Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 FindGridIndices

Source Notebook

Find the indices of approximate grid points

Contributed by: Sander Huisman

ResourceFunction["FindGridIndices"][pts,{i1→{j1,k1},i2→{j2,k2},i3→{j3,k3},…}]

finds the grid indices of the points pts given the indices for the is.

ResourceFunction["FindGridIndices"][pts,{i1→{j1,k1},i2→{j2,k2},i3→{j3,k3},…},delta]

finds the grid indices with a maximum deviation delta in the index.

ResourceFunction["FindGridIndices"][pts,{i1→{j1,k1},i2→{j2,k2},i3→{j3,k3},…},delta,max]

finds the grid indices with a maximum index distance of max allowed to jump 'gaps'.

Details

The purpose of FindGridIndices is to index a group of points that form an approximate, but not necessarily perfect, grid. Imagine assigning seat and row numbers to seats in an auditorium with known spatial locations.
The value of delta is given in units of the indices, and the default value is 0.25, which can also be given as Automatic.
The value of max is given in units of the indices, and the default value is ∞, which can also be given by Automatic.
At least 3 known coordinates are needed, and, for the best result, these should be a close as possible together.
The algorithm follows the following steps. Starting from the points with known indices: 
1. Find the center of these points 2. Find the point with unknown indices closest to this center 3. Find the nearest known points to this unknown point which can form an orthogonal basis 4. Express the unknown point in terms of this basis and express the location of this point in terms of new (unrounded) indices 5. Check whether these new indices are close enough to an integer 6. Assign new indices to this unknown point such that is becomes a known point 7. Repeat until all points are either assigned or discarded because they are not close enough to integer indices.

Examples

Basic Examples (3) 

Make a grid of points and figure out the grid-indices:

In[1]:=
points = {{1.25, 1.25}, {1.25, 2.25}, {1.25, 3.25}, {2.25, 1.25}, {2.25, 2.25}, {2.25, 3.25}, {3.25, 1.25}, {3.25, 2.25}, {3.25, 3.25}}; known = {1 -> {0, 0}, 2 -> {0, 1}, 4 -> {2, 0}}; out = ResourceFunction["FindGridIndices"][points, known]
Out[3]=

For some given points, and known indices, find all the indices:

In[4]:=
points = {{2.2457107617837337`, 3.151739383363539}, { 1.2194058590709853`, 4.177855965476845}, {0.18612053291149167`, 5.255740242248263}, {3.265717355719692, 4.302069883742035}, { 2.230859571348209, 5.330745256117548}, {1.3451808496165396`, 6.274072003786259}, {4.169600297056586, 5.34635094414414}, { 3.3000345340888666`, 6.1767441646481815`}, {2.21976647442846, 7.186465699338501}, {5.279614167307427, 6.223906792689673}, { 4.282957106416612, 7.196190176911716}, {3.2616964858975876`, 8.310921559418997}, {6.156158838009473, 7.2418246145486025`}, { 5.256388542385789, 8.258148926045028}, {4.216236774163855, 9.26524266508286}}; known = {1 -> {0, 0}, 2 -> {0, 1}, 7 -> {2, 0}}; out = ResourceFunction["FindGridIndices"][points, known]
Out[5]=

Given some point and known indices, visualize the configuration:

In[6]:=
testpts = CompressedData[" 1:eJw1WQdczfv7RySENEihhIjsrIzekj2Ln5u9cm9khIQG2rvOPqfTOquyZWeV lU1U14iSVWSVjLq67v/5nE//59Wr5zzz83z283y+PVdv9ljbolmzZgPpH8MO S5OufK/PRG6PiA/uThIUrnaOe/89G742l4NbrolEJ1ejnuGflXCuSt12f3sy yKTZg2HperzSPESPoczWY1VZOOzGzxtz/nM2vNgPeQz+R2AxicubHQ7BY63v +Lar0uDYUCBaYRmCLwqrJ2Pl2RiXZfgjcXskHi53DWyTno2RLo4ehs5hqFrj O8HwbRaWx5RK95G94HKDZapLk79pYTBkIM5GaAiBRTROZB1y8EnWwHXFgy7f tiRjSU6rJTk3BCjqG1zU96wcH6/N+XgtPhUtBj5oMbBbCLa2Gno29J2OY08B kiyMen0amw1d9wEmUyrD8dGdRuaYBmx4cmeEIjbtScyy1zr081/d1z9WjAJy N+etDkStzs4TYcCM173MJmXC/sHY7QODxQhs86nPyV+ZaEtIXi+Br+evyFNb BAiYIchumRCCqFObPX8hEz4vcn60ny4BE29WZaJocpdlfmMlmP/mxKYhmkx0 myUsPLdZjGenNz5bOFILUe/pIp0sFNPph4dLJs5sfHZ6oLkUfYjhkJ4JEvdu YyiFXcgZG1N/ASo+RbUsXxyGXz2JkZsC8yrnKVvsQiAebO3UEKHDshv97m+/ LUKr8t3TKtwycehbyZUZx0QYQuLjC7LwdW2Y/Qo3Od5ezV0hj9Jhqsr74OV/ hRixMxEP3+vwckyLyUt6CaFMJpiSCfHtqC35b4T4vC78bXpeCs5ld5IFbkzG PkJWfXTImOL5ed1jEUyJXLVXh96OP52uDJGiF8NVOj6/tUJUpk/x3DQ+E1e2 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-> {Black, Red, Green, Cyan}, PlotRange -> All]
Out[7]=

