def METHOD_NAME(self, monitorTileCounts):
    # Create a test tile with the default options
    params = {'encoding': 'JPEG'}
    source = monitorTileCounts(None, **dict(
        params,
        tileWidth=256, tileHeight=256,
        sizeX=256 * 2 ** 9, sizeY=256 * 2 ** 9, levels=10,
    ))
    meta = source.getMetadata()
    utilities.checkTilesZXY(source, meta, params)
    # We should have generated tiles
    assert large_image_source_test._counters['tiles'] > 0
    counter1 = large_image_source_test._counters['tiles']
    # Running a second time should take entirely from cache
    utilities.checkTilesZXY(source, meta, params)
    assert large_image_source_test._counters['tiles'] == counter1
    # Test most of our parameters in a single special case
    params = {
        'minLevel': 2,
        'maxLevel': 5,
        'tileWidth': 160,
        'tileHeight': 120,
        'sizeX': 5000,
        'sizeY': 3000,
        'encoding': 'JPEG',
    }
    source = monitorTileCounts(None, **dict(
        params,
        tileWidth=160, tileHeight=120,
        sizeX=5000, sizeY=3000, levels=6,
        minLevel=2,
    ))
    meta = source.getMetadata()
    meta['minLevel'] = 2
    utilities.checkTilesZXY(source, meta, params)
    # We should have generated tiles
    assert large_image_source_test._counters['tiles'] > counter1
    counter2 = large_image_source_test._counters['tiles']
    # Running a second time should take entirely from cache
    utilities.checkTilesZXY(source, meta, params)
    assert large_image_source_test._counters['tiles'] == counter2
    # Test the fractal tiles with PNG
    params = {'fractal': 'true'}
    source = monitorTileCounts(None, **dict(
        params,
        tileWidth=256, tileHeight=256,
        sizeX=256 * 2 ** 9, sizeY=256 * 2 ** 9, levels=10,
    ))
    meta = source.getMetadata()
    utilities.checkTilesZXY(source, meta, params, utilities.PNGHeader)
    # We should have generated tiles
    assert large_image_source_test._counters['tiles'] > counter2
    counter3 = large_image_source_test._counters['tiles']
    # Running a second time should take entirely from cache
    utilities.checkTilesZXY(source, meta, params, utilities.PNGHeader)
    assert large_image_source_test._counters['tiles'] == counter3