Fastest way to find the sign of different square - performance

Question

Given an image I and two matrices m_1 ;m_2 (same size with I). The function f is defined as:
Because my goal design wants to get the sign of f . Hence, the function f can rewritten as following:
I think that second formula is faster than first formula because: It
can ignore the square term
It can compute the sign directly, instead of two steps in first equation: compute the f and check sign.
Do you agree with me? Do you have another faster formula for f
I =[16 23 11 42 10
11 21 22 24 30
16 22 154 155 156
25 28 145 151 156
11 38 147 144 153];
m1 =[0 0 0 0 0
0 0 22 11 0
0 23 34 56 0
0 56 0 0 0
0 11 0 0 0];
m2 =[0 0 0 0 0
0 0 12 11 0
0 22 111 156 0
0 32 0 0 0
0 12 0 0 0];
The ouput f is
f =[1 1 1 1 1
1 1 -1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1]
I implemented the first way, but I did not finish the second way by matlab. Could you check help me the second way and compare it
UPDATE: I would like to add code of chepyle and Divakar to make clearly question. Note that both of them give the same result as above f
function compare()
I =[16 23 11 42 10
11 21 22 24 30
16 22 154 155 156
25 28 145 151 156
11 38 147 144 153];
m1 =[0 0 0 0 0
0 0 22 11 0
0 23 34 56 0
0 56 0 0 0
0 11 0 0 0];
m2 =[0 0 0 0 0
0 0 12 11 0
0 22 111 156 0
0 32 0 0 0
0 12 0 0 0];
function f=first_way()
f=sign((I-m1).^2-(I-m2).^2);
f(f==0)=1;
end
function f= second_way()
f = double(abs(I-m1) >= abs(I-m2));
f(f==0) = -1;
end
function f= third_way()
v1=abs(I-m1);
v2=abs(I-m2);
f= int8(v1>v2) + -1*int8(v1<v2); % need to convert to int from logical
f(f==0) = 1;
end
disp(['First way : ' num2str(timeit(#first_way))])
disp(['Second way: ' num2str(timeit(#second_way))])
disp(['Third way : ' num2str(timeit(#third_way))])
end
First way : 1.2897e-05
Second way: 1.9381e-05
Third way : 2.0077e-05

Answer 1

This seems to be comparable and might be a wee bit faster at times than the original approach -
f = sign(abs(I-m1) - abs(I-m2)) + sign(abs(m1-m2)) + ...
sign(abs(2*I-m1-m2)) - 1 -sign(abs(2*I-m1-m2) + abs(m1-m2))
Benchmarking Code
%// Create random inputs
N = 5000;
I = randi(1000,N,N);
m1 = randi(1000,N,N);
m2 = randi(1000,N,N);
num_iter = 20; %// Number of iterations for all approaches
%// Warm up tic/toc.
for k = 1:100000
tic(); elapsed = toc();
end
disp('------------------------- With Original Approach')
tic
for iter = 1:num_iter
out1 = sign((I-m1).^2-(I-m2).^2);
out1(out1==0)=-1;
end
toc, clear out1
disp('------------------------- With Proposed Approach')
tic
for iter = 1:num_iter
out2 = sign(abs(I-m1) - abs(I-m2)) + sign(abs(m1-m2)) + ...
sign(abs(2*I-m1-m2)) - 1 -sign(abs(2*I-m1-m2) + abs(m1-m2));
end
toc
Results
------------------------- With Original Approach
Elapsed time is 1.751966 seconds.
------------------------- With Proposed Approach
Elapsed time is 1.681263 seconds.

Answer 2

There is a problem with the accuracy of second formula, but for the sake of comparison, here's how I would implement it in matlab, along with a third approach to avoid squaring and the sign() function, inline with your intent. Note that the matlab's matrix and sign functions are pretty well optimized, the second and third approaches are both slower.
function compare()
I =[16 23 11 42 10
11 21 22 24 30
16 22 154 155 156
25 28 145 151 156
11 38 147 144 153];
m1 =[0 0 0 0 0
0 0 22 11 0
0 23 34 56 0
0 56 0 0 0
0 11 0 0 0];
m2 =[0 0 0 0 0
0 0 12 11 0
0 22 111 156 0
0 32 0 0 0
0 12 0 0 0];
function f=first_way()
f=sign((I-m1).^2-(I-m2).^2);
end
function f= second_way()
v1=(I-m1);
v2=(I-m2);
f= int8(v1<=0 & v2>0) + -1* int8(v1>0 & v2<=0);
end
function f= third_way()
v1=abs(I-m1);
v2=abs(I-m2);
f= int8(v1>v2) + -1*int8(v1<v2); % need to convert to int from logical
end
disp(['First way : ' num2str(timeit(#first_way))])
disp(['Second way: ' num2str(timeit(#second_way))])
disp(['Third way : ' num2str(timeit(#third_way))])
end
The output:
First way : 9.4226e-06
Second way: 1.2247e-05
Third way : 1.1546e-05