Find the indices and annotate the points with their coordinates shown on top. Note that the maximum distance is used to remove very remote points (i.e. it can't jump big gaps):

In[8]:=
result = ResourceFunction["FindGridIndices"][testpts, known, Automatic, 3]; Graphics[{Map[{Text[ Style[#["GridIndices"], 6], #["Position"] + {0, 20}], Point[#["Position"]]} &, result]}, ImageSize -> 800]
Out[9]=

Scope (2) 

Create data points in an arc and randomly remove some datapoints:

In[10]:=
SeedRandom[1337]; pts = Catenate@ Table[AngleVector[{r, \[Theta] Degree}] + RandomReal[0.03 {-1, 1}, 2], {r, 10, 20, 0.5}, {\[Theta], -120, -10, 2.5}]; b = {1, 2, 46}; pts = Delete[pts, List /@ RandomSample[Range[47, Length[pts]], 200]]; ListPlot[{pts, pts[[b]]}, AspectRatio -> Automatic] b = Rule @@@ Transpose[{b, {{0, 0}, {1, 0}, {0, 1}}}];
Out[11]=

Find the indices with a very small delta and visualize the indices with the color indicating the order:

In[12]:=
result = ResourceFunction["FindGridIndices"][pts, b]; Graphics[{MapIndexed[{Blend[{Red, Blue}, #2[[1]]/Length[result]], Text[Style[#1["GridIndices"], 6], #1["Position"] + {0, 0.2}], Point[#1["Position"]]} &, result]}, ImageSize -> 800]
Out[13]=

Possible Issues (2) 

Sometimes the delta has to be increased to capture all points:

In[14]:=
SeedRandom[1337]; pts = Catenate@ Table[AngleVector[{r, \[Theta] Degree}] + {0.2 r, 0.09 \[Theta]}, {r, 10, 20, 0.5}, {\[Theta], 45, 135, 2.5}]; pts += RandomReal[0.04 {-1, 1}, Dimensions[pts]]; b = {1, 2, 38}; b = Rule @@@ Transpose[{b, {{0, 0}, {1, 0}, {0, 1}}}]; result = ResourceFunction["FindGridIndices"][pts, b]; Graphics[{Map[{Text[ Style[#["GridIndices"], 6], #["Position"] + {0, 0.2}], Point[#["Position"]]} &, result]}, ImageSize -> 800]
Out[20]=

Increasing the maximum delta gives all the points:

In[21]:=
result = ResourceFunction["FindGridIndices"][pts, b, 0.5]; Graphics[{Map[{Text[ Style[#["GridIndices"], 6], #["Position"] + {0, 0.2}], Point[#["Position"]]} &, result]}, ImageSize -> 800]
Out[22]=

With a very large delta one can get bizarre results with duplicate indices:

In[23]:=
SeedRandom[1337]; pts = Catenate@ Table[AngleVector[{r, \[Theta] Degree}], {r, 10, 20, 0.5}, {\[Theta], 45, 135, 2.5}]; pts += RandomReal[0.07 {-1, 1}, Dimensions[pts]]; b = {1, 2, 38}; b = Rule @@@ Transpose[{b, {{0, 0}, {1, 0}, {0, 1}}}]; result = ResourceFunction["FindGridIndices"][pts, b, 0.8, 3]; Graphics[{Map[{Text[ Style[#["GridIndices"], 6], #["Position"] + {0, 0.2}], Point[#["Position"]]} &, result]}, ImageSize -> 800]
Out[29]=

Test whether there are duplicate indices:

In[30]:=
DuplicateFreeQ[result[[All, "GridIndices"]]]
Out[30]=

Publisher

SHuisman

Requirements

Wolfram Language 12.3 (May 2021) or above

Version History

  • 1.0.0 – 14 May 2025

Related Resources

License Information