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/*
emd.c
Last update: 3/14/98
An implementation of the Earth Movers Distance.
Based of the solution for the Transportation problem as described in
"Introduction to Mathematical Programming" by F. S. Hillier and
G. J. Lieberman, McGraw-Hill, 1990.
Copyright (C) 1998 Yossi Rubner
Computer Science Department, Stanford University
E-Mail: rubner@cs.stanford.edu URL: http://vision.stanford.edu/~rubner
*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include "emd.h"
#define DEBUG_LEVEL 0
/*
DEBUG_LEVEL:
0 = NO MESSAGES
1 = PRINT THE NUMBER OF ITERATIONS AND THE FINAL RESULT
2 = PRINT THE RESULT AFTER EVERY ITERATION
3 = PRINT ALSO THE FLOW AFTER EVERY ITERATION
4 = PRINT A LOT OF INFORMATION (PROBABLY USEFUL ONLY FOR THE AUTHOR)
*/
#define MAX_SIG_SIZE1 (MAX_SIG_SIZE+1) /* FOR THE POSIBLE DUMMY FEATURE */
/* NEW TYPES DEFINITION */
/* node1_t IS USED FOR SINGLE-LINKED LISTS */
typedef struct node1_t {
int i;
double val;
struct node1_t *Next;
} node1_t;
/* node1_t IS USED FOR DOUBLE-LINKED LISTS */
typedef struct node2_t {
int i, j;
double val;
struct node2_t *NextC; /* NEXT COLUMN */
struct node2_t *NextR; /* NEXT ROW */
} node2_t;
/* GLOBAL VARIABLE DECLARATION */
static int _n1, _n2; /* SIGNATURES SIZES */
static float _C[MAX_SIG_SIZE1][MAX_SIG_SIZE1];/* THE COST MATRIX */
static node2_t _X[MAX_SIG_SIZE1*2]; /* THE BASIC VARIABLES VECTOR */
/* VARIABLES TO HANDLE _X EFFICIENTLY */
static node2_t *_EndX, *_EnterX;
static char _IsX[MAX_SIG_SIZE1][MAX_SIG_SIZE1];
static node2_t *_RowsX[MAX_SIG_SIZE1], *_ColsX[MAX_SIG_SIZE1];
static double _maxW;
static float _maxC;
/* DECLARATION OF FUNCTIONS */
static float init(signature_t *Signature1, signature_t *Signature2,
float (*Dist)(feature_t *, feature_t *));
static void findBasicVariables(node1_t *U, node1_t *V);
static int isOptimal(node1_t *U, node1_t *V);
static int findLoop(node2_t **Loop);
static void newSol();
static void russel(double *S, double *D);
static void addBasicVariable(int minI, int minJ, double *S, double *D,
node1_t *PrevUMinI, node1_t *PrevVMinJ,
node1_t *UHead);
#if DEBUG_LEVEL > 0
static void printSolution();
#endif
/******************************************************************************
float emd(signature_t *Signature1, signature_t *Signature2,
float (*Dist)(feature_t *, feature_t *), flow_t *Flow, int *FlowSize)
where
Signature1, Signature2 Pointers to signatures that their distance we want
to compute.
Dist Pointer to the ground distance. i.e. the function that computes
the distance between two features.
Flow (Optional) Pointer to a vector of flow_t (defined in emd.h)
where the resulting flow will be stored. Flow must have n1+n2-1
elements, where n1 and n2 are the sizes of the two signatures
respectively.
If NULL, the flow is not returned.
