiley and Sons, New York, NY, second edition.

We proceed to establish the desired result regarding the Mahalanobis norm. To begin, note that the (squared) Euclidean distance de (u, w) between two vectors u = Tv and w = Ts in T(RD) is given by d2(u,w) = #u-w#2 = (u-w)t(u-w). (39) Because T is a linear transform, we have (u- = (Tv-Ts) = T (v-s), and hence we may write d2(u,w) = (u-w)t(u-w) (40) = (T(u-s))tT(v-s) (41) <BR> <BR> = (#-1/2S(U-S))t(#-1/2S(u-s)) (42)<BR> <BR> <BR> <BR> = = (v-s)tSt#-1S(u-s). (43) Now equation (11) above may be written more succinctly as SZ(#)St=#, from which Z(#) = St#S, (44) where we have used the fact that S-1 = St, demonstrated above. Now z (e) is an invertible matrix, so inverting both sides of (44), we have Z Z(#)-1 = (St#S)-1 = S-1#-1 (St)-1 St#-1S. (45) Substituting in (43) above, we have #u-w#2 = (v-s)tZ(#)-1(v-s),

which is the Mahalanobis norm in the original space, with respect to the covariance matrix z (-9.

Enhancements In this section we explain a series of enhancements of the basic method, which reduce the coding distortion. The development proceeds in three parts.

We first explain the rotation method This is a refinement of the basic method, which may reduce distortion, but is not guaranteed to do so. A key advantage of this enhancement is that the method, while requiring either no increase, or only a very modest increase, in the memory and computation requirements of the compression system, and with no increase in the size of the compressed data (or equivalently, the bandwidth needed to transmit the compressed data), can reduce the coding distortion.

Next, we explain the multiple hypercube method. This is likewise a refinement of the basic method, which is guaranteed to reduce the coding distortion, unless the distortion is zero. However, this method, which is based upon augmenting the (virtual) codebook, does increase the system's memory, bandwidth and computation requirements. Fortunately, as we will demonstrate, these all grow very slowly-only logarithmically with the total number of hypercube code points.

Finally, we explain the alternating search method, which combines these two techniques, and can offer still lower distortion.

Each method has various associated algorithms, for compression and decompression, and for determination of various parameters that appear in the

method, which are explained in detail below. These algorithms are novel and constitute part of the invention.

The Rotation Method In developing the fundamental idea that underlies our method above, we used a symmetry argument to establish the following : if the data to be compressed are uniformly distributed within a sphere, there is no preferred orientation to a hypercube codebook in the sense of yielding a lower average distortion.

This frees us to choose an orientation that makes it especially easy to compute the compressioin function g#(#).

While correct, this argument depends upon the unrealistic assumption that the distribution of data to be compressed, even after application of the symmetrizing transform T, will in fact be spherically symmetric. Thus, the question arises if rotating the hypercube codebook can lower the distortion. The answer to this question is yes, as we proceed to demonstrate in this section, and then systematically exploit.

The rotated hypercube codebook will no longer be perfect-that is, the boundaries of the hypercube's associated Voronoi diagram, which are necessarily hyperplanes that are mutually orthogonal to one another, will no longer be orthogonal to the axes of the natural coordinate system of R'.

Because it is this property that permits the extremely efficient computation of the compression function g#(#). using any other orientation would seem to eliminate the method's key advantage. But as we shall demonstrate shortly, in fact it is possible to accommodate a hypercube of arbitrary orientation, with no impact on the method's computational efficiency or memory usage.

Motivating Example In this section we motivate the idea of rotating the hypercube codebook. This is done by exhibiting an example data set, computing its symmetrizing transform T, finding the optimal radius vector a for the transformed data set TE, and then explicitly showing that rotating the hypercube codebook can substantially lower the distortion.

Figures 13a and 13b exhibit such an example. The starting data set #, displayed in Figure 13a, has no particular symmetry. Following the procedures above, we find the zero-mean and symmetrizing transforms, yielding ##, displayed in Figure 13b. Not surprisingly, 7 has a more spherical appearance to the eye; moreover by construction its covariance matrix is the identity.

We proceed to compute the optimal radius vector a for 7£, by the method above; by construction the resulting hypercube codebook zizis perfect. For the data in Figures 13a and 13b, we obtain #= #0.8431, 0. 8041) ; this yields an average coding distortion for ## of 0.6405 squared distortion units per element.

