Method and apparatus for encoding and decoding using wavelet-packets
This is a continuation of U.S. application Ser. No. 07/525,973 filed May 19, 1990 now U.S. Pat. No. 5,384,725. 
    
    
     BACKGROUND OF THE INVENTION 
     This invention relates to a method and apparatus for encoding and decoding signals which may represent any continuous or discrete values. 
     It is well established that various types of signals can be efficiently encoded and subsequently decoded in a manner which substantially reduces the size of the information required (e.g. number of bits, bandwidth, or memory) without undue or noticeable degradation of the decoded signal. Examples are the types of audio and video bandwidth compression schemes that are currently in widespread use. 
     In signal analysis, it is often useful to recognize the appearance of characteristic frequencies, but this knowledge generally has to be coupled with the location of the time (or space) interval giving rise to the frequency. Such questions can usually be tackled by use of the windowed Fourier transform, with different size windows corresponding to the scale of the transient feature. This analysis can be achieved by correlating the signal to all windowed exponentials and checking for large correlations. Unfortunately, due to lack of independence, information obtained can be redundant and inefficient for feature reconstruction purposes. 
     It is among the objects of the present invention to provide an improved encoding and decoding method and apparatus which overcomes limitations of prior art techniques and provides improved and more efficient operation. 
     SUMMARY OF THE INVENTION 
     The present invention involves, in part, the use of a library of modulated wavelet-packets which are effective in providing both precise frequency localization and space localization. An aspect of the invention involves feature extraction by determination of the correlations of a library of waveforms with the signal being processed, while maintaining orthogonality of the set of waveforms selected (i.e. a selected advantageous basis). 
     In accordance with an embodiment of the invention, a method is provided for encoding and decoding an input signal, comprising the following steps: applying combinations of dilations and translations of a wavelet to the input signal to obtain processed values; computing the information costs of the processed values; selecting, as encoded signals, an orthogonal group of processed values, the selection being dependent on the computed information costs; and decoding the encoded signals to obtain an output signal. As used,herein, wavelets are zero mean value orthogonal basis functions which are non-zero over a limited extent and are used to transform an operator by their application to the operator in a finite number of scales (dilations) and positions (translations) to obtain transform coefficients. [In the computational context, very small non-zero values may be treated as zero if they are known not to affect the desired accuracy of the solution to a problem.] A single averaging wavelet of unity mean is permitted. Reference can be made, for example, to: A. Haar, Zur Theorie der Orthogonalen Functionsysteme, Math Annal. 69 (1910); K. G. Beauchamp, Walsh Functions And Their Applications, Academic Press (1975); I. Daubechies, Orthonormal Bases of Compactly Supported Wavelets, Comm. Pure Appl. Math XL1 (1988). 
     In a preferred embodiment of the invention, the wavelet has a plurality of vanishing moments. In this embodiment, the step of applying combinations of dilations and translations of the wavelet to the input signal to obtain processed values comprises correlating said combinations of dilations and translations of the wavelet with the input signal. The combinations of dilations and translations of the wavelet are designated as wavelet-packets, and in a disclosed embodiment the step of applying wavelet-packets to the input signal to obtain processed values includes: generating a tree of processed values, the tree having successive levels obtained by applying to the input signal, for a given level, wavelet-packets which are combinations of the wavelet-packets applied at a previous level. Also in a disclosed embodiment, the steps of computing information costs and selecting an orthogonal group of processed values includes performing said computing at a number of different levels of said tree, and performing said selecting from among the different levels of the tree to obtain an orthogonal group having a minimal information cost (the &#34;best basis&#34;). Also in this embodiment, the step of selecting an orthogonal group of processed values includes generating encoded signals which represent said processed values in conjunction with their respective locations in said tree. 
    
