Patent Publication Number: US-2005129332-A1

Title: Method and system for processing multidimensional data

Description:
RELATED APPLICATION  
      This application is a continuation under 37 C.F.R. 1.53(b) of U.S. application Ser. No. 09/615,445 filed Jul. 13, 2000, which claimed priority under 35 U.S.C. 119(e) from U.S. Provisional Application Ser. No. 60/143,784 filed Jul. 14, 1999, which applications are incorporated herein by reference and made a part hereof. 
    
    
     GOVERNMENT SUPPORT  
      This invention was made with U.S. Government support under Navy, Grant No. N00014-91-J-1152. The Government has certain rights in the invention. 
    
    
     FIELD  
      The present invention relates to processing data, and more particularly to processing multidimensional data.  
     BACKGROUND  
      Many multidimensional data processing algorithms are based on multiresolution decompositions. These algorithms include, for example, compression algorithms, noise removal algorithms, and algorithms for the reconstruction of images. The more efficiently these algorithms operate, the better the modem communications and information processing systems in which they are embedded operate. For example, efficient compression algorithms permit fast transmission of information in communication systems. Without efficient compression algorithms, multidimensional data requires an unacceptable amount of bandwidth for transmission and an unacceptable amount of storage for archiving.  
      Consider, for example, a medical image, such as a mammographic screening image, which may be represented by four six-thousand pixel by six-thousand pixel arrays. A mammographic screening image consists of four images, two images for each of two breasts. Of the two images associated with each breast, one image is a top image and one image is a side image. A pixel is a “picture element,” which is an elementary unit of information contained in an image and is typically represented by an intensity level. If each pixel in the mammographic screening image is represented by sixteen bits, then each pixel may be encoded at one of 65,536 possible intensity levels. To transmit the mammographic screening image without compression, 2.3 billion bits must be sent over a communication link. A typical telephone line is capable of transmitting about 56,000 bits per second, so transmission of a mammographic screening image would require more than ten hours. A ten hour transmission time is unacceptable for transmitting a mammographic screening image, so image compression processing is used to reduce the transmission time.  
      Prior to processing multidimensional data using some compression methods, such as wavelet compression, the multidimensional data is approximated at several resolution levels. For example, two-dimensional image data is initially divided into two rows and two columns. Each row and each column is subsequently divided into two rows and two columns.  FIG. 1  is an illustration of a sequence of images  100 , including images  101 ,  103 ,  105 , and  107 , of multidimensional data partitioned into rows and columns. The images  101 ,  103 ,  105 , and  107  illustrate partitioning a first dimensions  109  (rows) and a second dimension  111  (columns) at a rate of one. Image  101  is partitioned in the first dimension  109  and the second dimension  111 . Each of the partitions in image  101  is partitioned or divided to form image  103 . Each of the partitions in image  103  are partitioned or divided to form image  105 . And each of the partitions in image  105  are partitioned or divided to form image  107 . The partitioning or subdividing of rows and columns continues until an acceptable resolution level is achieved. An acceptable resolution level is a level at which data can be compressed, transmitted, and decompressed, such that the decompressed data includes the information contained in the original data required by a viewer of the received data. For example, in the mammographic screening example described above, the decompressed data must contain enough information related to a cancerous tumor to allow a radiologist to identify the cancerous tumor by viewing the mammographic screening images reconstructed from the compressed data.  
      Isotropic decomposition is one type of decomposition used in some multi-dimensional data processing algorithms. To perform isotropic decomposition, one begins with a function φ of one variable such that the set 
 