Figure 13b shows the vertices of and the associated Voronoi boundaries.

To demonstrate the dependence of the coding distortion upon the orientation of the hypercube, we explicitly rotate the hypercube through the range [-#/2,#/2], computing the average coding distortion of 7 for each orientation considered. It is not necessary to search the nominal full rotation space, of # E, 7r3, due to the symmetry of the hypercube: rotating it about its center through exactly uz radians, in any plane that is parallel to any of the hypercube faces, changes the indices of the code points, but not their locations. Thus a clockwise rotation by 0 radians, where # s [4, is equivalent to a rotation by 0-X radians, where the latter

quantity lies in [-X/2*n]. Thus, the interval [7T/2, n] need not be examined, and likewise for the interval it suffices to search l-21.

The resulting data are graphed in Figure 14, which shows a per-element distortion vs. hypercube rotation angle. Note that the local minimum at about-1 radian is not quite as deep as the true minimum, at about +0. 28 radians. The distortion exhibits a total variation of about 11%, from minimum to maximum. B y explicit search, we determine that the lowest distortion orientation corresponds to a counterclockwise rotation of the hypercube through ß = 0. 2827 radians. This brings the distortion to 0.6132 squared distortion units per element, a reduction of about 4%, with respect to the unrotated hypercube. The rotated hypercube is depicted in Figure 15a.

It is worth noting here the following simple but essential duality. With respect to coding distortion, if k is the optimal rotation of the hypercube, we can just as easily rotate the data by-6*. The residual, which is the difference between a vector v and its reconstructione, will be the same length in either case, and hence the distortion will be identical. It is only necessary, in constructing a complete end-to-end data compression system, to incorporate an appropriate rotation of the data to its rotated location in the compression algorithm, and an inverse rotation of the reconstructed vector in the decompression algorithm. As we demonstrate below, both can be accomplished with zero computational or storage overhead.

The advantage of rotating the data, as opposed to the hypercube, is that the Voronoi diagram of the unrotated hypercube is perfect, and so we accrue all the advantages of the algorithm already described. Figures 15a and 15b show rotation of hypercube ; equivalent rotation of data. Figure 15a shows rotation of #(#), for #=#0.8431,0.8041#, through an angle 0* = 0. 2827 radians (the optimal

rotation); Figure 15b shows equivalent rotation of data, through an angle-@*.

Rotation of the data, as opposed to the codebook, is depicted in Figure 15b.

We will discuss this duality further below.

An extension to this idea is to vary not just the orientation of the hypercube, but also the radius vector at a given orientation. There is no reason to expect that the optimal s computed for the case of no rotation will remain so at other orientations. This question can be investigated by the same technique that yielded Figure 14 above, with the addition that at each sampled rotation, the optimal is determined. The resulting graph appears in Figure 16, which shows a per-element distortion vs. hypercube rotation angle. This graph was prepared with the same data and methods as used in Figure 14 above, with the exception that the optimal cl is computed at each orientation. Note that the two minima are of equal depth. At least in this example, the effect is small : the value of ak is unchanged, and the distortion at this rotation drops from 0.6132 to 0.6130 squared distortion units. However, in principle this extension can yield some improvement, and so we record it here.

Implementation of The Rotation Method When implementing the rotation method, the desired rotation R, which nominally follows the symmetrizing transform T, may be incorporated into T. That is, instead of compressing the symmetrized vector u = Tv, where T is the symmetrizing transform discussed earlier, the compression takes place on a rotated, symmetrized vector u = RTv. This equation nominally denotes two operations: multiplying the vector v by the matrix T, and then multiplying the result of that computation by another matrix R. This second multiplication nominally requires both additional storage and computation.

However, by forming a single matrix product M=RT in advance of performing any compression, and then compressing by the basic method a rotated, symmetrized vector u = Mv, the desired reduction in distortion can be achieved with no increase in memory, computation or bandwidth. This is because the memory associated with separately representing matrices R and T has now been coalesced in representing the single matrix M=RT, at the same cost as representing Talone in the basic method. Likewise, the computation associated with separately performing the multiplication by T followed by the multiplication by R has now been coalesced in multiplication by a single matrix M, at the same cost as multiplying by Talone in the basic method.