    
     Further features and advantages of the invention will become more readily apparent from the following detailed description when taken in conjunction with the accompanying drawings. 
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block diagram of an apparatus in accordance with an embodiment of the invention, and which can be used to practice the method of the invention. 
     FIG. 2 is a diagram illustrating Haar functions and combinations of dilations and translations of such functions. 
     FIG. 3 illustrates a tree of nodes. 
     FIGS. 4A, 4B 4C and 4D illustrate examples of possible orthogonal bases. 
     FIG. 5 illustrates a wavelet having two vanishing moments. 
     FIGS. 6-13 illustrates examples of wavelet-packets. 
     FIG. 14 illustrates an example of how information cost can be computed. 
     FIGS. 15A and 15B illustrate a procedure for reconstruction in accordance with an embodiment of the invention. 
     FIG. 16 shows a flow diagram which is suitable for controlling the encoder processor to implement an embodiment of the encoding apparatus and method in accordance with the invention. 
     FIG. 17 is a flow diagram of a routine for generating processed values from the sampled signal using a wavelet-packet basis. 
     FIG. 18 is a flow diagram of a routine for computing information cost. 
     FIG. 19 is a routine for selecting an advantageous or best orthogonal basis. 
     FIG. 20 is a flow diagram of a routine for generating output encoded words. 
     FIG. 21 is a flow diagram of the decoder routine for processing frames of words and reconstructing the orthogonal basis indicated by the words of a frame. 
     FIG. 22 is a further portion of the routine for reconstruction in the decoder. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Referring to FIG. 1, there is shown a block diagram of an apparatus in accordance with an embodiment of the invention for encoding and decoding an input signal which can be any continuous or discrete signal or sequence of numbers representing values in one or more dimensions (e.g. audio, still or moving pictures, atmospheric measurement data, etc.) and which, for purposes of illustration, can be considered as an audio signal x(t). At the encoder 100 the signal is coupled to an analog-to-digital converter 102, which produces signal samples x 1 , x 2 , x 3  . . . , a sequence of which can be envisioned as a vector x. The digital samples are coupled to an encoder processor 105 which, when programmed in the manner to be described, can be used to implement an embodiment of the invention and to practice an embodiment of the method of the invention. The processor 105 may be any suitable processor, for example an electronic digital or analog processor or microprocessor. It will be understood that any general purpose or special purpose processor, or other machine or circuitry that can perform the computations described herein, electronically, optically, or by other means, can be utilized. The processor 105, which for purposes of the particular described embodiments hereof can be considered as the processor or CPU of a general purpose electronic digital computer, such as a SUN-3/50 Computer sold by Sun Microsystems, will typically include memories 125, clock and timing circuitry 130, input/output functions 135 and display functions 140, which may all be of conventional types. 
     With the processor appropriately programmed, as described hereinbelow, a compressed output signal x c  is produced which is an encoded version of the input signal, but which requires less bandwidth. In the illustration of FIG. 1, the encoded signals x c  are shown as being coupled to a transmitter 140 for transmission over a communications medium (air, cable, fiber optical link, microwave link, etc.) 150 to a receiver 160. The encoded signals are also illustrated as being coupled to a storage medium 142, which may alternatively be part of the processor subsystem 105, and are also illustrated as being manipulated such as by multiplication by a sparse matrix M wp , as described in the abovereferenced copending U.S. patent application Ser. No. 525,974, now abandoned, filed of even date herewith. [See also Appendix III.] The matrix M wp  can obtain be obtained using the wavelet-packet best basis selection hereof (see also Appendix V). The signal itself may, of course, also be in the form of a matrix (i.e., a collection of vectors). The stored and/or manipulated signals can be decoded by the same processor subsystem 105 (suitably programmed, as will be described) or other decoding means. 
     In the illustrated embodiment, another processor 175, which is shown as being similar to the processor 105, also includes memories 225, clock and timing circuitry 230, input/output functions 235, and display functions 240, which may again be of conventional types. Processor 175 is employed, when suitably programmed as described, to decode the received encoded signal x c , and to produce an output digital signal x 1  &#39;, x 2  &#39;, x 3  &#39; . . . , (or vector x&#39;) which is a representation of the input digital signal, and which can be converted, such as by digital-to-analog converter 195, to obtain an analog representation x&#39;(t) of the original input signal x(t). As will become understood, the accuracy of the representation will depend upon the encoding process and the degree of bandwidth compression. 
     Before describing the underlying theory of the invention, it is noted that reference can be made to Appendices I-IV, appended hereto, for supplemental description of the theoretical foundations and further approaches. 
     A well known wavelet basis, which has a single vanishing moment, as defined hereinbelow, is the Haar basis [see, for example, A. Haar, Zur Theorie Der Orthogonalen Funktionensysteme, Math Annal. 69, 1910, and Appendix I of the abovereferenced copending U.S. patent application Ser. No. 525,974 now abandoned, filed of even date herewith]. Consider the Haar basis as applied to a simplified case of eight samples x 1 , x 2  . . . x 8 . For uniform amplitudes, and ignoring normalizing coefficients (which are multiples of 1/√2 for Haar wavelets), a set of waveforms can be developed from combinations of Haar wavelets, as illustrated in FIG. 2 and in accordance with the following relationships: 
     