FlowSize (Optional) Pointer to an integer where the number of elements in
Flow will be stored
******************************************************************************/
float emd(signature_t *Signature1, signature_t *Signature2,
float (*Dist)(feature_t *, feature_t *),
flow_t *Flow, int *FlowSize)
{
int itr;
double totalCost;
float w;
node2_t *XP;
flow_t *FlowP;
node1_t U[MAX_SIG_SIZE1], V[MAX_SIG_SIZE1];
w = init(Signature1, Signature2, Dist);
#if DEBUG_LEVEL > 1
printf("/nINITIAL SOLUTION:/n");
printSolution();
#endif
if (_n1 > 1 && _n2 > 1) /* IF _n1 = 1 OR _n2 = 1 THEN WE ARE DONE */
{
for (itr = 1; itr < MAX_ITERATIONS; itr++)
{
/* FIND BASIC VARIABLES */
findBasicVariables(U, V);
/* CHECK FOR OPTIMALITY */
if (isOptimal(U, V))
break;
/* IMPROVE SOLUTION */
newSol();
#if DEBUG_LEVEL > 1
printf("/nITERATION # %d /n", itr);
printSolution();
#endif
}
if (itr == MAX_ITERATIONS)
fprintf(stderr, "emd: Maximum number of iterations has been reached (%d)/n",
MAX_ITERATIONS);
}
/* COMPUTE THE TOTAL FLOW */
totalCost = 0;
if (Flow != NULL)
FlowP = Flow;
for(XP=_X; XP < _EndX; XP++)
{
if (XP == _EnterX) /* _EnterX IS THE EMPTY SLOT */
continue;
if (XP->i == Signature1->n || XP->j == Signature2->n) /* DUMMY FEATURE */
continue;
if (XP->val == 0) /* ZERO FLOW */
continue;
totalCost += (double)XP->val * _C[XP->i][XP->j];
if (Flow != NULL)
{
FlowP->from = XP->i;
FlowP->to = XP->j;
FlowP->amount = XP->val;
FlowP++;
}
}
if (Flow != NULL)
*FlowSize = FlowP-Flow;
#if DEBUG_LEVEL > 0
printf("/n*** OPTIMAL SOLUTION (%d ITERATIONS): %f ***/n", itr, totalCost);
#endif
/* RETURN THE NORMALIZED COST == EMD */
return (float)(totalCost / w);
}
/**********************
init
**********************/
static float init(signature_t *Signature1, signature_t *Signature2,
float (*Dist)(feature_t *, feature_t *))
{
int i, j;
double sSum, dSum, diff;
feature_t *P1, *P2;
double S[MAX_SIG_SIZE1], D[MAX_SIG_SIZE1];
_n1 = Signature1->n;
_n2 = Signature2->n;
if (_n1 > MAX_SIG_SIZE || _n2 > MAX_SIG_SIZE)
{
fprintf(stderr, "emd: Signature size is limited to %d/n", MAX_SIG_SIZE);
exit(1);
}
/* COMPUTE THE DISTANCE MATRIX */
_maxC = 0;
for(i=0, P1=Signature1->Features; i < _n1; i++, P1++)
for(j=0, P2=Signature2->Features; j < _n2; j++, P2++)
{
_C[i][j] = Dist(P1, P2);
if (_C[i][j] > _maxC)
_maxC = _C[i][j];
}
/* SUM UP THE SUPPLY AND DEMAND */
sSum = 0.0;
for(i=0; i < _n1; i++)
{
S[i] = Signature1->Weights[i];
sSum += Signature1->Weights[i];
_RowsX[i] = NULL;
}
dSum = 0.0;
for(j=0; j < _n2; j++)
{
D[j] = Signature2->Weights[j];
dSum += Signature2->Weights[j];
_ColsX[j] = NULL;
}
/* IF SUPPLY DIFFERENT THAN THE DEMAND, ADD A ZERO-COST DUMMY CLUSTER */
diff = sSum - dSum;
if (fabs(diff) >= EPSILON * sSum)
{
if (diff < 0.0)
{
for (j=0; j < _n2; j++)
_C[_n1][j] = 0;
S[_n1] = -diff;
_RowsX[_n1] = NULL;
_n1++;
}
else
{
for (i=0; i < _n1; i++)
_C[i][_n2] = 0;