Finding and transmitting the index of the closed hypercube vertex to the vector u=Mv, proceeds exactly as in the basic method, and hence the cost associated with those steps is unchanged.

The order of rotation and symmetrization may be interchanged, though the actual rotation and symmetrizing transform applied may therefore be different.

The Multiple Hypercubes Method In this section we describe a method that is guaranteed to lower the expected average coding distortion, providing it is not already zero. Unlike the rotation method of the preceding section, this technique does increase the system memory and computation requirements, as well as the upstream bandwidth.

The idea of the method can be described in a single sentence: to decrease the distortion, increase the number of code points, by increasing the number of hypercubes. This requires memory to store the additional hypercube radius

vectors, and also forces us to add an explicit distance computation to the compression algorithm. However, the rate of growth of the additional memory and computation requirements is exceedingly small, compared to the rate at which additional code points are added; specifically, they grow only logarithmically with the effective size of the codebook.

This discussion is organized as follows : First, we describe the method, and supply the compression and decompression algorithms.

Then we show how it guarantees to lower the average coding distortion, and also demonstrate that it requires only logarithmically growing memory and computation resources.

Finally, we describe an efficient algorithm for finding a family of hypercube codebooks that may be used in the method.

Description and Basic Algorithms The method works as follows. Instead of compressing and decompressing with a single D-hypercube codebook #(#), the vertices of which are determined by a single radius vector, compression and decompression take place with respect to a family of K distinct D-hypercube codebooks, defined by the set A = (al, a2,..., aK I of their associated radius vectors. Note that the total number of (implicit) code points associated with the set A is K-2. This follows because each radius vector describes a D-hypercube, each vertex of which is a code point, and there are 2° vertices of a D-hypercube. Because there are more code

points, the volume of each finite Voronoi cell is smaller, and hence the average coding distortion is reduced.

Compressing a vector v with respect to a collection of hypercube codebooks requires a slightly more complicated algorithm than the basic one given above.

As when compressing with a single hypercube codebook, we begin by applying a symmetrizing transform T, obtaining u = Tv. Likewise as when compressing with a single hypercube codebook, the orthant of the nearest code point to u is known. Specifically, it must be the orthant of u itself, because all the hypercubes are perfect, and hence share the same Voronoi cell structure, which consists of precisely the orthants themselves. However, there are now K distinct hypercube codebooks, each of which has a vertex that lies within the selected orthant. The identity of the closest hypercube vertex within the orthant, among the K possibilities, must be established, and thus a small amount of explicit search is now required for compression.

Anecdotally, the compression algorithm now becomes: 1. Given vector vto compress find u = Tv ; 2. Find the orthant of u, encoded as i= m (uD) m (ut1)... m (u1), exactly as in Figure 11; 3. Find, via explicit search within the orthant, the index k of the closest hypercube ogk EA (The search may also be conducted in the positive orthant, by mapping the transformed vector u into this orthant, via the map p defined below, and conducting the search for the index k of the closest hypercube among the vertices in the positive orthant. The result will be identical.)

The result of this search, along with the identity i of the orthant, must be transmitted to the receiver; hence the requirement for slightly greater upstream bandwidth. To decompress, the receiver uses the index k to select the appropriate hypercube radius vector ñl. Inspecting the coded orthant i then yields the appropriate vertex of the dk hypercube. This vertex is taken as", from which is computed.

The algorithms for compression and decompression with multiple hypercubes are described in detail in Figures 18 and 19 respectively, where Figure 18 shows an algorithm to compress a vector v with a multiple hypercube method (the function m (x) is defined herein); and where Figure 19 shows an algorithm to decompress a hypercube, index pair #k-1,i# (the functions ) are defined herein).

The descriptions make use of some special nomenclature, detailed here. We define 1, the set of non-negative real numbers; and we refer to the set PD as the positive orthant. We define the map p:RD # PD, for any , by <BR> and t he map #:IRD # IRD, for any #v1, ..., vD# # IRD, by<BR> #((v1, ..., vD))=##(u1), ..., #(vD)#. (48) Given two vectors v,w #RD, we define their pointwise product, by

tt 2 Ls 49) The composition of the compression and decompression algorithms, operating with respect to the set A of hypercube radius vectors, defines a quantization function t.. That is, #A. (u) denotes the result of compressing and then decompressing u, with respect to the radius vector set A.