         s.sub.1 =x.sub.1 +x.sub.2 
    
     
         s.sub.2 =x.sub.3 +x.sub.4 
    
     
         s.sub.3 =x.sub.5 +x.sub.6 
    
     
         s.sub.4 =x.sub.7 +x.sub.8 
    
     
         d.sub.1 =x.sub.1 -x.sub.2 
    
     
         d.sub.2 =x.sub.3 -x.sub.4 
    
     
         d.sub.3 =x.sub.5 -x.sub.6 
    
     
         d.sub.4 =x.sub.7 -x.sub.8                                  (1) 
    
     
         ss.sub.1 =s.sub.1 +s.sub.2 
    
     
         ss.sub.2 =s.sub.3 +s.sub.4 
    
     
         ds.sub.1 =s.sub.1 -s.sub.2 
    
     
         ds.sub.2 =s.sub.3 -s.sub.4 
    
     
         sd.sub.1 =d.sub.1 +d.sub.2 
    
     
         sd.sub.2 =d.sub.3 +d.sub.4 
    
     
         dd.sub.1 =d.sub.1 -d.sub.2 
    
     
         dd.sub.2 =d.sub.3 -d.sub.4                                 (2) 
    
     
         sss.sub.1 =ss.sub.1 +ss.sub.2 
    
     
         dss.sub.1 =ss.sub.1 -ss.sub.2 
    
     
         sds.sub.1 =ds.sub.1 +ds.sub.2 
    
     
         dds.sub.1 =ds.sub.1 -ds.sub.2 
    
     
         ssd.sub.1 =sd.sub.1 +sd.sub.2 
    
     
         dsd.sub.1 =sd.sub.1 -sd.sub.2 
    
     
         sdd.sub.1 =dd.sub.1 +dd.sub.2 
    
     
         ddd.sub.1 =dd.sub.1 -dd.sub.2                              (3) 
    
