The document discusses computing the sum of pixel values in a region of an image using an integral image or summed area table. It shows how to construct the integral image by cumulatively summing the pixel values in rows from left to right and top to bottom. This allows the sum over any rectangular region to be computed with just four references to the integral image, rather than having to sum the pixels individually. Computing the integral image upfront makes summing regions efficient when needed to be computed repeatedly for different areas.

1. Integral Image
Randy Cahya Wihandika
Universitas Brawijaya

2. Summed Area Table (Integral Image)
✘ If we need to compute the sum of a region in an
image repeatedly, it will be more efficient if we
compute the integral image

3. Summed Area Table (Integral Image)
✘ Consider this image4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6

4. Summed Area Table (Integral Image)
✘ Now if want to
compute the sum of the
highlighted area, we
normally do this:
2+6+8+9+7+4+4+6+8+6+
4+5+0+5+7+5+4+6+7+9+
6+4+3+2 = 127
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6

5. Summed Area Table (Integral Image)
✘ If we need to
compute another area,
we do the calculation all
over again:
4+6+8+6+4+5+0+5+7+5+
4+6+7+9+6+4+3+2+6+8+
5+6+8+9 = 133
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6

6. Summed Area Table (Integral Image)
✘ Now let’s compute the integral image

7. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4

8. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7

9. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14

10. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22

11. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22 25 27 32 39 48 49

12. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22 25 27 32 39 48 49
7

13. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22 25 27 32 39 48 49
7
12

14. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22 25 27 32 39 48 49
7
12
17

15. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22 25 27 32 39 48 49
7
12
17
17
18
23
30

16. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22 25 27 32 39 48 49
7 16
12
17
17
18
23
30

17. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22 25 27 32 39 48 49
7 16 27
12
17
17
18
23
30

18. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22 25 27 32 39 48 49
7 16 27 40
12
17
17
18
23
30

19. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22 25 27 32 39 48 49
7 16 27 40 50
12
17
17
18
23
30

20. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22 25 27 32 39 48 49
7 16 27 40 50 60 74 84 99 101
12
17
17
18
23
30

21. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22 25 27 32 39 48 49
7 16 27 40 50 60 74 84 99 101
12 27 42 62 74 89 107 120 140 148
17 39 58 80 98 121 148 168 192 206
17 42 63 89 113 144 177 201 230 247
18 47 77 103 132 170 208 236 271 293
23 55 90 123 161 205 247 278 315 342
30 65 101 140 186 235 283 322 368 401

22. Summed Area Table (Integral Image)
✘ Now let’s compute
the sum of the
highlighted area

23. Summed Area Table (Integral Image)
✘ Area =
•

24. Summed Area Table (Integral Image)
✘ Area =
•-•

25. Summed Area Table (Integral Image)
✘ Area =
•-•-•

26. Summed Area Table (Integral Image)
✘ Area =
•-•-•+•

27. Summed Area Table (Integral Image)
✘ Recall this
calculation:
2+6+8+9+7+4+4+6+8+6+
4+5+0+5+7+5+4+6+7+9+
6+4+3+2 = 127
✘ But now let’s use our
integral image!
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6

28. Summed Area Table (Integral Image)
4 7 14 22 25 27 32 39 48 49
7 16 27 40 50 60 74 84 99 101
12 27 42 62 74 89 107 120 140 148
17 39 58 80 98 121 148 168 192 206
17 42 63 89 113 144 177 201 230 247
18 47 77 103 132 170 208 236 271 293
23 55 90 123 161 205 247 278 315 342
30 65 101 140 186 235 283 322 368 401
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
Area =

29. Summed Area Table (Integral Image)
4 7 14 22 25 27 32 39 48 49
7 16 27 40 50 60 74 84 99 101
12 27 42 62 74 89 107 120 140 148
17 39 58 80 98 121 148 168 192 206
17 42 63 89 113 144 177 201 230 247
18 47 77 103 132 170 208 236 271 293
23 55 90 123 161 205 247 278 315 342
30 65 101 140 186 235 283 322 368 401
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
Area = 315

30. Summed Area Table (Integral Image)
4 7 14 22 25 27 32 39 48 49
7 16 27 40 50 60 74 84 99 101
12 27 42 62 74 89 107 120 140 148
17 39 58 80 98 121 148 168 192 206
17 42 63 89 113 144 177 201 230 247
18 47 77 103 132 170 208 236 271 293
23 55 90 123 161 205 247 278 315 342
30 65 101 140 186 235 283 322 368 401
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
Area = 315-140

