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discrete numerical 2D gradient

• To: mathgroup at smc.vnet.net
• Subject: [mg113084] discrete numerical 2D gradient
• From: Sebastian Schmitt <sschmitt at physi.uni-heidelberg.de>
• Date: Tue, 12 Oct 2010 13:49:53 -0400 (EDT)

Dear all!

I have 2D data represented as a matrix. I would like to obtain the
gradient. I searched quite some time in the Mathematica documentation to
find a function that does the job without success. Therefore I came up
with something myself. It is more for illustrative purposes than real
life application, e.g. I'm taking a simply forward difference. Could you
please take a look at my solution? (it should be copy'n'pastable)

------------------------------------------------------------------------

nrows = 10; ncolumns = 10;

dataMatrix = Table[i*j, {i, nrows}, {j, ncolumns}];

dataMatrix//MatrixPlot

fwdDifferenceMatrix =
  SparseArray[{Band[{1, 1}] -> 1, Band[{1, 2}] -> -1}, {nrows,
    ncolumns}, 0] ;

fwdDifferenceMatrix//MatrixForm

dxMatrix = dataMatrix.fwdDifferenceMatrix;

dyMatrix = Transpose[Transpose[dataMatrix].fwdDifferenceMatrix];

dxyMatrix = Function[{u, v}, List[u, v], Listable][dxMatrix, dyMatrix];

subdxyMatrix = dxyMatrix[[2 ;;, 2 ;;]];

Rotate[ListVectorPlot[subdxyMatrix],-90Degree]

------------------------------------------------------------------------

Is there a better (both mathematical and Mathematical) to construct the
y-derivatives (dyMatrix)?

I had also expected to find a built-in function to combine dxMatrix and
dyMatrix to a matrix of vectors. But maybe I just have to accept to use
pure function a lot.

The last problem I have is that ListVectorPlot has its origin in the
lower-left corner while the matrix counts from upper-left to
lower-right. I simply rotated the plot. Could I have done something else?

Thanks in advance,

Sebastian

• Follow-Ups:
  • Re: discrete numerical 2D gradient
    • From: Mark McClure <mcmcclur@unca.edu>