# Patent

## Title
Linearized data structure ordering images based on their attributes

## Abstract

A method of displaying images is based on both a first linearization and a second linearization. In one embodiment, the linearizations are performed by traversing two space-filling curves. In another embodiment, the linearizations are performed by traversing a cluster data structure. More than two linearizations may be displayed.

## Background
The present application is a divisional of U.S. patent application Ser. No. 09/777,165 filed Feb. 5, 2001, entitled “Creating A Linearized Data Structure For Ordering Images Based On Their Attributes ”, now U.S. Pat. No. 6,628,846, which is a divisional of U.S. patent application Ser. No. 09/150,496 filed Sep. 9, 1998, entitled Multi- Linearization Data Structure For Image Browsing”, now U.S. Pat. No. 6,233,367. The U.S. patent application Ser. No. 09/777,165 is hereby entirely incorporated by reference. The present application is also related to U.S. patent application Ser. No. 09/777,164 filed Feb. 5, 2001, entitled “Displaying Ordered Images Based On A Linearized Data Structure”, now U.S. Pat. No. 6,556,723.


A portion of the disclosure of this patent document contains material which is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent disclosure, as it appears in the Patent and Trademark Office patent files or records, but otherwise reserves all copyright rights whatsoever.
BACKGROUND OF THE INVENTION
1. Field of the Invention
The described invention relates to the field of data management. In particular, the invention relates to image categorization and display.
2. Description of Related Art
Browsing is a technique that users employ to view information rapidly for deciding whether the content is relevant to their needs. Browsing should give the user quick access to view the content and focus on sections of further interest. Multiple images are typically displayed. The user is able to scroll through the images until the desired image is found.
It is difficult, however, to characterize and sort images based on attributes of the images. While certain images may be closely related by one attribute, they may not be closely related by other attributes. Additionally, if many images are stored in a data structure, the amount of time to either place an image into the data structure or retrieve an image from the data structure can become excessive, such that a user will not want to wait.
SUMMARY OF THE INVENTION
A method of displaying images is based on both a first linearization and a second linearization. In one embodiment, the linearizations are performed by traversing two space-filling curves. In another embodiment, the linearizations are performed by traversing a cluster data structure. More than two linearizations may be displayed.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 shows one embodiment of a flowchart of the process of creating dual linearizations for each image.
FIG. 2 shows an example of a space-filling curve.
FIG. 3 shows an embodiment in which a second curve is employed to improve the reliability of providing images that are close together in the multi-dimensional space.
FIG. 4 shows an example of a browser interface based on a dual linearization.
FIG. 5 shows an example of a browser interface based on a triple linearization.
FIG. 6 shows an example of a system implemented on an SQL server that communicates with a middleware server to construct and serve web pages to an end user.
FIG. 7 shows the 1st, 2nd, 3rd and 4th orders of a two-dimensional Hilbert curve.
FIG. 8 shows the 1st, 2nd, and 3rd orders of a two-dimensional Peano curve.



DETAILED DESCRIPTION
A method of characterizing images that scales well for large amounts of images is described. Due to the large number of attributes by which an image may be categorized, up until this disclosure, it has been difficult to categorize and display like images in an efficient manner. The present invention provides for display of images based on multiple linearizations of the images. “Linearization,” in this application, implies not just, an ordering of a set of points, but also storing their positions, or at least relative distances between their positions.
The method described herein is scaleable, so that the amount of time to categorize and display a portion of a set of images does not change significantly whether the set of images is relatively small (e.g., a few hundred images) or relatively large (e.g., greater than 50,000 images).
Two effective methods for linearizing the images are described: a space-filling curve method and a clustering method. Other methods, however, can also be used to linearize the images.
Space Filling Curves
An image may be categorized by various attributes, or components, of the image. For example, an image can be broken down and characterized by color information, shape information, size information, texture information, objects represented, or based on other content of the image. The image may also be categorized by tag information, which is information associated with each image, but which may not be part of the image itself. Tag information may include annotation information, audio information, time stamp information, global position information (GPS), and so forth.
In one example, a histogram of an image is generated using 600 different color bins. The histogram includes the number of pixels corresponding to each of the 600 colors used.
In one embodiment, each of the components of the histogram is used to map the image to a coordinate in N-dimensional space. For this example, N is 600. Image 1, for example, is mapped to the coordinate:

    
    
        <attribute 1, attribute 2, attribute 3, . . . attribute N> in N-dimensional space, where attribute 1 is the number of pixels corresponding to a first color, attribute 2 is the number of pixels corresponding to a second color, and so forth.
    
    


Multiple images may be mapped into the N-dimensional space. Images that are located close together in the N-dimensional space are closely related by their attributes. In contrast, those that are located farther apart in the N-dimensional space are less closely related by their attributes.
One method that has been used for ordering a set of images mapped into N-dimensional space is by mapping all of the images to an arbitrary line. The position on the line to which the images are mapped indicates the linearization of the images. This linearization, however, does not preserve the relationship of similarly attributed images. This is because the mapping of the N-dimensional space into a one dimension line does not maintain information about how closely images are related in the N-dimensional space.
One embodiment of the present invention linearizes the images using a space-filling curve to help maintain the relationship of like images. A space-filling curve is a continuous path that visits every point in a k-dimensional space. The space-filling curve provides a way to linearize the points in the multi-dimensional space.
Space-filling curves have been studied for many years, and more recently have interested computer scientists. For example, see “Fractals for Secondary Key Retrieval” by C. Faloutsos and S. Roseman (Technical Report CS-TR-2242, University of Maryland Institute for Advanced Computer Studies, May 1989) for an example of the use of space-filling curves. There are several different types of space filling curves including the Hilbert curve, the Peano curve, the Sierpinski curve, the Lebesgue curve, and the Schoenberg curve. See Hans Sagan, “Space-Filling Curves,” Springer-Verlag, 1994, for more information on space-filling curves.
FIG. 7 shows the 1st, 2nd, 3rd and 4th orders of a two-dimensional Hilbert curve. The procedure to derive higher orders of the Hilbert curve is to rotate and reflect the curve at vertex 0 and at vertex 3. The curve can keep growing recursively by following the same rotation and reflection pattern at each vertex of the curve.
FIG. 8 shows the 1st, 2nd, and 3rd orders of a two-dimensional Peano curve. The procedure to derive higher orders of the Peano curve is to replace each vertex of the basic curve with the previous order curve. The Hilbert curve and Peano curves can be extended into multiple dimensions, as is well-known in the art.
Space filling curves have the property that the points that are close to each other on the space-filling curve are also close to each other in the multi-dimensional space. However, the opposite is not true: the points that are close to each other in the multi-dimensional space are not necessarily close together on the space-filling curve.
FIG. 2 shows an example of a two-dimensional and three-dimensional space-filling curve. In this case, an N-dimensional Hilbert curve is employed. Two points 10 and 11 that are close to each other in the multi-dimensional space lie close to a partition boundary separating two sections of the multi-dimensional space. When the curve is traversed as shown at the bottom of FIG. 2, the points that were close in space map to points that lie far apart in linearization. This undesirable distancing of close points occurs at, and because of, partition boundaries of the Hilbert and Peano curves.
FIG. 3 shows an embodiment in which a second curve is employed to improve the reliability of providing images that are close together in the multi-dimensional space. In one embodiment, a Peano curve is used to complement a Hilbert curve. The Peano and Hilbert curves are constructed such that their partition boundaries, where undesirable results are likely to happen, occur in different locations. Points 20 and 21 that lie close in space will likely end up close together in either the Hilbert curve or the Peano curve, as shown in FIG. 3.
Table 1 shows an example of a computer program written in C language that maps a point in N-dimensional space to a position on a one-dimensional traversal of a Hilbert curve and a position on a one-dimensional traversal of a Peano curve. In one embodiment, the code maps the point in N-dimensional space to an interval between 0 and 1. The output of the program is a bit pattern identifying a number between 0 and 1 for each image being analyzed. As other images are converted into other bit patterns between 0 and 1, a database is set up with a first linearization based upon the Hilbert curve and a second linearization based upon the Peano curve.
In one embodiment, for simplification, images that map to a partition boundary are adjusted slightly so that they do not fall directly on the partition boundary. This is done because a curve may hit points on a partition boundary multiple times. In a Hilbert curve, the partition boundaries are at points that are any multiple of ±1⁄2, 1⁄4, 1⁄8, 1/16, etc, of the N-dimensional space. In a Peano curve, the partition boundaries are at points that are any multiple of 1⁄3, 1/9, 1/27, etc., of the N-dimensional space.







