Extract four (von Neumann) neighbors of a matrix entry

Question

Assume I have a $m\times n$ matrix and I would like to extract four neighbors of randomly selected entry of a matrix. I have handled if location is not on the boundary. Any suggestion how to handle it if entry is on the boundary or handle it all in once? Thanks.

Here is an example.

SeedRandom[123];
{n, m} = {4, 6};
mat = RandomInteger[{1, 5}, {n, m}];
MatrixForm@mat

$\text{mat}= \left( \begin{array}{cccccc} 4 & 2 & 3 & 1 & 1 & 3 \\ 4 & 2 & 2 & 5 & 1 & 2 \\ 5 & 4 & 3 & 3 & 5 & 5 \\ 3 & 5 & 2 & 2 & 5 & 3 \\ \end{array} \right)$

loc = {RandomInteger[{2, n - 1}], RandomInteger[{2, m - 1}]}
fourNeighbor = (Extract[mat, # + loc] &) /@ {{0, -1}, {0, 1}, {-1, 0}, {1, 0}}

Edit:

Here is Moore Neighbors

nf = Nearest[Tuples@Range@Dimensions@mat -> Flatten[mat], 
   DistanceFunction -> ChessboardDistance];

neighbors[pt_] := nf[pt, {All, 1}][[1 ;;]]

Or

   mooreNeighborPositions =  AdjacencyList[NearestNeighborGraph[Tuples@Range@Dimensions@#,
 DistanceFunction -> ChessboardDistance], #2] &; 

 mooreNeighbors = Extract[#, mooreNeighborPositions@##] &;  mooreNeighbors[mat, {1, 1}]

Answer 1

You could use Nearest for this.

nf = Nearest[Tuples @ Range @ Dimensions @ mat -> Flatten[mat]];

neighbors[pt_] := nf[pt, {All, 1}][[2;;]]

Some examples:

neighbors[{3, 3}]
neighbors[{1, 1}]
neighbors[{4, 6}]

{2, 4, 3, 2}

{2, 4}

{5, 5}

Answer 2

vNNeighborPositions = AdjacencyList[
   NearestNeighborGraph @ Tuples @ Range @ Dimensions @ #, #2] &;

vNNeighbors = Extract[#, vNNeighborPositions @ ##] &;

Examples:

Row[Labeled[Style[#, 20] & @ MatrixForm @
      MapAt[Highlighted[#, Background -> Red] &, 
       MapAt[Highlighted, mat, vNNeighborPositions[mat, #]], #], 
    Grid[{{"pos:", #}, {"neighbors:", vNNeighbors[mat, #]}}], Top] & /@ 
 {{1, 1}, {1, 4}, {3, 1}, {4, 6}, {3, 5}}, 
 Spacer[10]]

SeedRandom[333]
mat = RandomInteger[10, {10, 15}];
poslist = RandomSample[Tuples @ Range @ Dimensions @ mat, 7];

Legended[MatrixPlot[ReplacePart[mat, 
   Join[Thread[poslist -> (ColorData[97] /@ Range[Length@poslist])], 
     Thread[vNNeighborPositions[mat, #] & /@ poslist -> Yellow],
     {{_, _} :> White}]], ImageSize -> 1 -> 40, Mesh -> All,
   Epilog ->  MapIndexed[Text[Style[#, 16, Black], #2 - .5] &, 
    Reverse /@ Transpose @ mat, {2}]], 
 Placed[SwatchLegend[(ColorData[97] /@ Range[Length @ poslist]), 
   Style[#, 14] & /@ ({Defer @ #, vNNeighbors[mat, #]} & /@ poslist), 
   LegendMarkerSize -> 20, LegendLabel -> "positions & neighbors"], Right]]

Answer 3

The four neighbors are

ij=Map[loc + # &, {{0, -1}, {0, 1}, {-1, 0}, {1, 0}}]

Now you must check the index range:

DeleteCases[Map[{Max[1,Min[n,#[[1]] ]],Max[1,Min[m,#[[2]] ]]}& ,ij],loc] (* index pairs*)
Extract[mat,%] (* neighbors  *)

Answer 4

nbrs[loc_?VectorQ, m_?MatrixQ] := Module[{nrows, ncols, pts},
  {nrows, ncols} = Dimensions[m];
  pts = Select[(loc + # &) /@ {{0, -1}, {0, 1}, {-1, 0}, {1, 0}},
    Between[#[[1]], {1, nrows}] && Between[#[[2]], {1, ncols}] &];
  Extract[mat, pts]
  ]