     The first group of relationships, (1) [the top eight waveforms in FIG. 2], are Haar functions, and are orthogonal. The last group of relationships, (3) [the bottom eight waveforms in FIG. 2], are the first eight of the well known Walsh functions. As is known in the art, the Walsh functions are orthogonal and complete, and can be advantageously used to transform and subsequently back-transform certain types of signals to achieve, inter alia, signal compression. It can be observed, however, that the set of the entire twenty-four functions of FIG. 2 is not orthogonal, which follows from the fact that some of the functions are derived from combinations of other functions. For example, sd 1  is the sum of d 1  and d 2 , and is not orthogonal to d 1  or to d 2 . 
     The sums and differences of relationships (1)-(3) are arranged in a tree of &#34;nodes&#34; in FIG. 3. Four &#34;levels&#34; are shown, level 0 being the sample data, and levels 1, 2 and 3 respectively corresponding to the groups of relationships (1), (2) and (3) above. The boxes (&#34;nodes&#34;) at each level contain respective sum and difference terms, and are connected by branches to the nodes from which they are derived. It is seen that level 1 has two nodes (labelled, from left to right, 1 and 2), level 2 has four nodes (labelled, from left to right, 1-4), and level 3 has eight nodes (labelled, from left to right, 1-8). It follows that a level k would have 2 k  nodes. The &#34;positions&#34; of the functions (or members) within a node are also numbered, from left to right, as illustrated in node 1 of level 1 (only). The nodes of level 1 each have four positions, the nodes of the level 2 each have two positions, and the nodes of level 3 each have one position. It is seen that each &#34;parent&#34; node has two &#34;child&#34; nodes, the members of the children being derived from those of the parent. Thus, for example, node 1 of level 2 and node 2 of level 2 are both children of node 1 of level 1. This follows from the relationships (2) set forth above. In particular, the members of node 1, level 2 (ss 1  and ss 2 ) are derived from sums of members of node 1, level 1, and the members of node 2, level 2 (ds 1  and ds 2 ) are derived from differences of members of node 1, level 1. 
     In accordance with an aspect of the present invention, a complete set of functions (that is, a set or &#34;basis&#34; which permits reconstruction of the original signal samples) is obtained from the tree, permitting selection of nodes from any level. The selection is made in a manner which minimizes the information cost of the basis; i.e., the selected basis can be represented with a minimum bandwidth requirement or minimum number of bits for a given quality of information conveyed. Orthogonality of the selected basis is maintained by following the rule that no ancestor (parent, grandparent, etc.) of a selected node is used. [Conversely, no descendant (child, grandchild, etc.) of a selected node is used.] For the simplified tree of FIG. 3, there are twenty five possible orthogonal basis selections. Using shaded boxes to indicate the nodes selected for a given basis, four examples of possible orthogonal bases are shown in FIGS. 4A-4D. If desired a basis which is the best level basis could alternatively be determined and used. 
     The Haar wavelet system, and wavelet-packets derived therefrom, has been used so far for ease of explanation. In accordance with an aspect of the present invention, advantageous wavelet-packets are generated using wavelets having a plurality of vanishing moments and, preferably, several vanishing moments. For description of the types of wavelets from which these wavelet-packets can be derived, reference can be made to I. Daubechies, Orthonormal Bases of Compactly Supported Wavelets, Comm. Pure, Applied Math, XL1, 1988; Y. Meyer Principe d&#39;Incertitude, Bases Hilbertiennes et Algebres d&#39;Operateurs, Seminaire Bourbaki, 1985-86, 662, Asterisque (Societe Mathematique de France); S. Mallat, Review of Multifrequency Channel Decomposition of Images and Wavelet Models, Technical Report 412, Robotics Report 178, NYU (1988), and the aboverefrenced copending U.S. patent application Ser. No. 525,974 now abandoned, filed of even date herewith. 
     The wavelet illustrated in FIG. 5 has two vanishing moments. As used herein, the number of vanishing moments, for a wavelet ψ(x) is determined by the highest integer n for which 
     
         -ψ(x)x.sup.k dx=0 
    
     where 0≦k≦n-1, and this is known as the vanishing moments condition. Using this convention, the Haar wavelet has 1 vanishing moment, and the wavelet of FIG. 5 has 2 vanishing moments. 
     [It is generally advantageous to utilize wavelets having as many vanishing moments as is practical, it being understood that the computational burden increases as the number of vanishing moments (and coefficients) increases. Accordingly, a trade-off exists which will generally lead to use of a moderate number of vanishing moments.] The wavelet of FIG. 5 has coefficients as follows (see e.g. Daubechies, supra): ##EQU1## 
     