TABLE 1







/************************************************************


© Intel Corporation, 1998


 Pseudo-code for collapsing points in a space to points in a line,


 via space-filling curve traversal.


 The “points” are arrays of plain old integer coordinates. These are


 combined to produce a single value, indicating a position along a line.


 Since we're dealing with high-dimensional spaces (DIMENSION ==


 ~600), the final value will have hundreds of digits, and we construct a


 special datatype to hold it.


************************************************************/


BEGIN









for each image I, BEGIN:









Compute I's n-bin color histogram, H



Normalize H to a standard number of pixels



Convert H to an n-dimensional vector, V



Call Hctol(V) to print out position of V along









a Hilbert curve traversal.









Call Pctol(V) to print out position of V along









a Peano curve traversal.









END



Sort images along Hilbert curve positions: output sorted list



Sort images along Peano curve positions: output sorted list



/* These two orderings can then be used to generate Browser



Windows */







END


/*


 * function: hilbert_cube_to_line --- Hctol








 *
Collapses m-dimensional coordinates to a line via


 *
Hilbert curve traversal







 */


function Hctol(point)


{


 integer tau, /* chunk of digits of final line position */









lasttau;  /* holder for previous value of tau  */








 integer i, j, k,   /* counters
*/









 lastbit=0,



  one, zero,



  r=0, b=0,



 DEGREE=19; /* DEGREE == number of bits precision per



 coordinate */







 FOR i = DEGREE . . . 1


{


 /* Extract tau from all the coordinate values */


 FOR j = 1 . . . DIMENSION


  {









 SET j'th bit of tau = i'th bit of point[j]







  }


  SWAP tau and lasttau;


 XOR tau with lasttau;


 ROLL tau r bits to the left;


 tau−>end = (tau−>end + roller)%DIMENSION;


 /* twiddle bits in tau */


  if(lastbit = 0)









TOGGLE first bit of tau: replace 1 with 0, 0 with 1;







 TOGGLE b'th bit (from the left) of tau;


 /* Avalanche and send digits to the output buffer */


 for(j = 1 . . . DIMENSION) {









COMPUTE k = the XOR of the first j bits of tau;



OUTPUT k;







  if(k==1) one=j;









else zero=j;







 }


 /* compute b */


 SET lastbit = k;


  if(lastbit == 0)









b=DIMENSION-one;









else









b=DIMENSION-zero;







  COMPUTE r = (r+DIMENSION-1-b)%DIMENSION;


 }


}


/*


 * function: peano_cube_to_line








 *
Collapses m-dimensional coordinates to a line via


 *
Peano curve traversal


 */







void Pctol(point)


{


 int i, j, trit, flag=0, DEGREE=12;


 int modulus, h=0;








 int flags[2048];
/* flags are used to determine which way



 the curve winds at any given point  */







 FOR all i, SET flags[i]=0;


 FOR i = DEGREE . . . 1


  FOR j = DIMENSION . . . 1









 {



 flag = flag XOR flags[j];



 trit = i'th digit of point[j]  /* This is in base 3, so trit









takes on values 0, 1, or 2 */









 if(trit=1) INVERT flag[j];



 else









if(flag is TRUE)









SET trit = 2-trit; /* Sets 2 to 0, or 0 to 2 */









OUTPUT trit;



SET flag = flag XOR flags[j];







  }


 }


}


/*


 Source code for collapsing points in a space to points in a line,


 via space-filling curve traversal.