{φ(x−j)|jεZ}
 
 forms a Riesz basis for the span of these functions. Assume that φ satisfies the rewrite rule  
               ϕ   ⁡     (   x   )       =       ∑   j     ⁢       a   j     ⁢     ϕ   ⁡     (     x   -   j     )                   (   1   )             
 
 for a finite set of coefficients a j . Let S k  be the space of all functions  
         S   k     :=     {         ∑   j     ⁢       c   j     ⁢     ϕ   ⁡     (         2   k     ⁢   x     -   j     )           |       c   j     ∈   R       }         
 
 and choose a bounded projection P k  from L p (R) to S k . Under certain conditions (see Daubechies) any ƒεL p (R) can be re-written as  
       f   =         lim     k   -&gt;   ∞       ⁢     P   k       =         P   0     ⁢   f     +       ∑     k   =   1     ∞     ⁢     (         P   k     ⁢   f     -       P     k   -   1       ⁢   f       )               
 
 where, because of the rewrite rule (1), P k ƒ−P k−1 ƒ is in S k . Thus, since P 0 ƒεS 0 ,  
       f   =           P   0     ⁢   f     +       ∑     k   =   1     ∞     ⁢     (         P   k     ⁢   f     -       P     k   -   1       ⁢   f       )         ⁢     
     ⁢           =         ∑     j   ∈   Z       ⁢       d   j     ⁢     ϕ   ⁡     (     ·     -   j       )           +       ∑     k   =   1     ∞     ⁢       ∑     j   ∈   Z       ⁢       d     j   ,   k       ⁢       ϕ   ⁡     (       2   k     ·     -   j       )       .                   
 
 For suitable functions φ and special projectors P k , one can find a function ψ, associated with φ, such that  
             P   k     ⁢   f     -       P     k   -   1       ⁢   f       =       ∑     j   ∈   Z       ⁢       c     j   ,     k   -   1         ⁢     ψ   ⁡     (       2     k   -   1       ·     -   j       )               
 
 (note the new scaling −2 k−1  instead of 2 k ) so that  
       f   =         ∑     j   ∈   Z       ⁢       d   j     ⁢     ϕ   ⁡     (     ·     -   j       )           +       ∑     k   =   0     ∞     ⁢       ∑     j   ∈   Z       ⁢       c     j   ,   k       ⁢       ψ   ⁡     (       2   k     ·     -   j       )       .                 
 
      For a function ƒ: R d →R a similar decomposition holds. Define a set Ψ of 2 d −1 functions defined for x=(x 1 , . . . , x d )εR d  by  
       Ψ   :=       {           ∏     i   =   1     d     ⁢       v   i     ⁡     (     x   i     )         |     v   i       =       ϕ   ⁢           ⁢   or   ⁢           ⁢     v   i       =   ψ       }     ⁢   \   ⁢     {       ∏     i   =   1     d     ⁢     ϕ   ⁡     (     x   i     )         }           
 
 together with the function  
         Φ   ⁡     (   x   )       =       ∏     i   =   1     d     ⁢       ϕ   ⁡     (     x   i     )       .           
 
 Then, under suitable conditions, any ƒ in L p (R d ) can be written as  
       f   =         ∑     j   ∈     Z   d         ⁢       d   j     ⁢     Φ   ⁡     (     ·     -   j       )           +       ∑     k   =   0     ∞     ⁢       ∑     j   ∈     Z   d         ⁢       ∑     ψ   ∈   Ψ       ⁢       c     j   ,   k       ⁢       ψ   ⁡     (       2   k     ·     -   j       )       .                   
 
 Note that, since x=(x 1 , . . . , x d ) and the multi-index j=(j 1 , . . . , j d ), 
 
ψ(2 k   x−j )=ψ(2 k   x   1   −j   1 , . . . , 2 k   x   d   −j   d ), 
 
 i.e., each of the components x i  of x has been scaled by the same amount, 2 k . 
 