         g.sub.1 =h.sub.4 
    
     
         g.sub.2 =-h.sub.3 
    
     
         g.sub.3 =h.sub.2 
    
     
         g.sub.4 =-h.sub.1 
    
     The vanishing moments condition, written in terms of defining coefficients, would be the following: ##EQU2## and L is the number of coefficients. 
     The procedure for applying this wavelet is similar to that for the Haar wavelet, but groups of four elements are utilized and are multiplied by the h coefficients and the g coefficients to obtain the respective terms of opposing polarity. To obtain wavelet-packets from the wavelets, the previously illustrated levels of sum and difference terms are obtained using the h and g coefficients, respectively. The &#34;sum&#34; correlation terms for the first level are computed as follows: ##EQU3## 
     The &#34;difference&#34; correlation terms for the first level are computed as follows: ##EQU4## 
     The four sets of second level sum and difference correlation terms can then be computed from the first level values as follows: ##EQU5## and so on. Extra values at end positions can be handled by &#34;wrap-around&#34;, truncation, or other known means of handling end conditions. It will be understood that the procedure described in conjunction with relationships (4)-(9) is operative to successively correlate the signal samples with the wavelet-packets for each successive level. If desired, the correlations can be implemented by generating wavelet-packets a priori (using the indicated coefficients, for this example), and then individually correlating wavelet-packets with the signal using analog (e.g. electronic or optical means), digital, or any suitable technique. If desired, a special purpose network could be used to perform the correlations. 
     In terms of the diagram of FIG. 3, the sums (4) and the differences (5) would occupy nodes 1 and 2, respectively, of level 1, and the sums (6), differences (7), sums (8) and differences (9) would occupy the nodes 1, 2, 3 and 4, respectively, of level 2. 
     It will be understood that in many practical situations the number of samples considered in a frame or window (level 0) will be larger than 8, and the tree will also be larger than those shown here for ease of illustration. FIGS. 6-10 show the first five wavelet-packets synthesized for sample length 1024, using a wavelet with six coefficients (six h&#39;s and six g&#39;s), and FIGS. 11-13 illustrate three of the higher frequency wavelet-packets. 
     In accordance with an aspect of the invention, the basis is selected in a manner which minimizes information cost. There are various suitable ways in which information cost can be computed. FIG. 14 shows a parent node and its two children nodes. As a measure of information, one can compute the number N p  of correlation values in the parent node that exceed the particular threshold and the total number N c  of correlation values in the child nodes that exceed the particular threshold. As represented in FIG. 14, if N p  is less than or equal to N c , the parent node will be preferred, whereas if N p  is greater than N c , the children nodes will be preferred. As higher level comparisons are made (with the ancestors of the parent) the selection may be supplanted by an ancestor. 
     Another measure of information cost that can be used is the entropy cost function that is well known in information theory, and is threshold independent (see Appendix IV). Suitable weighting of coefficients can also be used. For example, if it is known or computed that certain values should be filtered, emphasized, or ignored, weighting can be appropriately applied for this purpose. 
     FIG. 15A illustrates a procedure for reconstruction which can be utilized at the decoder processor. The shaded boxes indicate the nodes defining the orthogonal basis that was selected at the encoder and is to be decoded. The arrows illustrate the reconstruction paths, and the cross-hatched boxes indicate reconstructed nodes, the last (level 0) reconstruction providing the reconstructed decoder output information. In particular, node 1, level 2 is reconstructed from node 1, level 3 and node 2, level 3. Node 1, level 1 is then reconstructed from node 1, level 2 and node 2, level 2, and so on. 
     FIG. 15B shows children nodes containing sy 1 , sy 2 , . . . sy n  and dy 1 , dy 2 , . . . dy n  being mapped into their parent node to obtain the reconstructed y 1 , y 2 , . . . y 2n . If the four coefficients h 1 , h 2 , h 3  and h 4  (with the corresponding g 1 , g 2 , g 3  and g 4 ) were used for encoding, the decoding relationships will be as follows: ##EQU6## 
     The values mapped into the parent mode are accumulated and, as in the encoder, extra values at the end positions (e.g. at y 0  above) can be handled by &#34;wrap-around&#34;, truncation, or other known means of handling end conditions. 
     Referring to FIG. 16, there is shown a flow diagram which, when taken together with the further flow diagrams referred to therein, is suitable for controlling the processor to implement an embodiment of the encoding apparatus and method in accordance with the invention. The block 1610 represents the generating of processed values by correlating wavelet-packets with the input signal samples. There are various ways in which this can be achieved, the routine of FIG. 17 describing an implementation of the present embodiment. The block 1620 represents the routine, described further in conjunction with FIG. 18, for computing the information costs of the processed values. As described further hereinbelow, there are various ways of computing the measure or cost of information contained in the processed values. In an illustrated embodiment, a thresholding procedure is utilized, and information cost is determined by the number of values which exceed a particular threshold. The block 1630 represents the routine of FIG. 19 for selection of an advantageous orthogonal basis from among the processed values, the selection of the basis being dependent on the computed information costs. The block 1640 represents compiling encoded frames from the selected processed values which constitute the basis, such as for subsequent recovery after processing, storage, and/or transmission. This routine is described in conjunction with FIG. 20. 
