How to determine the area of a polygon in an Image - c++

Question

If I am having a convex polygon vertices, then my area calculation in image is not coming accurate by the standard formula.
(For simiplicity) if I am having 3x3 square, and the vertices are (1,1) (1,3) (3,3) (3,1)
by polygon area calculation method depicted here
and dividing the summation by 2 we get the Area.
So for the 3 x 3 data above, we'll get area as 4 instead of 9.
This is happening because vertices are not points but a pixel.
this is the corresponding code. The coordinates are cyclic.
int X[] = { 1, 1, 3, 3, 1};
int Y[] = { 1, 3, 3, 1, 1};
double Sum1 = 0;
double Sum2 = 0;
int numElements = 5;
for (int k = 0; k < numElements-1; k++)
{
Sum1 += X[k] * Y[k + 1];
Sum2 += Y[k] * X[k + 1];
}
double area = std::abs((double)(Sum1 - Sum2))/2;
For square, we can do +1 to width and height and get the area correct. But what about the irregular polygons in the image?
I hope the question makes sense.

Answer 1

If you don't want to work with pixel corners as vertices, consider next method (works for simple figures - all convex ones, some concave ones):
Add additional fake pixel at right side of every right-border pixel, at bottom side of every bottom-border pixel, at right-bottom of right-bottom corner pixel. Here gray pixels are initial, light blues ones - fake.

Answer 2

The area can be calculated in following steps:
1) fetching the pixels between the vertices
2) sorting the pixels by x coordinates (or y coordinates)
3) taking the difference between min and max y coordinates (or x) for a particular x (or y) value and adding one to the difference
4) summing up the total difference
NOTE: the area might vary (if there are slanted edges in the polygon) depending on the line drawing method chosen
int compare(const void * a, const void * b)
{
return (((Point*)a)->x() - ((Point*)b)->x());
}
double CalculateConvexHullArea(vector<int> ConvexHullX, vector<int> ConvexHullY)
{
float Sum1 = 0;
float Sum2 = 0;
std::vector<Point> FillPoints;
for (int k = 0; k < ConvexHullX.size() - 1; k++)
{
drawLine(ConvexHullX[k], ConvexHullX[k+1], ConvexHullY[k], ConvexHullY[k+1], FillPoints);
}
//sorting coordinates
qsort(FillPoints.data(), FillPoints.size(), sizeof(Point), compare);
double area = 0;
int startY = FillPoints[0].y(), endY = FillPoints[0].y();
int currX = FillPoints[0].x();
// traversing x and summing up diff of min and max Y
for (int cnt = 0; cnt < FillPoints.size(); cnt++)
{
if (FillPoints[cnt].x() == currX)
{
startY = startY > FillPoints[cnt].y() ? FillPoints[cnt].y() : startY;
endY = endY < FillPoints[cnt].y() ? FillPoints[cnt].y() : endY;
}
else
{
int diffY = endY - startY + 1;
area += diffY;
currX = FillPoints[cnt].x();
startY = endY = FillPoints[cnt].y();
}
}
return area + endY - startY + 1;
}