diff --git a/sources/ometiff/large_image_source_ometiff/__init__.py b/sources/ometiff/large_image_source_ometiff/__init__.py
index 02a7d9609..bb78f2d5e 100644
--- a/sources/ometiff/large_image_source_ometiff/__init__.py
+++ b/sources/ometiff/large_image_source_ometiff/__init__.py
@@ -16,6 +16,7 @@
 #  limitations under the License.
 ##############################################################################
 
+import copy
 import math
 import numpy
 import PIL.Image
@@ -101,30 +102,7 @@ def __init__(self, path, **kwargs):
             raise TileSourceException('Not an OME Tiff')
         self._omeinfo = info['OME']
         self._checkForOMEZLoop(largeImagePath)
-        if isinstance(self._omeinfo['Image'], dict):
-            self._omeinfo['Image'] = [self._omeinfo['Image']]
-        for img in self._omeinfo['Image']:
-            if isinstance(img['Pixels'].get('TiffData'), dict):
-                img['Pixels']['TiffData'] = [img['Pixels']['TiffData']]
-            if isinstance(img['Pixels'].get('Plane'), dict):
-                img['Pixels']['Plane'] = [img['Pixels']['Plane']]
-        try:
-            self._omebase = self._omeinfo['Image'][0]['Pixels']
-            if ((not len(self._omebase['TiffData']) or (
-                    len(self._omebase['TiffData']) == 1 and
-                    self._omebase['TiffData'][0] == {})) and
-                    len(self._omebase['Plane'])):
-                self._omebase['TiffData'] = self._omebase['Plane']
-            if len({entry.get('UUID', {}).get('FileName', '')
-                    for entry in self._omebase['TiffData']}) > 1:
-                raise TileSourceException('OME Tiff references multiple files')
-            if (len(self._omebase['TiffData']) != int(self._omebase['SizeC']) *
-                    int(self._omebase['SizeT']) * int(self._omebase['SizeZ']) or
-                    len(self._omebase['TiffData']) != len(
-                        self._omebase.get('Plane', self._omebase['TiffData']))):
-                raise TileSourceException('OME Tiff contains frames that contain multiple planes')
-        except (KeyError, ValueError, IndexError):
-            raise TileSourceException('OME Tiff does not contain an expected record')
+        self._parseOMEInfo()
         omeimages = [
             entry['Pixels'] for entry in self._omeinfo['Image'] if
             len(entry['Pixels']['TiffData']) == len(self._omebase['TiffData'])]
@@ -206,6 +184,39 @@ def _checkForOMEZLoop(self, largeImagePath):
         info['Image']['Pixels']['PlanesFromZloop'] = 'true'
         info['Image']['Pixels']['SizeZ'] = str(zloop)
 
+    def _parseOMEInfo(self):
+        if isinstance(self._omeinfo['Image'], dict):
+            self._omeinfo['Image'] = [self._omeinfo['Image']]
+        for img in self._omeinfo['Image']:
+            if isinstance(img['Pixels'].get('TiffData'), dict):
+                img['Pixels']['TiffData'] = [img['Pixels']['TiffData']]
+            if isinstance(img['Pixels'].get('Plane'), dict):
+                img['Pixels']['Plane'] = [img['Pixels']['Plane']]
+        try:
+            self._omebase = self._omeinfo['Image'][0]['Pixels']
+            if ((not len(self._omebase['TiffData']) or
+                    len(self._omebase['TiffData']) == 1) and
+                    len(self._omebase['Plane'])):
+                if not len(self._omebase['TiffData']) or self._omebase['TiffData'][0] == {}:
+                    self._omebase['TiffData'] = self._omebase['Plane']
+                elif (int(self._omebase['TiffData'][0].get('PlaneCount', 0)) ==
+                        len(self._omebase['Plane'])):
+                    planes = copy.deepcopy(self._omebase['Plane'])
+                    for idx, plane in enumerate(planes):
+                        plane['IFD'] = plane.get(
+                            'IFD', int(self._omebase['TiffData'][0].get('IFD', 0)) + idx)
+                    self._omebase['TiffData'] = planes
+            if len({entry.get('UUID', {}).get('FileName', '')
+                    for entry in self._omebase['TiffData']}) > 1:
+                raise TileSourceException('OME Tiff references multiple files')
+            if (len(self._omebase['TiffData']) != int(self._omebase['SizeC']) *
+                    int(self._omebase['SizeT']) * int(self._omebase['SizeZ']) or
+                    len(self._omebase['TiffData']) != len(
+                        self._omebase.get('Plane', self._omebase['TiffData']))):
+                raise TileSourceException('OME Tiff contains frames that contain multiple planes')
+        except (KeyError, ValueError, IndexError):
+            raise TileSourceException('OME Tiff does not contain an expected record')
+
     def getMetadata(self):
         """
         Return a dictionary of metadata containing levels, sizeX, sizeY,
diff --git a/sources/tiff/large_image_source_tiff/__init__.py b/sources/tiff/large_image_source_tiff/__init__.py
index a4c7a855d..c47ef6cb1 100644
--- a/sources/tiff/large_image_source_tiff/__init__.py
+++ b/sources/tiff/large_image_source_tiff/__init__.py
@@ -65,6 +65,14 @@ class TiffFileTileSource(FileTileSource):
         'image/x-ptif': SourcePriority.PREFERRED,
     }
 