     Referring to FIG. 17, there is shown a flow diagram of a routine for generating the processed values from the sampled signal, or signal portion, using a wavelet-packet basis. The block 1710 represents the reading in of the selected coefficients h i , g i . The block 1720 is then entered, this block representing the initializing of a level index at 0, the initializing of a node index at 1, and the initializing of a position index at 1. The sample data, considered as level 0 is then read in, as represented by the block 1725. The sample data may consist, for example, of 256 sequential samples of an acoustical signal to be compressed and transmitted. The level index is then incremented, as represented by block 1730, and the first level processed values are computed and stored in accordance with the relationships (4) and (5) set forth above (block 1735 and loops 1748 and 1760). For example, for the first position of the first node of level 1, s 1  will be computed. If the wavelet employed is representable by a filter having four coefficients, as in the example above, s 1  will be computed as the sum of h 1  x 1 , h 2  x 2 , h 3  x 3 , h 4  x 4 . If a wavelet of more vanishing moments is used, more coefficients will be employed. In general, it will be preferable to utilize a wavelet having several coefficients, greater than four, the above examples being set forth for ease of illustration. 
     In loop 1748, inquiry is made (diamond 1740) as to whether the last position of the current node has been reached. If not, the position index is incremented (block 1745), and the block 1735 is re-entered for computation of the next processed value of the current node and level. The loop 1748 is then continued until all processed values have been computed for the current node, whereupon inquiry is made (diamond 1750) as to whether the last node of the current level has been reached. If not, the node index is incremented (block 1755) and the loop 1760 is continued until the processed values have been computed for all nodes of the current level. For the first level, there will be only two nodes, with the values thereof being computed in accordance with the relationships (4) and (5) set forth above. 
     When the inquiry of diamond 1750 is answered in the affirmative, diamond 1770 is entered, and inquiry is made as to whether the last level has been processed. If not, the block 1730 is re-entered, to increment the level index, and the loop 1780 is continued until processed values have been obtained for the nodes at all levels of the tree. 
     Referring to FIG. 18, there is shown a flow diagram of the routine for computing the information cost of the nodes of the tree, so that an advantageous orthogonal basis can be selected. The block 1810 represents initializing the level index to the highest level (e.g., the last level in the illustration of FIG. 3). The node index and the position index are initialized at 1 (blocks 1815 and 1820). A node content count, which is used in the present embodiment to keep track of the number of processed values in a node that exceed a predetermined threshold, is initialized at zero, as represented by the block 1825. Inquiry is then made (diamond 1830) as to whether the value at the current position is less than a predetermined threshold value. If not, the node content count is incremented (block 1835), and the diamond 1840 is entered. If, however, the processed value at the current position is less than the threshold value, the diamond 1840 is entered directly. [At this point, the processed value could be set to zero prior to entry of diamond 1840 from the &#34;yes&#34; output branch of diamond 1830, but it is more efficient to handle this later.] Inquiry is then made (diamond 1840) as to whether the last position of the node has been reached. If not, the position index is incremented (block 1850), diamond 1830 is re-entered, and the loop 1855 is continued until all processed values of the current node have been considered. When this occurs, the node content count is stored for the current node (of the current level), as represented by the block 1860. Inquiry is then made (diamond 1865) as to whether the last node of the level has been processed. If not, the block 1870 is entered, the node index is incremented, the block 1820 is re-entered, and the loop 1875 is continued until all nodes of the current level have been considered. Inquiry is then made (diamond 1880) as to whether the current level is the highest level. If so, there is no higher level against which comparison of parent-to-children node comparisons can be made. In such case, the level index is incremented (block 1885), block 1815 is re-entered, and the procedure just described is repeated to obtain and store node content counts for each node of the next-to-highest level. When this has been done, the inquiry of diamond 1880 will be answered in the negative, and the next routine (FIG. 19) will be operative to compare the level (which has just been processed to compute information cost of each node) with the children nodes of the previously processed higher level. 