 The “points” are arrays of plain old integer coordinates. These are


 combined to produce a single value, indicating a position along a line.


 Since we're dealing with high-dimensional spaces (DIMENSION ==


 ~600), the final value will have hundreds of digits, and we must


 construct a special datatype to hold it.


 */


 #include <stdio.h>


 #define MAXD 700


 #define DIMENSION length


 #define SIDELENGTH 531441








/*
 */







/* Code to handle the numeral datatype */








/*
 */







typedef struct numeral{










 char d[MAXD];
/* At most MAXD digits  */








   int end;
/* position of highest digit */







} * numeral;


char hopper[10];


int hopsize, hopcount, hopmod;


/* Ninit --- allocates a numeral and sets its value to 0 */


numeral ninit(int length)


{


 numeral done; int i;


 if((done=(numeral)malloc(sizeof(struct numeral)))==NULL)


  {fprintf(stderr,“Numeral allocation problem.\n”); exit(1);}


 for(i=0; i<MAXD; i++) done−>d[i]=‘0’;


 done−>end=DIMENSION-1;


 return done;


}


/* Hopperhead --- initializes the “hopper.” More on this below */


void hopperhead(int size, int mod)


{


 hopsize=size; hopmod=mod; hopcount=0;


}


/* Hop --- pours digits into a “hopper,” emptying the whole thing








 *
 when it becomes full.







 *


 * The hopper is a small buffer of digits in some small base b, such


 * as 2 or 3. These digits are generated by the curve-mapping functions,


 * in the form of very large binary/ternary numbers. We don't want to


 * store all that in a file, so the hop function conglomerates N base-b


 * digits into a single digit base N*b before printing it to standard out.


 *


 */


void hop(int c)


{


 int i,j;


 if(c != 0) hopper[hopcount++]=(char) c;


 if(hopcount >= hopsize ∥ c==0) {/* Dump hopper in base-n format */


  for(i=j=0; j<hopcount; j++)









 i = i*hopmod + hopper[j]-‘0’;







  printf(“%c”, i+‘ ’+1);


  hopcount=0;


 }


}


/*


 * function: hilbert_cube_to_line








 *
Collapses m-dimensional coordinates to a line via


 *
Hilbert curve traversal


 *


 *
This function accepts an array of integers (coordinates)


 *
and converts it to a position along a line, sending the


 *
digits to an output buffer which eventually prints to


 *
standard output.


 */







void Hctol(int * v, int length)


{


 numeral tau, lasttau;


 int i, j, k, one, zero, lastbit=0, lastbyte=0, roller=0, DEGREE=19;


 tau=ninit(DIMENSION); lasttau=ninit(DIMENSION);








 hopperhead(6,2);
/* Set up hopper: Base 2, */









/* Buffer of six digits max */







 for(i=1<<(DEGREE−1); i>0; i>>=1)


 {


  /* Extract tau from all the coordinate values */


  for(j=0; j<DIMENSION; j++)









 tau−>d[j] = ((v[j]&i)==0) ? ‘0’ : ‘1’;







  tau−>end=DIMENSION−1;


  /* Perform: tau {circumflex over ( )}= lasttau; lasttau {circumflex over ( )}= tau; */


  for(j=0; j<DIMENSION; j++) {









 k=tau−>d[j];



 tau−>d[j]{circumflex over ( )}= (lasttau−>d[j] & 1);



 lasttau−>d[j]=k;







