How to get the original Image matrix from the Integral Image of a N*N matrix?

IF the original image matrix is represented by r=[ 1 2 3 4;5 6 7 8;9 10 11 12;13 14 15 16]. Then it's integral image using the inbuilt 'integralImage' function is given by

0 0 0 0 0

0 1 3 6 10

0 6 14 24 36

0 15 33 54 78

0 28 60 96 136

Excluding first row and first column we get

1 3 6 10

6 14 24 36

15 33 54 78

28 60 96 136

My requirement is to obtain the original image from the given integral image provided.

Answer 1

[Stephen Cobeldick]

A simple solution based on the explanation given on the integralImage help page:

>> II = [0,0,0,0,0;0,1,3,6,10;0,6,14,24,36;0,15,33,54,78;0,28,60,96,136]
II =
     0     0     0     0     0
     0     1     3     6    10
     0     6    14    24    36
     0    15    33    54    78
     0    28    60    96   136
>> II(2:end,2:end)-II(2:end,1:end-1)-II(1:end-1,2:end)+II(1:end-1,1:end-1)
ans =
    1    2    3    4
    5    6    7    8
    9   10   11   12
   13   14   15   16

[Karthik Nagaraj]

Thank you so much!

Answer 2

[Diarmaid Cualain]

To supplement stephens answer, you can also use the Matlab function "diff":

>>II = [0,0,0,0,0;0,1,3,6,10;0,6,14,24,36;0,15,33,54,78;0,28,60,96,136]
>>diff(diff(II,1,2),[])

ans =

     1     2     3     4
     5     6     7     8
     9    10    11    12
    13    14    15    16

[Karthik Nagaraj]

Thank You so much

[Diarmaid Cualain]

Glad it helped!