+    # When getting tiles for otherwise empty directories (missing powers of
+    # two), we composite the tile from higher resolution levels.  This can use
+    # excessive memory if there are too many missing levels.  For instance, if
+    # there are six missing levels and the tile size is 1024 square RGBA, then
+    # 16 Gb are needed for the composited tile at a minimum.  By setting
+    # _maxSkippedLevels, such large gaps are composited in stages.
+    _maxSkippedLevels = 3
+
     def __init__(self, path, **kwargs):
         """
         Initialize the tile class.  See the base class for other available
@@ -272,9 +280,13 @@ def getTile(self, x, y, z, pilImageAllowed=False, numpyAllowed=False,
         try:
             allowStyle = True
             if self._tiffDirectories[z] is None:
-                if sparseFallback:
-                    raise IOTiffException('Missing z level %d' % z)
-                tile = self.getTileFromEmptyDirectory(x, y, z, **kwargs)
+                try:
+                    tile = self.getTileFromEmptyDirectory(x, y, z, **kwargs)
+                except Exception:
+                    if sparseFallback:
+                        raise IOTiffException('Missing z level %d' % z)
+                    else:
+                        raise
                 allowStyle = False
                 format = TILE_FORMAT_PIL
             else:
@@ -327,10 +339,14 @@ def getTileFromEmptyDirectory(self, x, y, z, **kwargs):
         :param z: original level.
         :returns: tile in PIL format.
         """
+        basez = z
         scale = 1
         while self._tiffDirectories[z] is None:
             scale *= 2
             z += 1
+        while z - basez > self._maxSkippedLevels:
+            z -= self._maxSkippedLevels
+            scale = int(scale / 2 ** self._maxSkippedLevels)
         tile = PIL.Image.new(
             'RGBA', (self.tileWidth * scale, self.tileHeight * scale))
         maxX = 2.0 ** (z + 1 - self.levels) * self.sizeX / self.tileWidth
@@ -361,8 +377,11 @@ def getPreferredLevel(self, level):
         :returns level: a level with actual data that is no lower resolution.
         """
         level = max(0, min(level, self.levels - 1))
+        baselevel = level
         while self._tiffDirectories[level] is None and level < self.levels - 1:
             level += 1
+        while level - baselevel > self._maxSkippedLevels:
+            level -= self._maxSkippedLevels
         return level
 
     def getAssociatedImagesList(self):
diff --git a/tox.ini b/tox.ini
index f636f4c7f..0c1edc087 100644
--- a/tox.ini
+++ b/tox.ini
@@ -26,6 +26,8 @@ deps =
   pytest-cov>=2.6
   pytest-girder>=3.0.4
   pytest-xdist
+  celery!=4.4.4
+# celery 4.4.4 is broken; avoid it until a new version is released
 whitelist_externals =
   rm
   npx