  }


  /* Roll the value of tau “roller” bits to the left */


  tau−>end = (tau−>end + roller)%DIMENSION;


  /* twiddle bits in tau */









 if(lastbit==0) tau−>d[(tau−>end+1)%DIMENSION] {circumflex over ( )}= 1;







  tau−>d[(tau−>end+1+lastbyte)%DIMENSION] {circumflex over ( )}= 1;


  /* Avalanche and send digits to the output buffer */


  for(j=k=0, one=zero=DIMENSION−1; j<DIMENSION; j++) {









 hop(tau−>d[(tau−>end+1+j)%DIMENSION] {circumflex over ( )}k);



 k {circumflex over ( )}= (tau−>d[(tau−>end+1+j)%DIMENSION] & 1);



 if(k) one=j; else zero=j;







  }


  /* compute lastbyte */









 if(lastbit=k) lastbyte=DIMENSION-1-zero;



  else lastbyte=DIMENSION-1-one;







  roller = (roller+DIMENSION-1-lastbyte)%DIMENSION;


 }


 hop(0);  /* flush the output buffer */


 free(tau); free(lasttau);


}


/*


 * function: peano_cube_to_line








 *
 Collapses m-dimensional coordinates to a line via


 *
 Peano curve traversal


 *


 *
 This function accepts an array of integers (coordinates)


 *
 and converts it to a position along a line, sending the


 *
 digits to an output buffer which eventually prints to


 *
 standard output.


 */







void Pctol(int * v, int length)


{


 int i, j, trit, flag=0, DEGREE=12;


 int modulus, h=0;


 int flags[2048];








 hopperhead(4,3);
/* Set up hopper: base-3,  */



/* Buffer of 4 digits max  */


 for(i=0; i<2048; i++) flags[i]=0;