      One disadvantage of isotropic decomposition is that not all data is isotropic, and anisotropic multidimensional data is not efficiently processed by algorithms based on isotropic decomposition.  
      For these and other reasons there is a need for the present invention.  
     SUMMARY  
      According to one aspect of the present invention, a method is described for forming multi-resolution representations of data. The method includes the operations of partitioning the data in a first dimension at a first rate, and partitioning the data in a second dimension at a second rate, wherein the first rate is not equal to the second rate. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       FIG. 1  is an illustration of a sequence of prior art images of multidimensional data being partitioned into rows and columns;  
       FIG. 2  is an illustration of a sequence of images of multidimensional data being partitioned according to the present invention; and  
       FIG. 3  is a block diagram of one embodiment of a system including a computer-readable medium having computer-executable instructions for performing a method according to the present invention. 
    
    
     DESCRIPTION  
      In the following detailed description of exemplary embodiments of the invention, reference is made to the accompanying drawings which form a part hereof, and in which is shown by way of illustration specific exemplary embodiments in which the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice the invention, and it is to be understood that other embodiments may be utilized and that logical, mechanical, electrical and other changes may be made without departing from the spirit or scope of the present invention. The following detailed description is, therefore, not to be taken in a limiting sense, and the scope of the present invention is defined only by the appended claims.  
      Some portions of the detailed descriptions which follow are presented in terms of algorithms and symbolic representations of operations on data bits within a computer memory. These algorithmic descriptions and representations are the means used by those skilled in the data processing arts to most effectively convey the substance of their work to others skilled in the art. An algorithm is here, and generally, conceived to be a self-consistent sequence of operations leading to a desired result. The operations are those requiring physical manipulations of physical quantities. Usually, though not necessarily, these quantities take the form of electrical or magnetic signals capable of being stored, transferred, combined, compared, and otherwise manipulated. It has proven convenient at times, principally for reasons of common usage, to refer to these signals as bits, values, elements, symbols, characters, terms, numbers, or the like. It should be borne in mind, however, that all of these and similar terms are to be associated with the appropriate physical quantities and are merely convenient labels applied to these quantities. Unless specifically stated otherwise as apparent from the following discussions, it is appreciated that throughout the present invention, discussions utilizing terms such as “processing” or “computing” or “calculating” or “determining” or “displaying” or the like, refer to the action and processes of a computer system, or similar electronic computing device, that manipulates and transforms data represented as physical (electronic) quantities within the computer system&#39;s registers and memories into other data similarly represented as physical quantities within the computer system memories or registers or other such information storage, transmission or display devices. As used herein, the term image is defined to be a collection of data, and an image may be displayed by a display device, such as a cathode ray tube, a liquid crystal display, or similar device.  
       FIG. 2  is an illustration of a sequence of images  201 - 204  being repeatedly partitioned according to the present invention. Each of the images in the sequence of images  201 - 204  is a representation of multidimensional data. In the illustration shown in  FIG. 2 , the multidimensional data is two-dimensional data, however, the present invention is not limited to processing two-dimensional data. The exemplary embodiments of the present invention described in the application are described in terms of two-dimensional data because two-dimensional data examples are most easily illustrated and understood. Those skilled in the art will recognize that the present invention is applicable to data having two or more dimensions.  
      The sequence of images  201 - 204  may represent a variety of data types. For example, in one embodiment of the present invention, the sequence of images  201 - 204  represents X-ray images. In an alternate embodiment, the sequence of images  201 - 204  represents magnetic resonance imaging (MRI) images. In another alternate embodiment, the sequence of images  201 - 204  represents positron emission tomography (PET) images. In another alternate embodiment, the sequence of images  201 - 204  represents multi-spectral images. In still another alternate embodiment, the sequence of images  201 - 204  represents video images in which one of the dimensions is time.  
      Each of the types of images described above, along with other types of images not described above, but which are also suitable for use in connection with the present invention, share a common characteristic which is that the smoothness of the data is anisotropic. Data smoothness is anisotropic when the data smoothness depends on the dimension of the data in which the smoothness is being estimated. For example, for two-dimensional data having a first dimension and a second dimension, data having a smoothness estimate of two in the first dimension and a smoothness estimate of three in the second dimension is anisotropic. Examples of data that is anisotropic include hyperspectral image data and video data. Hyperspectral image data has the same smoothness in each of the two spatial directions in a fixed frequency band, but has a different smoothness in the frequency direction for a given spatial point. Video data has isotropic smoothness within a frame for a fixed time, but has a different smoothness in the time dimension.  
      In one embodiment of the present invention, Besov spaces are used to estimate the smoothness in a particular dimension and to determine whether the multidimensional data is anisotropic. The use of Besov spaces to estimate the smoothness in a particular dimension is described in detail below. However, the present invention is not limited to the use of Besov spaces in estimating the smoothness of multidimensional data. Any method, algorithm, or numerical calculation that is capable of estimating data smoothness is suitable for use in connection with the present invention.  
      Besov spaces can be used to measure the smoothness of data. Let the data be represented by a function ƒ:Ω→R, where Ω is an open subset of R d , d-dimensional Euclidean space. Assume that there are smoothness parameters α=(α 1 , . . . , α d ), with α i &gt;0, and to be specific, that we want to measure the smoothness of the data in the space L p  (R d , so that the data has smoothness α i  in the e i  direction, i=1, . . . , d, with e i =(0, . . . , 0, 1, 0, . . . , 0) being the ith coordinate vector, which has a 1 as its ith element and 0 as all other elements.  
      For any h=(h 1 , . . . , h d ) in R d , define the rth difference of ƒ in the direction h at the point xεR d  recursively as 
 
