Grid-type shapefile from raster GeoTIFF

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I need to produce a shapefile from a raster. I need to be able to specify either an interval, or manually define the intervals. The input file is a GeoTIFF. I would like to use GDAL and/or Python.

Example raster:



I tried using gdal_contour, which at first glance would have been perfect. However, I want the result to have the same "pixel" definition - that is, the "contours" should maintain the block appearance of the pixels at their extents. For this reason, I can't use gdal_contour, as the results look like this:



Is there another way to do this? My output should be a shapefile that looks like a grid, with only pixels in the same interval connected to each other.

Thanks



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