The operation of the compression algorithm is depicted graphically in Figure 17.

Figure 17 shows the orthant of u = Tv shaded; this corresponds to the common Voronoi cell of each of the perfect hypercubes. Thus only the implicit code points within this orthant need be searched to find a'. (u). The shaded orthant shows the region within which T' (u) must lie ; hence only these points need be searched explicitly to find the closest code point.

Distortion Reduction; Logarithmic Resource Requirements We now show that by an appropriate choice of hypercubes, this method can guarantee a reduction in the average expected coding distortion. We use the words"an appropriate choice of hypercubes"advisedly because the technique we will now present, while ensuring a reduction in distortion, is somewhat inefficient. In the discussion below, we describe a superior method of choosing hypercubes, the orthant K-means algorithm. However, the latter method, while offering significant advantages, does not guarantee a reduction in distortion. This point is discussed further, in the development of the latter algorithm.

A = {#1. #2. .... #K} Let j be the collection of D-hypercube radius vectors, with associated quantization function 14. Recall that the average coding distortion, with respect to, the transformed example data set, is Suppose then that > ; then 3t etZ such that ##A(#)-## = ## > 0. Let #K+1 = p(#), and set A' = A#{#K+1}. Clearly then #A'(#) = #, and hence ##A'(#)-## = 0 Moreover, for any we have ##A(v)-v#2 # ##A'(v)-v#2, because A2A and hence #A'(v) either matches #A(v), or yields a code point that is closer to v.

Now A (A) may be rewritten as

where M is is with all instances of # deleted, and n# is the number of times # appears in. (Note that the sigma-denoted sum in (55) proceeds only over the elements of #.) Likewise A (A) may be written in exactly the same way, but with #A replaced by #A'. Then taking the difference of A (A) and A (A'), each written this way, we have

The last inequality is strict, and hence A (A) < A (A) as claimed.

The cost of this method, which adjoins 2D (implicitly defined) code points to the codebook, is just the cost of finding a single u E a that does not correspond to any vector in A, the collection of K distinct D-hypercube radius vectors. This asymptotic cost is O (U D K) operations, the time to search through (at most) all elements of U for a suitable tz Though the asymptotic cost is low, this method is inefficient, in that it uses a full hypercube to account for a single element of , and thereby reduce the coding distortion. No attempt is made to adjust the other vectors in A, or to account for the coding distortion of other elements of. These shortcomings are remedied by the algorithm of the next section, the orthant K-means algorithm.

The Orthant K-Means Algorithm We now describe the orthant K-means algorithm, which efficiently finds a collection of K perfect hypercube codebooks that yield low average coding distortion for the transformed example data a = T£ In the text that follows we explain the intuition behind the method. The exact description of the algorithm appears in Figure 23. This algorithm determines a collection of K distinct D- hypercube codebooks, specified by their radius vectors, that cover the transformed example data set u well. This algorithm is novel-and constitutes part of the invention.

We begin by stating the objective. We have already seen how to choose a single perfect D-hypercube codebook that is optimal, with respect to the minimum-distortion criterion, for coding transformed example data . Now our aim is to find a collection of K > 1 perfect D-hypercubes, which when used to code, further reduce the distortion.

This problem is akin to the one facing the designer of a conventional vector quantization codebook, who will typically use the K-means algorithm, described above in Figure 2, to establish a set of K codepoints. The problem with that algorithm, for our purposes, is that the codepoints selected will not in general lie at the vertices of perfect hypercubes, even if the number of codepoints sought (that is, the value of K), is an appropriate multiple of a power of 2. This is evident in the example depicted in Figures 20a and 20b, which show the effect of applying the conventional K-means algorithm to the data shown, to generate 8 code points. Figures 20a and 20b show a conventional K-means algorithm, where Figure 20a shows transformed example data , and Figure 20b shows a result of the K-means algorithm, for K= 8.

The obstacle to applying the conventional K-means algorithm is that the code points that it finds are not constrained lie on the vertices of one or more perfect hypercubes. Indeed, it is not immediately obvious how to discover even a single hypercube within the elements of the training data. That is, to find a group of points of, the elements of which all lie close to the vertices of some perfect hypercube codebook. The property of a point set, that its elements all lie in a hypercube arrangement with respect to one another, is decidedly nonlocal. That is, this is not a property that is discoverable by selecting a point of u, and then examining that point and other elements that lie close to it. The orthant K-means algorithm overcomes this obstacle by exploiting the following observation.

