Finding average of attributed lines

Question

I've been given some data in a particularly (in my opinion) odd way, namely an array of points connected to each other via lines, whereby each line has for a lack of a better word, "attribute" assigned to it. Being a numerical value between {1,4}.

In poorly hacked up code, I've managed to simply draw the lines connecting the points in a plane for visualizations sake.

Each blue line represents a value between 1 and 4, and the black lines are a simple grid. For clarity. The grid goes from [0,9] in y and [0,8] in x at 0.5 steps.

Graphics[{{Blue, Line[pts]}, lho, lver}, Frame -> True, 
PlotRange -> {{0, 8}, {0, 9}}]

I would like to build up an average of each 1x1 grid square of the values of each line inside said square. This way each square will become a single point value that I can now attribute to a {x,y,z} form with z being the average and x, y being coordinates in the grid.

For example, Square 1:1, starting at the bottom left has 8 lines within it, 4 of them having the value 2.45556, and the other 4 having the value 1.6690. Doing the math (4*2.45556+4*1.6690)/8 = 2.06228

Doing this for another square, such as 1:4 (starting from the left corner bottom corner again) would only see two lines two of which are shared from 1:1, so assuming those two lines both have the value 2.45556, it's average would be also 2.4556.

I can now have the points at {0.5 ,1.5 ,2.06228} and {3.5,1.5,2.45556} for further analysis. This particular example as a demonstration has 64 squares in total, however this will scale up to several hundred later.

First, is it possible to give a line a numerical value? If such a thing is possible, how would I attempt turning each square and its segment(s) of a line(s) into an average to be used later?

My first thought would be to give each line a specific colour from a function that scales {1,4} into a gradient from blue to red, then somehow using those values. But this is far beyond my skills in mathematica.

In the end I would like to turn this array of data of mine into a density plot similar to here.

3D heatmap density plot

pts, lve, and lho added updated lver which now has all square boundries

Thank you in advance.

pts = {{0, 1}, {8, 1}, {0, 1}, {8, 2}, {0, 1}, {8, 3}, {0, 1}, {8, 
2}, {0, 1}, {8, 4}, {0, 1}, {8, 5}, {0, 1}, {8, 6}, {0, 1}, {8, 
7}, {0, 1}, {8, 8}, {0, 2}, {8, 1}, {0, 2}, {8, 2}, {0, 2}, {8, 
3}, {0, 2}, {8, 2}, {0, 2}, {8, 4}, {0, 2}, {8, 5}, {0, 2}, {8, 
6}, {0, 2}, {8, 7}, {0, 2}, {8, 8}, {0, 3}, {8, 1}, {0, 3}, {8, 
2}, {0, 3}, {8, 3}, {0, 3}, {8, 2}, {0, 3}, {8, 4}, {0, 2}, {8, 
5}, {0, 3}, {8, 6}, {0, 3}, {8, 7}, {0, 3}, {8, 8}, {0, 4}, {8, 
1}, {0, 4}, {8, 2}, {0, 4}, {8, 3}, {0, 4}, {8, 2}, {0, 4}, {8, 
4}, {0, 4}, {8, 5}, {0, 4}, {8, 6}, {0, 4}, {8, 7}, {0, 4}, {8, 
8}, {0, 5}, {8, 1}, {0, 5}, {8, 2}, {0, 5}, {8, 3}, {0, 5}, {8, 
2}, {0, 5}, {8, 4}, {0, 5}, {8, 5}, {0, 5}, {8, 6}, {0, 5}, {8, 
7}, {0, 5}, {8, 8}, {0, 6}, {8, 1}, {0, 6}, {8, 2}, {0, 6}, {8,
3}, {0, 6}, {8, 2}, {0, 6}, {8, 4}, {0, 6}, {8, 5}, {0, 6}, {8,
6}, {0, 6}, {8, 7}, {0, 6}, {8, 8}, {0, 7}, {8, 1}, {0, 7}, {8,
2}, {0, 7}, {8, 3}, {0, 7}, {8, 2}, {0, 7}, {8, 4}, {0, 7}, {8,
5}, {0, 7}, {8, 6}, {0, 7}, {8, 7}, {0, 7}, {8, 8}, {0, 8}, {8,
1}, {0, 8}, {8, 2}, {0, 8}, {8, 3}, {0, 8}, {8, 2}, {0, 8}, {8,
4}, {0, 8}, {8, 5}, {0, 8}, {8, 6}, {0, 8}, {8, 7}, {0, 8}, {8, 8}};

lho = {Line[{{0, 0.5}, {9, 0.5}}], Line[{{0, 1.5}, {9, 1.5}}], 
   Line[{{0, 2.5}, {9, 2.5}}], Line[{{0, 3.5}, {9, 3.5}}], 
   Line[{{0, 4.5}, {9, 4.5}}], Line[{{0, 5.5}, {9, 5.5}}], 
   Line[{{0, 6.5}, {9, 6.5}}], Line[{{0, 7.5}, {9, 7.5}}], 
   Line[{{0, 8.5}, {9, 8.5}}]};

lver = {Line[{{0, 0}, {0, 9}}], Line[{{1, 0}, {1, 9}}], 
   Line[{{2, 0}, {2, 9}}], Line[{{3, 0}, {3, 9}}], 
   Line[{{4, 0}, {4, 9}}], Line[{{5, 0}, {5, 9}}], 
   Line[{{6, 0}, {6, 9}}], Line[{{7, 0}, {7, 9}}], 
   Line[{{8, 0}, {8, 9}}]};

Answer 1

I am not sure but I think it does what you ask for. I just create a constant list lineweights of weights for the lines. (I would discourage storing the weight within the Line object in any way.)

lines = Flatten[Table[Line[{{0, i}, {8, j}}], {i, 1, 8}, {j, 1, 8}]];
lineweights = ConstantArray[1., Length[lines]];
rectangles = Flatten[Outer[
    Rectangle @@ Transpose[List[##]] &, 
    Partition[N[lver[[All, 1, 1, 1]]], 2, 1], 
    Partition[lho[[All, 1, 1, 2]], 2, 1], 
    1
  ]];
A = 1 - Outer[
  Boole@*RegionDisjoint, 
    DiscretizeRegion /@ lines, 
    rectangles
  ];
means = Mean[lineweights A];
Show[
 Graphics[Transpose[{GrayLevel /@ Rescale[1 - means], rectangles}]],
 Graphics[{Darker@Blue, lines}]
 ]