D[_n2] = diff;
_ColsX[_n2] = NULL;
_n2++;
}
}
/* INITIALIZE THE BASIC VARIABLE STRUCTURES */
for (i=0; i < _n1; i++)
for (j=0; j < _n2; j++)
_IsX[i][j] = 0;
_EndX = _X;
_maxW = sSum > dSum ? sSum : dSum;
/* FIND INITIAL SOLUTION */
russel(S, D);
_EnterX = _EndX++; /* AN EMPTY SLOT (ONLY _n1+_n2-1 BASIC VARIABLES) */
return sSum > dSum ? dSum : sSum;
}
/**********************
findBasicVariables
**********************/
static void findBasicVariables(node1_t *U, node1_t *V)
{
int i, j, found;
int UfoundNum, VfoundNum;
node1_t u0Head, u1Head, *CurU, *PrevU;
node1_t v0Head, v1Head, *CurV, *PrevV;
/* INITIALIZE THE ROWS LIST (U) AND THE COLUMNS LIST (V) */
u0Head.Next = CurU = U;
for (i=0; i < _n1; i++)
{
CurU->i = i;
CurU->Next = CurU+1;
CurU++;
}
(--CurU)->Next = NULL;
u1Head.Next = NULL;
CurV = V+1;
v0Head.Next = _n2 > 1 ? V+1 : NULL;
for (j=1; j < _n2; j++)
{
CurV->i = j;
CurV->Next = CurV+1;
CurV++;
}
(--CurV)->Next = NULL;
v1Head.Next = NULL;
/* THERE ARE _n1+_n2 VARIABLES BUT ONLY _n1+_n2-1 INDEPENDENT EQUATIONS,
SO SET V[0]=0 */
V[0].i = 0;
V[0].val = 0;
v1Head.Next = V;
v1Head.Next->Next = NULL;
/* LOOP UNTIL ALL VARIABLES ARE FOUND */
UfoundNum=VfoundNum=0;
while (UfoundNum < _n1 || VfoundNum < _n2)
{
#if DEBUG_LEVEL > 3
printf("UfoundNum=%d/%d,VfoundNum=%d/%d/n",UfoundNum,_n1,VfoundNum,_n2);
printf("U0=");
for(CurU = u0Head.Next; CurU != NULL; CurU = CurU->Next)
printf("[%d]",CurU-U);
printf("/n");
printf("U1=");
for(CurU = u1Head.Next; CurU != NULL; CurU = CurU->Next)
printf("[%d]",CurU-U);
printf("/n");
printf("V0=");
for(CurV = v0Head.Next; CurV != NULL; CurV = CurV->Next)
printf("[%d]",CurV-V);
printf("/n");
printf("V1=");
for(CurV = v1Head.Next; CurV != NULL; CurV = CurV->Next)
printf("[%d]",CurV-V);
printf("/n/n");
#endif
found = 0;
if (VfoundNum < _n2)
{
/* LOOP OVER ALL MARKED COLUMNS */
PrevV = &v1Head;
for (CurV=v1Head.Next; CurV != NULL; CurV=CurV->Next)
{
j = CurV->i;
/* FIND THE VARIABLES IN COLUMN j */
PrevU = &u0Head;
for (CurU=u0Head.Next; CurU != NULL; CurU=CurU->Next)
{
i = CurU->i;
if (_IsX[i][j])
{
/* COMPUTE U[i] */
CurU->val = _C[i][j] - CurV->val;
/* ...AND ADD IT TO THE MARKED LIST */
PrevU->Next = CurU->Next;
CurU->Next = u1Head.Next != NULL ? u1Head.Next : NULL;
u1Head.Next = CurU;
CurU = PrevU;
}
else
PrevU = CurU;
}
PrevV->Next = CurV->Next;
VfoundNum++;
found = 1;
}
}
if (UfoundNum < _n1)
{
/* LOOP OVER ALL MARKED ROWS */
PrevU = &u1Head;
for (CurU=u1Head.Next; CurU != NULL; CurU=CurU->Next)
{
i = CurU->i;
/* FIND THE VARIABLES IN ROWS i */
PrevV = &v0Head;
for (CurV=v0Head.Next; CurV != NULL; CurV=CurV->Next)
{
j = CurV->i;
if (_IsX[i][j])
{
/* COMPUTE V[j] */
CurV->val = _C[i][j] - CurU->val;
/* ...AND ADD IT TO THE MARKED LIST */
PrevV->Next = CurV->Next;
CurV->Next = v1Head.Next != NULL ? v1Head.Next: NULL;
v1Head.Next = CurV;
CurV = PrevV;
}
else
PrevV = CurV;
}