 for(modulus=SIDELENGTH/3, i=DEGREE−1; i>=0; i--,


 modulus /= 3){


  for(j=DIMENSION−1; j>=0; j--){










 flag {circumflex over ( )}= flags[j];
/* XOR-out current flag  */



 trit = (v[j]/modulus)%3;
  /* Get current base-3 digit */



 if(trit==1) flags[j] {circumflex over ( )}= 1;
 /* Invert current flag, or: */



 else if(flag) trit = 2-trit;
 /*  maybe invert the digit */



 hop(trit+‘0’);
/* Send digit to output  */



 flag {circumflex over ( )}= flags[j];
/* XOR-in the new flag value */







  }


 }








 hop(0);
/* Flush output buffer  */







}









FIG. 1 shows one embodiment of a flowchart of the process of creating dual linearizations for each image. At step 101, a histogram H(Ij) is created of image Ij. One implementation of the histogram is by using color information. Other feature information like shape and size can also be implemented, and can be in the form of histogram or other feature representation.
Essentially, any image content that can be represented as a vector (a point in space) may be used. Points “close” to one another are similar in that particular kind of content. Once the set of images is converted into a set of points, the rest of the process is the same: traverse the “space” along the two curves, and output the data structure.
At step 102, the histogram of k attributes is mapped as a point p(j) in k-dimensional space, where k is any positive integer. At step 103, the position of p(j) is computed in a Hilbert traversal of the k-space. In one embodiment, the position of p(j) is computed along an interval [0,1] in a Hilbert traversal of the k-space. At step 104, the position of p(j) is computed along interval [0,1] in a Peano traversal of the k-space. At step 105, p(j)'s four nearest neighbors based on the values obtained from the Hilbert and Peano curve traversals are located. Their links are updated according to the positions of the images.
In one embodiment, a linked data structure is used to store each image as it is linearized. For example, a new image is placed into a first linked structure by its linearization based on the Hilbert curve traversal. The new image is linked to its nearest neighbor, i.e., the images closest to the new image in linearization. The new image is similarly placed into a second linked structure by its linearization based on the Peano curve traversal.
Once all the images are added to the data structure, it is possible to follow a first set of links to traverse the linearization based on the Hilbert traversal. It is also possible to follow a second set of links to traverse the linearization based on the Peano traversal. In another embodiment, linearizations based on other methods can be employed. Additionally more than two linearizations may be used.
One advantage of using the described data structure is that it does not need to be completely recalculated each time an image is added or deleted from the data structure. Instead, only a few links need to be modified for each addition or deletion of an image. A second advantage of the data structure is that it is data independent, so that two existing databases can be combined into one with little more than a simple list merge.
Clustering
Clustering is another method of providing a linearization to images. A cluster is a grouping of images using a tree-like structure. The nature of clusters is such that it is not unusual that like images are placed into the data structure at the same time. By performing a linearization based upon a smart traversal of the cluster, like images are more likely to be placed next to each other in the linearization.
One way of linearizing a cluster is to add an entire first group of images before traversing the cluster to add another group of images. For example, if a first group of images was all added to the cluster at the same time, these images should be linked together prior to moving on to a group of images that was added to the cluster at a different time than the first group.
In another embodiment, the attributes of the target image are compared with attributes of all groups of the cluster. The groups that are the closest in attributes are linearized first. Other groups are added based on their similarity in attributes. A centroid, or representative item having the average characteristics of an entire group, can be used to compare the target image with an entire group within the cluster.
Another way to perform a linearization is to traverse the cluster in an orderly fashion. For example, a first linearization can traverse the cluster taking all right branches prior to taking any left branches. A second linearization can traverse the cluster taking all left branches. Alternatively, a first linearization can alterate between right and left branches, and a second linearization can alternate in the opposite manner.
In another embodiment, multiple linearizations can be created by first performing a first traversal of a cluster by traversing a first random traversal of the cluster linearizing each of the images of a section before moving on to a different section. A second traversal of the entire cluster is then performed by a different random traversal of the cluster.
Display of Multi-Linearizations
FIG. 4 shows an example of a browser interface based on a dual linearization. In this implementation, a total of 53 images are displayed at a time. The current image of focus, or the target image, is centered in the display. The images immediately to the left and right are the nearest neighbors 42a in one linearization. The images immediately to the top and bottom are the nearest neighbors 42b in a second linearization.
The nearest neighbors are the images closest to the target image in linearization. In the case of the space-filling curves, the nearest neighbors correspond to the points that are closest in position in a linearization to the point corresponding to the target image. In one embodiment, the first linearization may be based on a Hilbert curve, and the second linearization may be based on a Peano curve. In a second embodiment, the first linearization is based on a first traversal of a cluster data structure, and the second lienarization is based on a second traversal of the cluster data structure.
In one embodiment, the rules of displaying nearest neighbors of an image are applied recursively to all other images. Additionally, in one embodiment, images further away from the center, or target image, are displayed with a smaller size to denote greater dissimilarity from those images closer to the center. In FIG. 3, the next nearest neighbors 43 to the target image are smaller than the nearest neighbors 42a and 42b. The next-next nearest neighbors 44 of the target image are even smaller then its next nearest neighbors 43.
FIG. 5 shows an example of a browser interface based on a triple linearization. In one embodiment, the images above and below an image are the nearest neighbors based on a first linearization. Two other images may correspond to nearest neighbors based on a second linearization, and the remaining two images may correspond to nearest neighbors based on a third linearization. In other embodiments, more linearizations may be added, and a mixture of linearizations based on different methods is possible.
FIG. 6 shows an example of a system implemented on an SQL server that communicates with a middleware server to construct and serve web pages to an end user. The SQL server maintains the data structure of the multiple linearizations. The web server allows easy selection of the target image from any of the displayed images. When one of the images is selected, it becomes the target image, and the nearest neighbor and next-nearest neighbor images are updated correspondingly.
Thus, a system and method for using multiple linearizations for image browsing is disclosed. The specific arrangements and methods described herein are merely illustrative of the principles of this invention. Numerous modifications in form and detail may be made without departing from the scope of the described invention. Although this invention has been shown in relation to a particular embodiment, it should not be considered so limited. Rather, the described invention is limited only by the scope of the appended claims.
The method steps of FIG. 1 may be performed by a computer processor executing instructions organized into a program module or a custom designed state machine. Storage devices suitable for tangibly embodying computer program instructions include all forms of non-volatile memory including, but not limited to: semiconductor memory devices such as EPROM, EEPROM, and flash devices; magnetic disks (fixed, floppy, and removable); other magnetic media such as tape; and optical media such as CD-ROM disks. Further, the methods described herein may be embodied in a hardware device such as a printed circuit board comprising discrete logic, integrated circuits, or specially designed application specific integrated circuits (ASIC).