     In particular, the level index is initialized (block 1905) to the highest level less one, and all nodes on the highest level are marked &#34;kept&#34;. The node index is initialized (block 1910) and the node content count of the current node of the current level is compared (block 1920) to the sum of the node content counts of the two nodes which are children of the current node. [For example, if the current node is N i  and the current level is L j , then the count for the current node is compared to the sum of the counts for nodes N 2i-1  and N 2i  of level L j+1 .] If the comparison shows that the parent has an equal or lower count, the parent is marked &#34;kept&#34;, and the two children nodes are marked &#34;not kept&#34; (as represented by the block 1930). Conversely, if the comparison shows that the sum of two children nodes has a lower count than the parent node, each of the children nodes keeps its current mark, and the current parent node is marked &#34;not kept&#34; (as represented by the block 1940). In the case where the Children nodes are preferred, the sum of the counts of the children nodes are attributed to the parent node (block 1945). By so doing, the lowest count will be used for subsequent comparisons as ancestors are examined. The attribution of the count to the parent node will not be problematic, since only &#34;kept&#34; nodes will be considered in the next stage of processing. Inquiry is then made (diamond 1950) as to whether the last node of the current level has been reached. If not, the node index is incremented (block 1955), block 1920 is re-entered, and the loop 1958 is continued until all nodes at the current level have been considered. Inquiry is then made (diamond 1960) as to whether the current level is level 1. If not, the level index is decremented (block 1965), block 1815 (FIG. 18) is re-entered, and the loop 1970 continues until all levels have been considered. At this point, the nodes which define the basis to be used have been marked &#34;kept&#34; [possibly together with some of their descendent nodes], and correspond, for example, to the shaded nodes of the FIG. 4 illustrations. 
     Referring to FIG. 20, there is shown a flow diagram of a routine for generating output encoded words which, in the present embodiment, are collected in a frame which represents the encoded form of the data x 1 , x 2 , x 3  . . . x n . For example, for an acoustical signal, the frame may represent a particular number of acoustical samples, for example 256 samples. As a further example, for a video signal, the frame may represent a frame of video, portion thereof, or transformed frequency components thereof. The number of encoded words in a frame will generally vary as the information being encoded varies, and will also generally depend upon the level of the threshold employed, if a threshold is employed as in the present embodiment. FIG. 20 illustrates an embodiment of a routine for generating a frame of words for the basis that was selected using the routines of FIGS. 18 and 19. A tree location index will be calculated which points to nodes in the tree in depth-first order (or so-called &#34;pre-order&#34;), as is well known in the art. The tree location index is initialized to 1 at level 0, node 1 (block 2010). Inquiry is made (diamond 2015) as to whether the node at that tree location is market &#34;kept&#34;, and, if not, diamond 2020 is entered directly, and inquiry is made as to whether the entire tree has been examined, as indicated by the tree location index. If the entire tree has been searched, block 2080 is entered and a &#34;frame complete&#34; indication can be generated. If not, then loop 2011 is continued until a node marked &#34;kept&#34; is encountered, or until the entire tree has been searched. If a node marked &#34;kept&#34; is encountered, block 2030 is entered, and the tree location index of this &#34;kept&#34; node is recorded in memory; suppose for example that it is called &#34;X&#34;. The position index in the node is initialized (block 2035). Inquiry is then made (diamond 2040) as to whether the value at the current position is above the predetermined threshold. If not, diamond 2055 is entered directly, and no word is generated for the value at the current position. If the value is above the threshold, block 2045 is entered, this block representing the generation of a word which includes the current level, node, and position, and the value at the position. The block 2050 is then entered, this block representing the loading of the just generated word into an output register. Inquiry is then made (diamond 2055) as to whether the last position in the current node has been reached. If not, the position is incremented (block 2060), diamond 2040 is re-entered, and the loop 2061 is continued until all positions in the node &#34;X&#34; have been considered. It will be understood that various formats can be used to represent the words. For example, a specific number of bits can be used for each of the level, node, position, and value. Alternatively, words could be of different length, e.g. depending on information content or entropy, with suitable delineation between words, as is known in the art. Also, if desired, all words in a particular node could be encoded with a single indication of level and node, with individual indications of position-value pairs. Inquiry is next made (diamond 2070), as to whether the last node location in the tree in depth-first order has been reached. If not, the tree location-index is incremented (block 2070), and inquiry is made as to whether the new node is a descendant of &#34;X&#34;, by a comparison of depth-first indices well known in the art. When this is the case, diamond 2065 is re-entered, and the loop 2071 is continued until the first node which is not a descendant of &#34;X&#34; is encountered, or until there are no more nodes. When a first non-descendant of &#34;X&#34; is encountered, diamond 2015 is re-entered and the loop 2081 is continued until all nodes which are both marked &#34;kept&#34; and have no ancestors marked &#34;kept&#34; have contributed to the frame. Such nodes contain a complete orthogonal group of wavelet-packet correlations (see also Appendix I, II and V). When either loop 2011 or the loop 2071 terminates by exhaustion of the nodes, block 2080 is entered and a &#34;frame complete&#34; indication can be generated. If desired, the frame can then be read out of the encoder register. However, it will be understood that the encoder register can serve as a buffer from which words can be read-out synchronously or asynchronously, depending on the application. 