PrevU->Next = CurU->Next;
UfoundNum++;
found = 1;
}
}
if (! found)
{
fprintf(stderr, "emd: Unexpected error in findBasicVariables!/n");
fprintf(stderr, "This typically happens when the EPSILON defined in/n");
fprintf(stderr, "emd.h is not right for the scale of the problem./n");
exit(1);
}
}
}
/**********************
isOptimal
**********************/
static int isOptimal(node1_t *U, node1_t *V)
{
double delta, deltaMin;
int i, j, minI, minJ;
/* FIND THE MINIMAL Cij-Ui-Vj OVER ALL i,j */
deltaMin = INFINITY;
for(i=0; i < _n1; i++)
for(j=0; j < _n2; j++)
if (! _IsX[i][j])
{
delta = _C[i][j] - U[i].val - V[j].val;
if (deltaMin > delta)
{
deltaMin = delta;
minI = i;
minJ = j;
}
}
#if DEBUG_LEVEL > 3
printf("deltaMin=%f/n", deltaMin);
#endif
if (deltaMin == INFINITY)
{
fprintf(stderr, "emd: Unexpected error in isOptimal./n");
exit(0);
}
_EnterX->i = minI;
_EnterX->j = minJ;
/* IF NO NEGATIVE deltaMin, WE FOUND THE OPTIMAL SOLUTION */
return deltaMin >= -EPSILON * _maxC;
/*
return deltaMin >= -EPSILON;
*/
}
/**********************
newSol
**********************/
static void newSol()
{
int i, j, k;
double xMin;
int steps;
node2_t *Loop[2*MAX_SIG_SIZE1], *CurX, *LeaveX;
#if DEBUG_LEVEL > 3
printf("EnterX = (%d,%d)/n", _EnterX->i, _EnterX->j);
#endif
/* ENTER THE NEW BASIC VARIABLE */
i = _EnterX->i;
j = _EnterX->j;
_IsX[i][j] = 1;
_EnterX->NextC = _RowsX[i];
_EnterX->NextR = _ColsX[j];
_EnterX->val = 0;
_RowsX[i] = _EnterX;
_ColsX[j] = _EnterX;
/* FIND A CHAIN REACTION */
steps = findLoop(Loop);
/* FIND THE LARGEST VALUE IN THE LOOP */
xMin = INFINITY;
for (k=1; k < steps; k+=2)
{
if (Loop[k]->val < xMin)
{
LeaveX = Loop[k];
xMin = Loop[k]->val;
}
}
/* UPDATE THE LOOP */
for (k=0; k < steps; k+=2)
{
Loop[k]->val += xMin;
Loop[k+1]->val -= xMin;
}
#if DEBUG_LEVEL > 3
printf("LeaveX = (%d,%d)/n", LeaveX->i, LeaveX->j);
#endif
/* REMOVE THE LEAVING BASIC VARIABLE */
i = LeaveX->i;
j = LeaveX->j;
_IsX[i][j] = 0;
if (_RowsX[i] == LeaveX)
_RowsX[i] = LeaveX->NextC;
else
for (CurX=_RowsX[i]; CurX != NULL; CurX = CurX->NextC)
if (CurX->NextC == LeaveX)
{
CurX->NextC = CurX->NextC->NextC;
break;
}
if (_ColsX[j] == LeaveX)
_ColsX[j] = LeaveX->NextR;
else
for (CurX=_ColsX[j]; CurX != NULL; CurX = CurX->NextR)
if (CurX->NextR == LeaveX)
{
CurX->NextR = CurX->NextR->NextR;
break;
}
/* SET _EnterX TO BE THE NEW EMPTY SLOT */
_EnterX = LeaveX;
}
/**********************
findLoop
**********************/
static int findLoop(node2_t **Loop)
{
int i, steps;
node2_t **CurX, *NewX;
char IsUsed[2*MAX_SIG_SIZE1];
for (i=0; i < _n1+_n2; i++)
IsUsed[i] = 0;
CurX = Loop;
NewX = *CurX = _EnterX;
IsUsed[_EnterX-_X] = 1;
steps = 1;
do
{
if (steps%2 == 1)
{
/* FIND AN UNUSED X IN THE ROW */
NewX = _RowsX[NewX->i];
while (NewX != NULL && IsUsed[NewX-_X])
NewX = NewX->NextC;
}
else
{
/* FIND AN UNUSED X IN THE COLUMN, OR THE ENTERING X */
NewX = _ColsX[NewX->j];