## Claims

1. A data structure stored on a computer-readable storage device comprising a first order of a plurality of images based on a first traversal of a multi-dimensional image attribute space containing points corresponding to the plurality of images with a first space-filling curve and a second order of the plurality of images based on a second traversal of the multi-dimensional image attribute space containing the points corresponding to the plurality of images with a second space-filling curve, wherein a partition boundary of the first space-filling curve occurs at a different location than a partition boundary of the second space-filling curve.

2. The data structure of claim 1, wherein the first space-filling curve comprises a curve that is selected from the group consisting of a Hilbert curve, a Peano curve, a Sierpinski curve, a Lebesgue curve, and a Schoenberg curve.

3. The data structure of claim 2, wherein the first space-filling curve is different than the second space-filling curve.

4. The data structure of claim 1, wherein the first space-filling curve comprises a Hilbert curve and the second space-filling curve comprises a Peano curve.

5. The data structure of claim 1, wherein the data structure includes links for an image to its nearest neighbors that are closest to the image in the first traversal.

6. The data structure of claim 1, wherein the data structure comprises a relative distance between a first point corresponding to a first image and a second point corresponding to a second image in the multi-dimensional image attribute space.

7. A data structure stored on a computer-readable storage device comprising a first order of a plurality of images based on a first traversal of a cluster corresponding to the plurality of images and a second order of the plurality of images based on a second traversal of the cluster corresponding to the plurality of images, wherein the first traversal and the second traversal are based on taking different branches.

8. The data structure of claim 7, wherein the data structure includes links for an image to its nearest neighbors that are closest to the image in the first order.

9. A data structure stored on a computer-readable storage device comprising a first linearization of a plurality of images including a first order for the plurality of images and a second linearization of the plurality of images including a second order for the plurality of images, wherein the first linearization is based on a first space-filling curve and the second linearization is based on a second space-filling curve that has a partition boundary at a different location than that of the first space-filling curve.

10. The data structure of claim 9, wherein the first linearization is based on a traversal of a multi-dimensional image attribute space with a curve having a property that points that are close to each other on the curve are also close to each other in the multi-dimensional image attribute space.

11. The data structure of claim 9, wherein the first linearization of the plurality of images is based on a space-filling curve.

12. The data structure of claim 9, wherein the first linearization of the plurality of images is based on a space-filling curve that is selected from the group consisting of a Hilbert curve, a Peano curve, a Sierpinski curve, a Lebesgue curve, and a Schoenberg curve.

13. The data structure of claim 9, wherein the first linearization is based on a first space-filling curve and the second linearization is based on a second space-filling curve that is different than the first space-filling curve.

14. The data structure of claim 9, wherein the first linearization is based on a Hilbert curve and the second linearization is based on a Peano curve.

15. The data structure of claim 9, wherein the data structure includes links for an image to its nearest neighbors that are closest to the image in the first linearization.

16. The data structure of claim 1, implemented in a computer comprising a flash memory.

17. The data structure of claim 7, implemented in a computer comprising a flash memory.

18. The data structure of claims 9, implemented in a computer comprising a flash memory.