31. Summed Area Table (Integral Image)
4 7 14 22 25 27 32 39 48 49
7 16 27 40 50 60 74 84 99 101
12 27 42 62 74 89 107 120 140 148
17 39 58 80 98 121 148 168 192 206
17 42 63 89 113 144 177 201 230 247
18 47 77 103 132 170 208 236 271 293
23 55 90 123 161 205 247 278 315 342
30 65 101 140 186 235 283 322 368 401
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
Area = 315-140-90

32. Summed Area Table (Integral Image)
4 7 14 22 25 27 32 39 48 49
7 16 27 40 50 60 74 84 99 101
12 27 42 62 74 89 107 120 140 148
17 39 58 80 98 121 148 168 192 206
17 42 63 89 113 144 177 201 230 247
18 47 77 103 132 170 208 236 271 293
23 55 90 123 161 205 247 278 315 342
30 65 101 140 186 235 283 322 368 401
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
Area = 315-140-90+42 = 127

33. Summed Area Table (Integral Image)
4 7 14 22 25 27 32 39 48 49
7 16 27 40 50 60 74 84 99 101
12 27 42 62 74 89 107 120 140 148
17 39 58 80 98 121 148 168 192 206
17 42 63 89 113 144 177 201 230 247
18 47 77 103 132 170 208 236 271 293
23 55 90 123 161 205 247 278 315 342
30 65 101 140 186 235 283 322 368 401
✘ Area =
315 – 140 – 90 + 42 = 127
✘ It is much more
efficient!

34. Summed Area Table (Integral Image)
✘ The computation of
the integral image can be
made efficient too!
•=

35. Summed Area Table (Integral Image)
✘ The computation of
the integral image can be
made efficient too!
•=•

36. Summed Area Table (Integral Image)
✘ The computation of
the integral image can be
made efficient too!
•=•+•

37. Summed Area Table (Integral Image)
✘ The computation of
the integral image can be
made efficient too!
•=•+•-•

38. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22 25 27 32 39 48 49
7 16 27 40 50 60 74 84 99 101
12 27 42 62 74 89 107 120 140 148
17 39 58 80 98 121 148 168 192 206
17 42 63 89 113 144 177 201 230 247
18 47 77 103 132 170 ?
23 55 90 123 161 205
30 65 101 140 186 235

39. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22 25 27 32 39 48 49
7 16 27 40 50 60 74 84 99 101
12 27 42 62 74 89 107 120 140 148
17 39 58 80 98 121 148 168 192 206
17 42 63 89 113 144 177 201 230 247
18 47 77 103 132 170 x
23 55 90 123 161 205
30 65 101 140 186 235
x = 177

40. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22 25 27 32 39 48 49
7 16 27 40 50 60 74 84 99 101
12 27 42 62 74 89 107 120 140 148
17 39 58 80 98 121 148 168 192 206
17 42 63 89 113 144 177 201 230 247
18 47 77 103 132 170 x
23 55 90 123 161 205
30 65 101 140 186 235
x = 177+170

41. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22 25 27 32 39 48 49
7 16 27 40 50 60 74 84 99 101
12 27 42 62 74 89 107 120 140 148
17 39 58 80 98 121 148 168 192 206
17 42 63 89 113 144 177 201 230 247
18 47 77 103 132 170 x
23 55 90 123 161 205
30 65 101 140 186 235
x = 177+170-144

42. Summed Area Table (Integral Image)
4 3 7 8 3 2 5 7 9 1
3 6 4 5 7 8 9 3 6 1
5 6 4 7 2 5 4 3 5 6
5 7 4 2 6 8 9 7 4 6
0 3 2 4 6 8 6 4 5 3
1 4 9 0 5 7 5 4 6 5
5 3 5 7 9 6 4 3 2 5
7 3 1 6 8 5 6 8 9 6
4 7 14 22 25 27 32 39 48 49
7 16 27 40 50 60 74 84 99 101
12 27 42 62 74 89 107 120 140 148
17 39 58 80 98 121 148 168 192 206
17 42 63 89 113 144 177 201 230 247
18 47 77 103 132 170 208
23 55 90 123 161 205
30 65 101 140 186 235
x = 177+170-144+5 = 208