while (NewX != NULL && IsUsed[NewX-_X] && NewX != _EnterX)
NewX = NewX->NextR;
if (NewX == _EnterX)
break;
}
if (NewX != NULL) /* FOUND THE NEXT X */
{
/* ADD X TO THE LOOP */
*++CurX = NewX;
IsUsed[NewX-_X] = 1;
steps++;
#if DEBUG_LEVEL > 3
printf("steps=%d, NewX=(%d,%d)/n", steps, NewX->i, NewX->j);
#endif
}
else /* DIDN'T FIND THE NEXT X */
{
/* BACKTRACK */
do
{
NewX = *CurX;
do
{
if (steps%2 == 1)
NewX = NewX->NextR;
else
NewX = NewX->NextC;
} while (NewX != NULL && IsUsed[NewX-_X]);
if (NewX == NULL)
{
IsUsed[*CurX-_X] = 0;
CurX--;
steps--;
}
} while (NewX == NULL && CurX >= Loop);
#if DEBUG_LEVEL > 3
printf("BACKTRACKING TO: steps=%d, NewX=(%d,%d)/n",
steps, NewX->i, NewX->j);
#endif
IsUsed[*CurX-_X] = 0;
*CurX = NewX;
IsUsed[NewX-_X] = 1;
}
} while(CurX >= Loop);
if (CurX == Loop)
{
fprintf(stderr, "emd: Unexpected error in findLoop!/n");
exit(1);
}
#if DEBUG_LEVEL > 3
printf("FOUND LOOP:/n");
for (i=0; i < steps; i++)
printf("%d: (%d,%d)/n", i, Loop[i]->i, Loop[i]->j);
#endif
return steps;
}
/**********************
russel
**********************/
static void russel(double *S, double *D)
{
int i, j, found, minI, minJ;
double deltaMin, oldVal, diff;
double Delta[MAX_SIG_SIZE1][MAX_SIG_SIZE1];
node1_t Ur[MAX_SIG_SIZE1], Vr[MAX_SIG_SIZE1];
node1_t uHead, *CurU, *PrevU;
node1_t vHead, *CurV, *PrevV;
node1_t *PrevUMinI, *PrevVMinJ, *Remember;
/* INITIALIZE THE ROWS LIST (Ur), AND THE COLUMNS LIST (Vr) */
uHead.Next = CurU = Ur;
for (i=0; i < _n1; i++)
{
CurU->i = i;
CurU->val = -INFINITY;
CurU->Next = CurU+1;
CurU++;
}
(--CurU)->Next = NULL;
vHead.Next = CurV = Vr;
for (j=0; j < _n2; j++)
{
CurV->i = j;
CurV->val = -INFINITY;
CurV->Next = CurV+1;
CurV++;
}
(--CurV)->Next = NULL;
/* FIND THE MAXIMUM ROW AND COLUMN VALUES (Ur[i] AND Vr[j]) */
for(i=0; i < _n1 ; i++)
for(j=0; j < _n2 ; j++)
{
float v;
v = _C[i][j];
if (Ur[i].val <= v)
Ur[i].val = v;
if (Vr[j].val <= v)
Vr[j].val = v;
}
/* COMPUTE THE Delta MATRIX */
for(i=0; i < _n1 ; i++)
for(j=0; j < _n2 ; j++)
Delta[i][j] = _C[i][j] - Ur[i].val - Vr[j].val;
/* FIND THE BASIC VARIABLES */
do
{
#if DEBUG_LEVEL > 3
printf("Ur=");
for(CurU = uHead.Next; CurU != NULL; CurU = CurU->Next)
printf("[%d]",CurU-Ur);
printf("/n");
printf("Vr=");
for(CurV = vHead.Next; CurV != NULL; CurV = CurV->Next)
printf("[%d]",CurV-Vr);
printf("/n");
printf("/n/n");
#endif
/* FIND THE SMALLEST Delta[i][j] */
found = 0;
deltaMin = INFINITY;
PrevU = &uHead;
for (CurU=uHead.Next; CurU != NULL; CurU=CurU->Next)
{
int i;
i = CurU->i;
PrevV = &vHead;
for (CurV=vHead.Next; CurV != NULL; CurV=CurV->Next)
{
int j;
j = CurV->i;
if (deltaMin > Delta[i][j])
{
deltaMin = Delta[i][j];
minI = i;
minJ = j;
PrevUMinI = PrevU;
PrevVMinJ = PrevV;
found = 1;
}
PrevV = CurV;
}
PrevU = CurU;
}
if (! found)
break;
/* ADD X[minI][minJ] TO THE BASIS, AND ADJUST SUPPLIES AND COST */
Remember = PrevUMinI->Next;