Consider the 2D vertices K of a perfect hypercube codebook, as defined b y equation (6): the ith element of any" (s) is either +-ai or-. ai, and hence the i th element of p (0 is αi. Thus for any vertex v # #(#), we have p (v) = C.

This has the following important consequence. Suppose we map the transformed training data into the positive orthant PD, defining U+ = #p(u) # u #U#.

Then clusters in M+ correspond to collections of points in Mall of which lie on or near the vertices of the same perfect hypercube. This is evident in Figures 20a and 21 a, respectively. Here, points in R2, not evidently distributed to lie on or near perfect 2-hypercube vertices, are seen to fall naturally into two clusters, when mapped by p into the positive orthant 2.

The centers of each of these clusters in, which are exhibited by a square and diamond respectively in Figure 21 b, can then be used as the radius vectors 1 and ã2 of two perfect 2-hypercubes. The full codebooks t () and (a2), which are thereby implicitly specified, are exhibited in Figure 22b. It is evident there that some hypercube vertices are effectively unused as code points. But this is no matter, as the distortion achieved is nevertheless quite low, and the use of the perfect hypercube technique has the great efficiencies of memory and computation requirements previously discussed.

This insight is formalized in the orthant K-means algorithm, which is a part of this invention, and which is described in detail in Figure 23. The transformed training data are mapped via the function p to the positive orthant yielding tu+ zu is referred to as folded data, because the effect of the map p in 22can be obtained by folding, in quarters along the coordinate axes, a paper graph of the plotted points of M. The folded data are then processed to find K clusters, b y an iterative technique akin to the one found in a conventional K-means algorithm.

The key difference between the conventional version and the orthant version is

that in the latter, the search proceeds in PD exclusively, and yields only K radius vectors These radius vectors then implicitly define the desired K 2D code points, which by construction must lie on perfect D-hypercube vertices.

Note that obtaining K 2D code points via a conventional K-means algorithm would require maintaining, and iteratively updating, a set of K 2D candidate code points; the asymptotic cost is O (L/' D-2°-<) operations. By comparison the orthant K-means algorithm, likewise generating a codebook of K # 2D implicit code points, has an asymptotic cost of just O (U- D-K) operations. As before we examine the ratio of costs of the algorithms being compared, in this case (cost of conventional K-means algorithm) / (cost of orthant K-means algorithm).

This ratio is (U#D#2D#K) / (U#D#K), and so we see that the orthant K-means algorithm is exponential more efficient than its conventional counterpart. A similar argument applies to the comparative memory requirements of the two algorithms. Hence, the orthant K-means algorithm will run substantially faster than its conventional counterpart. This is a significant advantage in adaptive, dynamic applications of the invention, wherein the recomputation of the implicit codebook must occur in real time.

Moreover, the orthant K-means algorithm is inherently efficient, insofar as for fixed Kand U, it generates 0 (2 implicit code points, at a cost that is only logarithmic (with respect to D) in the effective size of the codebook.

The effect of applying both the conventional and orthant K-means algorithms, to generate 8 code points in both cases, is exhibited in Figures 20a, 20b; 21 a, 21 b ; and 22a, 22b.

Figures 20a and 20b show the effect of a conventional K-means algorithm, where Figure 20a shows transformed example data It, and Figure 20b shows a result of the conventional K-means algorithm, for K= 8.

Figures 21 a and 21 b show the effect of the orthant K-means algorithm, where Figure 21 a shows folded example data, = M, and Figure 21b shows the result of the orthant K-means algorithm, for K = 2 (the markers correspond to the a vectors of the desired Khypercubes).

Figures 22a and 22b show a comparison of conventional K-means and orthant K- Means, where Figure 22a shows code points from conventional K-means, and Figure 22b shows code points from orthant K-means. Note that the points in Figure 22b lie on 2-hypercube vertices; those in Figure 22a do not.

Although the invention is described herein with reference to the preferred embodiment, one skilled in the art will readily appreciate that other applications may be substituted for those set forth herein without departing from the spirit and scope of the present invention. Accordingly, the invention should only be limited by the Claims included below.