     Referring to FIG. 21, there is shown a flow diagram of the decoder routine for processing frames of words and reconstructing the orthogonal basis indicated by the words of a frame. The block 2110 represents the reading in of the next frame. In the described embodiment, it is assumed that the frames of words are read into a buffer (e.g. associated with decoder processor subsystem 170 of FIG. 1), and the individual words processed sequentially by placement into appropriate addresses (which can be visualized as the selected basis nodes of a tree--as in FIG. 4), from which reconstruction is implemented in the manner to be described. However, it will be understood that individual words can be received synchronously or asynchronously, or could be output in parallel into respective tree nodes, if desired. Also, as was the case with the encoder, parallel processing or network processing could be employed to implement reconstruction, consistent with the principles hereof. In the routine of FIG. 21, the next word of the frame is read (block 2115), and a determination is made as to whether the node and level of the word is occurring for the first time (diamond 2117). If so the node (and its level) is added to the list of nodes (block 2118). The value indicated in the word is stored (block 2120) at a memory location indicated by the level, node, and position specified in the word. It will be understood that memory need be allocated only for positions within the nodes designated by the read-in words. Inquiry is then made (diamond 2130) as to whether the last word of the frame has been reached. If not, the block 2115 is re-entered, and the loop 2135 is continued until all words of the frame have been stored at appropriate locations. It will be understood that, if desired, the word locations (level, node, and position) could alternatively be stored, and the values subsequently recovered by pointing back to their original storage locations. 
     During the next portion of the decoder routine, as shown in FIG. 22, the values in the nodes on the list are utilized to implement reconstruction as in the diagram of FIGS. 15A and 15B, with parent nodes being reconstructed from children nodes until the level zero information has been reconstructed. During this procedure, when a parent node is reconstructed from its children nodes, the parent node is added to the list of nodes, so that it will be used for subsequent reconstruction. This part of the procedure begins by initializing to the first node on the list (block 2210). Next, the block 2215 represents going to the memory location of the node and initializing to the first position in the node. Inquiry is then made (diamond 2220) as to whether there is a non-zero value at the position. If not, diamond 2240 is entered directly. If so, the value at the position is mapped into the positions of the parent node, with accumulation, as described above in conjunction with relationships (10) and (11). Inquiry is then made (diamond 2240) as to whether the last position of the node has been reached. If not, the next position in the node is considered (block 2245), diamond 2220 is re-entered, and the loop 2250 continues until all positions in the node have been considered. It will be understood that, if desired, a marker or vector can be used to indicate strings of blank positions in a node, or to point only to occupied positions, so that a relatively sparse node will be efficiently processed. In this regard, reference can be made to the abovereferenced copending U.S. patent application Ser. No. 525,974 now abandoned, filed of even date herewith. When the last position of the node has been considered, the node is removed from the list of nodes, as represented by block 2255, and inquiry is made (diamond 2260) as to whether the parent node is at level 0. If so, diamond 2270 is entered directly. If, however, the parent node is not at level 0, the parent node is added to the list of nodes (block 2265). Inquiry is then made (diamond 2270) as to whether the last node on the list has been reached. If not, the next node on the list is considered (block 2275), block 2215 is re-entered, and the loop 2280 is continued until processing is complete and the reconstructed values have been obtained. The decoder output values can then be read-out (block 2290). 
     It will be understood that similar techniques can be employed at higher dimensions and in other forms (see e.g. Appendix V). More complicated tree structures, such as where a node has more than two children (e.g. Appendices II and V) can also be utilized. 
     The invention has been described with reference to particular preferred embodiments, but variations within the spirit and scope of the invention will occur to those skilled in the art. For example, it will be recognized that the wavelet upon which the wavelet-packets are based can be changed as different parts of a signal are processed. Also, the samples can be processed as sliding windows instead of segments. ##SPC1## PG,37