addBasicVariable(minI, minJ, S, D, PrevUMinI, PrevVMinJ, &uHead);
/* UPDATE THE NECESSARY Delta[][] */
if (Remember == PrevUMinI->Next) /* LINE minI WAS DELETED */
{
for (CurV=vHead.Next; CurV != NULL; CurV=CurV->Next)
{
int j;
j = CurV->i;
if (CurV->val == _C[minI][j]) /* COLUMN j NEEDS UPDATING */
{
/* FIND THE NEW MAXIMUM VALUE IN THE COLUMN */
oldVal = CurV->val;
CurV->val = -INFINITY;
for (CurU=uHead.Next; CurU != NULL; CurU=CurU->Next)
{
int i;
i = CurU->i;
if (CurV->val <= _C[i][j])
CurV->val = _C[i][j];
}
/* IF NEEDED, ADJUST THE RELEVANT Delta[*][j] */
diff = oldVal - CurV->val;
if (fabs(diff) < EPSILON * _maxC)
for (CurU=uHead.Next; CurU != NULL; CurU=CurU->Next)
Delta[CurU->i][j] += diff;
}
}
}
else /* COLUMN minJ WAS DELETED */
{
for (CurU=uHead.Next; CurU != NULL; CurU=CurU->Next)
{
int i;
i = CurU->i;
if (CurU->val == _C[i][minJ]) /* ROW i NEEDS UPDATING */
{
/* FIND THE NEW MAXIMUM VALUE IN THE ROW */
oldVal = CurU->val;
CurU->val = -INFINITY;
for (CurV=vHead.Next; CurV != NULL; CurV=CurV->Next)
{
int j;
j = CurV->i;
if(CurU->val <= _C[i][j])
CurU->val = _C[i][j];
}
/* If NEEDED, ADJUST THE RELEVANT Delta[i][*] */
diff = oldVal - CurU->val;
if (fabs(diff) < EPSILON * _maxC)
for (CurV=vHead.Next; CurV != NULL; CurV=CurV->Next)
Delta[i][CurV->i] += diff;
}
}
}
} while (uHead.Next != NULL || vHead.Next != NULL);
}
/**********************
addBasicVariable
**********************/
static void addBasicVariable(int minI, int minJ, double *S, double *D,
node1_t *PrevUMinI, node1_t *PrevVMinJ,
node1_t *UHead)
{
double T;
if (fabs(S[minI]-D[minJ]) <= EPSILON * _maxW) /* DEGENERATE CASE */
{
T = S[minI];
S[minI] = 0;
D[minJ] -= T;
}
else if (S[minI] < D[minJ]) /* SUPPLY EXHAUSTED */
{
T = S[minI];
S[minI] = 0;
D[minJ] -= T;
}
else /* DEMAND EXHAUSTED */
{
T = D[minJ];
D[minJ] = 0;
S[minI] -= T;
}
/* X(minI,minJ) IS A BASIC VARIABLE */
_IsX[minI][minJ] = 1;
_EndX->val = T;
_EndX->i = minI;
_EndX->j = minJ;
_EndX->NextC = _RowsX[minI];
_EndX->NextR = _ColsX[minJ];
_RowsX[minI] = _EndX;
_ColsX[minJ] = _EndX;
_EndX++;
/* DELETE SUPPLY ROW ONLY IF THE EMPTY, AND IF NOT LAST ROW */
if (S[minI] == 0 && UHead->Next->Next != NULL)
PrevUMinI->Next = PrevUMinI->Next->Next; /* REMOVE ROW FROM LIST */
else
PrevVMinJ->Next = PrevVMinJ->Next->Next; /* REMOVE COLUMN FROM LIST */
}
/**********************
printSolution
**********************/
static void printSolution()
{
node2_t *P;
double totalCost;
totalCost = 0;
#if DEBUG_LEVEL > 2
printf("SIG1/tSIG2/tFLOW/tCOST/n");
#endif
for(P=_X; P < _EndX; P++)
if (P != _EnterX && _IsX[P->i][P->j])
{
#if DEBUG_LEVEL > 2
printf("%d/t%d/t%f/t%f/n", P->i, P->j, P->val, _C[P->i][P->j]);
#endif
totalCost += (double)P->val * _C[P->i][P->j];
}
printf("COST = %f/n", totalCost);
}
#ifndef _EMD_H
#define _EMD_H
/*
emd.h
Last update: 3/24/98
An implementation of the Earth Movers Distance.