Δ h   r (ƒ, x ):=Δ h   r−1 (ƒ, x+h )−Δ h   r−1 (ƒ, x ) 
 
 and 
 
Δ h   0 (ƒ, x ):=ƒ( x ) 
 
 Δ h   r (ƒ,x) is defined on the set 
 
Ω rh   :={xεΩ|x+khεΩ,k= 1 , . . . , r} 
 
      Let t=(t 1 , . . . , t d ), t i &gt;0 for all i, and define the rth modulus of smoothness of ƒ in L p (R d ) to be  
               ω   r     ⁡     (     f   ,   t     )       p     :=           ω   r     ⁡     (     f   ,   t   ,   …   ⁢           ,     t   d       )       p     :=       sup            h   i          ≤     t   i         ⁢              Δ   h   r     ⁡     (     f   ,   ·     )                L   p     ⁡     (     Ω   rh     )               ,   where       
              g            L   p     ⁡     (   I   )         :=         (       ∫   I     ⁢              f   ⁡     (   x   )            p     ⁢     ⅆ   x         )       1   /   p       .         
 
      The anisotropic Besov space B q   α (L p (Ω)) for 0&lt;α 1 , . . . , α d &lt;r is defined to be the set of all functions ƒ for which  
              f            B   q   α     ⁡     (       L   p     ⁡     (   Ω   )       )         :=       (       ∑     k   =   0     ∞     ⁢       [       2   k     ⁢         ω   r     ⁡     (     f   ,     2       -   k     /     α   1         ,   …   ⁢           ,     2       -   k     /     α   d           )       p       ]     q       )       1   /   q           
 
 is finite, and 
 
∥ƒ∥ B     q       α     (L     p     (Ω)) :=∥ƒ∥ L     p     (Ω) +|ƒ| B     q       α     (L     p     (Ω)) . 
 
 Heuristically, a function ƒ in B q   α (L p (Ω)) has α i  “derivatives” in L p  in the ith coordinate direction. 
 
