# Upsampling matrix by inserting zeros inbetween elements

I want to essentially take an image with $n \times n$ pixels and expand it into a sparse array, spacing out the pixels by a factor $2$ or $3$. Each pixel in the input image occupies one of the corners of a cell of $2 \times 2$ or $3 \times 3$ new pixels in the expanded image, with the other $n \times (n - 1)$ pixels (in each cell) set to $0$.

So, I want to create a padded array where:

If one line of data is

WXYZ


then the output should be

W0 X0 Y0 Z0
00 00 00 00


I think Mathematica's SparseArray might be the way to go, but I'm not sure how to go about implementing this.

-

l = {{2, 3, 4}, {4, 5, 6}};
a = SparseArray[{i_, j_} /; (Mod[{i, j}, 3] == {1, 1}) :>
l[[Sequence @@ (Quotient[{i, j}, 3] + 1)]], 3 Dimensions@l]
MatrixPlot /@ {l, a}

(*Or the same thing by using Replace[], without SparseArray[] *)

f[p_] := Replace[p, x_ -> Sequence @@ {x, 0, 0}]
MatrixPlot@Transpose@f@Transpose@f@l


Edit

Answering your comment "How should I modify the code to step through the positions in each cell and fill each of the 3x3 pixels i.e. rather than the 1,1 location"

l = {{2, 3, 4}, {4, 5, 6}};
a = SparseArray[{i_, j_} :> l[[Sequence @@ (1 + Quotient[{i, j} - 1, 3])]], 3 Dimensions@l]
MatrixPlot /@ {l, a}


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Thanks that worked perfectly :). Question - How should I modify the code to step through the positions in each cell and fill each of the 3x3 pixels i.e. rather than the 1,1 location. – Abhi Nov 26 '13 at 4:46
@Abhi See edit, please – Dr. belisarius Nov 26 '13 at 4:59
Oops I meant in successive steps. So essentially creating a1, a2, ... a9 different matrices. And in each Matrix the value of the pixel will be in the following cell. Sorry for the confusion. – Abhi Nov 26 '13 at 5:16
@Abhi Have you tried writing it yourself? – Dr. belisarius Nov 26 '13 at 6:09
I've been fighting with it for the past hour :). I've only used Mathematica very sparingly and the symbolic scripting has me very confused. – Abhi Nov 26 '13 at 6:12

A straightforward way to tackle this problem is to use Upsample, a function designed to do exactly this kind of array manipulation. Suppose m is a 2 by 3 array. These can be spaced out by padding each entry with zeros in both the x- and y-directions.

m = RandomInteger[{0, 10}, {2, 3}];
ArrayPlot[Upsample[m, {3, 3}]]


The {3,3} parameter tells the function how much padding to do in each direction. Looking at the documentation for Upsample shows that there is a third parameter (the offset) that allows placement of the nonzero terms in any desired location. For instance, to {2,2} places it in the middle of each 3x3 block:

{ArrayPlot[m], ArrayPlot[Upsample[m, {3, 3}, {2, 2}]]}


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Wow thanks. This gave me a factor of 25 speed boost. I was surprised that I couldn't locate something like Upsample. – Abhi Nov 27 '13 at 6:07