A Linear Approximate Algorithm for Earth Mover's Distance with Thresholded Ground Distance

Algorithm 3

EMD⁺ between two histograms.

     Input: 𝑃, 𝑄: histogram; 𝜇: threshold; 𝐵: qualified bins

       of original point.

     Output: Distance between 𝑃 and 𝑄

(1)      cost=0;

(2)      𝜔=0;

   //  Part 1, corresponding bin

(3)    foreach  𝑝∈𝑃  do

(4)       if  corresponding 𝑞 exists  then

(5)         mass←min(𝑤_𝑝,𝑤_𝑞); // 𝑤_𝑝: weight of bin 𝑝

(6)         𝜔←𝜔+mass;

(7)         𝑤_𝑝←𝑤_𝑝−mass;

(8)         𝑤_𝑞←𝑤_𝑞−mass;

(9)        if  𝑤_𝑝=0  then

(10)          remove 𝑝 from 𝑃;

(11)       if  𝑤_𝑞=0  then

(12)          remove 𝑞 from 𝑄;

   //  Part 2, cross bin

(13)  for  𝑖=1 to 𝐵.size  do

(14)   foreach  𝑝∈𝑃  do

(15)   if  𝑖th qualified bin 𝑞 exists  then

(16)       mass←min(𝑤_𝑝,𝑤_𝑞);

(17)       𝜔←𝜔+mass;

(18)       cost←cost+mass×distance(𝑝,𝑞);

(19)       𝑤_𝑝←𝑤_𝑝−mass, 𝑤_𝑞←𝑤_𝑞−mass;

(20)        if  𝑤_𝑝=0  then

(21)            remove 𝑝 from 𝑃;

(22)        if  𝑤_𝑞=0  then

(23)            remove 𝑞 from 𝑄;

   //  Part 3, compute the EMD⁺

(24) EMD⁺←cost+(max(𝑤(𝑃),𝑤(𝑄))−𝜔)×𝜇;