      If α 1 = . . . =α d , then  
              f            B   q   α     ⁡     (       L   p     ⁡     (   Ω   )       )         :=       (       ∑     k   =   0     ∞     ⁢       [       2   k     ⁢         ω   r     ⁡     (     f   ,     2       -   k     /     α   1         ,   …   ⁢           ,     2       -   k     /     α   1           )       p       ]     q       )       1   /   q           
 
 which is equivalent to the usual semi-norm  
              f            B   q   α     ⁡     (       L   p     ⁡     (   Ω   )       )         :=       (       ∑     k   =   0     ∞     ⁢       [       2       α   1     ⁢   k       ⁢         ω   r     ⁡     (     f   ,     2   k       )       p       ]     q       )       1   /   q           
 
 of the isotropic Besov space B q   α     1   (L p (Ω)) . A more detailed description of Besov spaces is provided in Technical Report #328 available from the Center for Applied Mathematics at Purdue University and is incorporated herein by reference. 
 
      In one embodiment of the present invention, multiresolution representations of data are formed by repeatedly partitioning the data in a first dimension at a first rate, and repeatedly partitioning the data in a second dimension at a second rate. The first rate is not equal to the second rate. For example, each of the images in the sequence of images  201 - 204  shown in  FIG. 2  is a two-dimensional image having a first dimension  209  and a second dimension  211 . The sequence of images  201 - 204  illustrates repeatedly partitioning data in dimension  209  at a first rate equal to one. The first line  213  in image  201  shows a partitioning of image  201  along the first dimension  209 . In the partition illustrated in image  202 , each partition in image  201  is divided into two partitions in dimension  209 . In the images  202 - 204  partition lines  215 - 228  are added to illustrate that each of the partitions in each subsequent image is divided into two partitions. For the sequence of images  201 - 204 , the partition rate in the first dimension is one, which means that partitioning occurs in each image in the sequence of images.  
      The sequence of images  201 - 204  also illustrates repeatedly partitioning data in second dimension  211  at a second rate equal to one-half. The first line  229  in image  202  shows a partitioning of image  201  along the second dimension  211 . In the partition illustrated in image  202 , image  201  in the second dimension is divided into two partitions. In the images  202 - 204 , partition lines  231 - 232  are added to illustrate that each of the partitions in alternating subsequent images is divided into two partitions.  
      In an alternate embodiment of the present invention, an estimate of smoothness in a first dimension is obtained and an estimate of smoothness in a second dimensions is obtained. The smoothness estimates may be obtained using Besov spaces, as described above, or any other method of estimating smoothness. The first partition rate is set to one, and if the first dimension smoothness estimate is less than the second dimension smoothness estimate, then the second partition rate is set to a ratio of a first dimension smoothness estimate to a second dimension smoothness estimate. If the first dimension smoothness estimate is greater than the second dimension smoothness estimate, then the second rate is set to one and the first rate is set to a ratio of the second dimension smoothness estimate to the first dimension smoothness estimate.  
      Multidimensional anisotropic data can be decomposed. If a function ƒ:R d →R is in the anisotropic Besov space 
 
B q   α (L p (R d )) 
 
 then  
              f            B   q   α     ⁡     (       L   p     ⁡     (   Ω   )       )         :=       (       ∑     k   =   0     ∞     ⁢       [       2   k     ⁢         ω   r     ⁡     (     f   ,     2       -   k     /     α   1         ,   …   ⁢           ,     2       -   k     /     α   d           )       p       ]     q       )       1   /   q           
 
 is finite. However, the quantity on the right is equivalent to  
                   (       ∑     k   =   0     ∞     ⁢       [       2     k   /     α   _         ⁢         ω   r     ⁡     (     f   ,     2       -   k     ⁢           ⁢       α   _     /     α   1           ,   …   ⁢           ,     2       -   k     ⁢           ⁢       α   _     /     α   d             )       p       ]     q       )       1   /   q       ⁢           ⁢   where     ⁢     
     ⁢       α   _     =         min     1   ≤   i   ≤   d       ⁢         α   i     .     
     ⁢   Note     ⁢           ⁢   that   ⁢           ⁢       α   _       α   i           ≤   1               (   2   )             
 
 with equality only for those i for which α i = α ; there is always at least one i for which this is true. Note also that (2) is equivalent to  
               (       ∑     k   =   0     ∞     ⁢       [       2     k   /     α   _         ⁢         ω   r     ⁡     (     f   ,     2     -     ⌊     k   ⁢           ⁢       α   _     /     α   1         ⌋         ,   …   ⁢           ,     2     -     ⌊     k   ⁢           ⁢       α   _     /     α   d         ⌋           )       p       ]     q       )       1   /   q             (   3   )             
 
 where └y┘ is the greatest integer ≦y, since this increases the size of each argument of the modulus of smoothness by at most a factor of 2. 
 
