How to efficiently upsample a greyscale image to generate a representative binary image

I want to create binary images by upsampling a series of greyscale images (192 x 144 resolution) by a factor of 10 with the new pixel values being a pseudorandom distribution of 0’s & 1’s based on the original grey level value. The new resolution would be 1920 x 1440.

Here is a small example of a greyscale image:

i = Image[{{0.16, 0.22, 0.32}, {0.73, 0.06, 0.13}, {0., 0.86, 0.18}}, ImageSize->100]

The following function gives the desired result for such small images/arrays:

SeedRandom[123]
resize[dat_] :=
 Map[SparseArray[
  Partition[
   RandomSample[
    ConstantArray[0, IntegerPart[(1 - #)*100]]~Join~
     ConstantArray[1, IntegerPart[100 #]]
  ], 10]] &, dat, {2}]

  s = resize[ImageData@i];
  bin = ArrayFlatten[s];
  Image[bin, ImageSize -> 100]

However, when I try to uses it for a larger image ArrayFlatten gets upset…

SeedRandom[123]
bigi = Image@RandomReal[1, {50, 50}, WorkingPrecision -> 2]

ArrayFlatten[resize[ImageData@bigi]]

Obviously, I am stepping outside of ArrayFlatten‘s defined parameters, but I am unable to decipher the meaning of the error message…

Why is this happening?

Is there a simpler way to achieve my desired result for larger greyscale images?

f[img_, k_] := Image[ArrayFlatten[
    Map[
     RandomChoice[{1 - #, #} -> {0, 1}, {k, k}] &,
     ImageData[img],
     {2}
     ]
    ]];

f[ExampleData[{"Texture", "Bark"}], 5]

ImageApply[RandomVariate@*BernoulliDistribution, ImageResize[img, {100, 100}]]

With[{dims = {1920, 1440}}, 
 Binarize[ImageSubtract[ImageResize[img, dims], RandomImage[1, dims]],0]]