Based of the solution for the Transportation problem as described in
"Introduction to Mathematical Programming" by F. S. Hillier and
G. J. Lieberman, McGraw-Hill, 1990.
Copyright (C) 1998 Yossi Rubner
Computer Science Department, Stanford University
E-Mail: rubner@cs.stanford.edu URL: http://vision.stanford.edu/~rubner
*/
/* DEFINITIONS */
#define MAX_SIG_SIZE 100
#define MAX_ITERATIONS 500
#define INFINITY 1e20
#define EPSILON 1e-6
/*****************************************************************************/
/* feature_t SHOULD BE MODIFIED BY THE USER TO REFLECT THE FEATURE TYPE */
typedef int feature_t;
/*****************************************************************************/
typedef struct
{
int n; /* Number of features in the signature */
feature_t *Features; /* Pointer to the features vector */
float *Weights; /* Pointer to the weights of the features */
} signature_t;
typedef struct
{
int from; /* Feature number in signature 1 */
int to; /* Feature number in signature 2 */
float amount; /* Amount of flow from "from" to "to" */
} flow_t;
float emd(signature_t *Signature1, signature_t *Signature2,
float (*func)(feature_t *, feature_t *),
flow_t *Flow, int *FlowSize);
#endif
Code for the Earth Movers Distance (EMD) | ||||||||
|
| ||||||||
| Introduction | ||||||||
| This is an implementation of the Earth Movers Distance, as described in [1]. The EMD computes the distance between two distributions, which are represented by signatures. The signatures are sets of weighted features that capture the distributions. The features can be of any type and in any number of dimensions, and are defined by the user. The EMD is defined as the minimum amount of work needed to change one signature into the other. The notion of "work" is based on the user-defined ground distance which is the distance between two features. The size of the two signatures can be different. Also, the sum of weights of one signature can be different than the sum of weights of the other (partial match). Because of this, the EMD is normalized by the smaller sum. The code is implemented in C, and is based on the solution for the Transportation problem as described in [2] Please let me know of any bugs you find, or any questions, comments, suggestions, and criticisms you have. If you find this code useful for your work, I would like very much to hear from you. Once you do, I'll inform you of any improvements, etc. Also, an acknowledgment in any publication describing work that uses this code would be greatly appreciated.
| ||||||||
| Usage | ||||||||
To use the code, perform the following steps:
The signature data type typedef struct
{
int n; /* Number of features in the distribution */
feature_t *Features; /* Pointer to the features vector */
float *Weights; /* Pointer to the weights of the features */
} signature_t;
The feature data type In special cases, the user might want to change some of the values in the definitions in
#define MAX_SIG_SIZE 100
#define MAX_ITERATIONS 100
#define INFINITY 1e20
#define EPSILON 1e-6
| ||||||||
| Examples | ||||||||
| ||||||||
| Log of changes | ||||||||
| ||||||||
| References | ||||||||
| [1] Y. Rubner, C. Tomasi, and L. J. Guibas. A Metric for Distributions with Applications to Image Databases. Proceedings of the 1998 IEEE International Conference on Computer Vision, Bombay, India, January 1998, pp. 59-66. [2] F. S. Hillier and G. J. Lieberman. Introduction to Mathematical Programming McGraw-Hill, 1990. |
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