      Define S k  to be the linear span of the functions  
           ϕ     j   ,   k       ⁡     (   x   )       :=       ∏     i   =   1     d     ⁢     ϕ   ⁡     (         2     ⌊     k   ⁢           ⁢       α   _     /     α   i         ⌋       ⁢     x   i       -     j   i       )             
 
 for j=(j 1 , . . . ,j d )εZ d . Note that the scaling in each variable is always an integer power of 2, so that S k  is, indeed, included in S k+1 , by the rewrite rule for φ; furthermore, since  α /α i =1 for at least one i, we know that S k  is strictly contained in S k+1 , i.e., when moving from S k  to S k+1 , one refines functions by a factor of two in at least one direction. In fact, going from S k  to S k+1  we refine in all directions e i  for which  
             α   _       α   i       ⁢   k     &lt;     m     k   ,   i       ≤         α   _       α   i       ⁢     (     k   +   1     )           
 
 for some integer m k,i,  and for no other directions. 
 
      Thus, if P k  is defined to be the projection onto the new S k , then  
         f   =         P   0     ⁢   f     +       ∑     k   =   1     ∞     ⁢     (         P   k     ⁢   f     -       P     k   -   1       ⁢   f       )           ,       
 
 and P k ƒ−P k−1 ƒ is again in S k . If there is a function ψ associated with φ and the projections P k , then P k ƒ−P k−1 ƒ can be written as a linear combination of the functions  
         ∏                 i   ∈     Λ   k                   η   i     =       ϕ   ⁢           ⁢   or   ⁢           ⁢     η   i       =   ψ                         not   ⁢           ⁢   all   ⁢           ⁢     η   i       =   ϕ             ⁢         η   i     ⁡     (         2     ⌊       (     k   -   1     )     ⁢       α   _     /     α   i         ⌋       ⁢     x   i       -     j   i       )       ⁢       ∏     i   ∉     Λ   k         ⁢     ϕ   ⁡     (         2     ⌊       (     k   -   1     )     ⁢       α   _     /     α   i         ⌋       ⁢     x   i       -     j   i       )               
 
 for all jεZ d , where Λ k  consists of the set of coordinates that are refined in going from S k−1  to S k . This method can be applied to all orthogonal and biorthogonal wavelets, wavelet frames, or like multiresolution methods. A more detailed description of multiresolution decomposition is provided in Technical Report #328 available from the Center for Applied Mathematics at Purdue University and is incorporated herein by reference. 
 
      The methods of the present invention may be realized, at least in part, as one or more programs or modules running on a computer—that is, as a program or module executed from a computer-readable medium such as a memory by a processor of a computer. The programs are desirably storable on a computer-readable medium such as a floppy disk or a CD-ROM, for distribution and installation and execution on another (suitably equipped) computer.  
       FIG. 3  is a block diagram of one embodiment of a computerized system  300  according to the present invention. In one embodiment of the present invention, the computerized system  300  is used to form multiresolution representations of data. The computerized system  300  includes a processor  301 , a storage medium  303 , and a module  305 . The processor  301  is coupled to the storage medium  303 , and module  305  is capable of being stored on storage medium  303 . In one embodiment, the processor  301  is a microprocessor, however the present invention is not limited to use in connection with a particular type of processor. Any processor, such as a digital signal processor (DSP), a reduced instruction-set computing (RISC) processor, or a complex instruction-set computing (CISC) processor, capable of processing information is suitable for use in connection with the present invention. In one embodiment, the storage medium  303  is a computer-readable storage medium, such as a CD-ROM, floppy disk, or a semiconductor storage device, such as a cache memory. Using a cache memory to store the module permits multiresolution representations of the data to be quickly generated. The module  305  is capable of executing on the processor  301  and capable of repeatedly partitioning data in a first dimension at a first rate, and repeatedly partitioning data in a second dimension at a second rate, wherein the first rate is not equal to the second rate.  
      In an alternate embodiment of the present invention, data having a time dimension is decomposed using multiresolution decomposition. Exemplary types of data having a time dimension include video images, seismic images, functional (time dependent) magnetic resonance imaging (fMRI) images, and functional (time dependent) positron emission tomography (fPET) images or kinetic positron emission tomography (PET). Functional MRI images and functional PET images are generally considered to be images obtained from techniques involving fast MRI scans, fast PET scans, or techniques for co-registering PET and MRI scans. However, in the present invention, fMRI and fPET images are any anisotropic images obtained using MRI imaging systems, PET imaging systems or combinations of MRI and PET imaging systems.  
      In one embodiment, a computer readable medium having computer-executable instructions for decomposing data having a time dimension includes a number of operations. First, a spatial dimension smoothness estimate for spatial dimension data is obtained. Second a time dimension smoothness estimate for the time dimension data is obtained. In one embodiment, the spatial dimension smoothness estimate and the time dimension smoothness estimate may be obtained by the use of Besov spaces. Third, a first data partition rate is set to one. Fourth, a second data partition rate is set to a ratio of the time dimension smoothness estimate to the spatial dimension smoothness estimate. Fifth, the spatial dimension data is repeatedly partitioned at the first rate. Sixth, the time dimension data is repeatedly partitioned at the second rate.  
      In another alternate embodiment, a method of compressing data having a first dimension and a second dimension includes forming a multiresolution representation of the data and compressing the multiresolution representation of the data. The method of forming the multiresolution representation of the data includes repeatedly partitioning the data in the first dimension at a first rate, and repeatedly partitioning the data in the second dimension at a second rate, wherein the first rate is not equal to the second rate. Repeatedly partitioning the data in the second dimension at a second rate comprises estimating a first smoothness in the first dimension, estimating a second smoothness in the second dimension, and computing the second rate by forming a ratio of the first smoothness to the second smoothness. In one embodiment, compressing the multiresolution representation of the data includes compressing the multiresolution representation of the data using wavelet compression.  
      In still another alternate embodiment of the present invention, a method includes forming a multiresolution representation of video data having a spatial dimension and a time dimension and processing the multiresolution representation of the video data to remove noise from the multiresolution representation of the video data. Processing the multiresolution representation of the video data to remove noise from the multiresolution representation of the video data comprises filtering the multiresolution representation of the video data. In one embodiment, processing the multiresolution representation of the video data comprises filtering the multiresolution representation of the data using a low-pass filter. In an alternate embodiment, processing the multi-resolution representation of the video data to remove noise from the multi-resolution representation of the video data comprises filtering the multiresolution representation of the video data using a band-pass filter. The filters used to process the multiresolution representation of the video data are typically digital filters.  
      A method and system for processing multidimensional data has been described. The method and system are based on anisotropic multidimensional decompositions, including anisotropic wavelet decompositions. The method provides for preparing multidimensional scalar or vector data with anisotropic smoothness for fuirther processing, such as data compression, noise removal, and reconstruction.  
      Although specific embodiments have been illustrated and described herein, it will be appreciated by those of ordinary skill in the art that any arrangement which is calculated to achieve the same purpose may be substituted for the specific embodiments shown. This application is intended to cover any adaptations or variations of the present invention. Therefore, it is intended that this invention be limited only by the following claims and equivalents thereof.