THE
WATERSHED
SEGMENTATION
NADINE GARAISY
1
GENERAL DEFINITION
A drainage basin or watershed
is an extent or an area of land
where surface
water from rain melting snow
or ice converges to a single
point at a lower elevation,
usually the exit of the basin,
where the waters join another
waterbody, such as
a river, lake, wetland, sea,
or ocean
2
INTRODUCTION
The Watershed transformation is
a powerful tool for image
segmentation, it uses the regionbased approach and searches for
pixel and region similarities.
The watershed concept was first
applied by Beucher and Lantuejoul
at 1979, they used it to segment
images of bubbles and SEM
metallographic pictures
3
IMAGE REPRESENTATION
We will represent a gray-tone image by a function:
𝑓: ℤ2 → ℤ
𝑓(𝑥) is the gray value of the image at point 𝑥
 A section of 𝑓 at level 𝑖 is a set 𝑋𝑖 (𝑓) defined as:
𝑋𝑖 𝑓 = 𝑥 ∈ ℤ2 : 𝑓 𝑥 ≥ 𝑖
And in the same way we define 𝑍𝑖 (𝑓) as:
𝑍𝑖 𝑓 = 𝑥 ∈ ℤ2 : 𝑓 𝑥 ≤ 𝑖
⟺ 𝑋𝑖 𝑓 =𝑍 𝑐 𝑖+1 𝑓
4
REMINDER-IMAGE GRADIENT
An image gradient is a
directional change in
the intensity or color
in an image. Image
gradients may be used
to extract information
from images.
5
IMAGE GRADIENT
an intensity image
a gradient image in the x
a gradient image in the y
direction measuring
direction measuring vertical
horizontal change in intensity
change in intensity
6
IMAGE GRADIENT
The morphological gradient of a picture is
defined as
𝑔 𝑓 = 𝑓⨁𝐵 − 𝑓 ⊝ 𝐵
Where 𝑓⨁𝐵 is the dilation of 𝑓 and 𝑓 ⊝ 𝐵 is
its erosion.
But because 𝑓 is continuously differentiable,
𝑔 𝑥 is nothing more than the modulus of the
gradient of 𝑓:
𝑔 𝑓 = 𝑔𝑟𝑎𝑑 𝑓 𝑥
=
𝜕𝑓
𝜕𝑥
2
𝜕𝑓
+
𝜕𝑦
2 1/2
7
GEODESIC DISTANCE
For two points 𝑥, 𝑦 𝜖𝑋 when 𝑋 ⊂ ℤ2
we define the geodesic distance
𝑑𝑋 (𝑥, 𝑦) as the length of the shortest
path (if any) included in 𝑋 and linking
𝑥 and 𝑦.
Let 𝑌 be any set included in 𝑋, then:
𝑅𝑋 𝑌 = {𝑥 ∈ 𝑋: ∃𝑦 ∈ 𝑌, 𝑑𝑋 𝑥, 𝑦 𝑓𝑖𝑛𝑖𝑡𝑒}
is the set of all points of 𝑋 that are at a
finite geodesic distance from 𝑌.
8
GEODESIC ZONE OF
INFLUENCE
The geodesic zone of influence of 𝑌𝑖 (when 𝑌 is composed
of 𝑛 connected components 𝑌𝑖 ) is the set of points
in 𝑋 whose finite distance is closest to 𝑌𝑖 (among all 𝑌
components)
𝑔𝑧𝑋 𝑌𝑖 = {𝑥 ∈ 𝑋: 𝑑𝑋 𝑥, 𝑌𝑖 finite
and ∀ 𝑗 ≠ 𝑖, 𝑑𝑋 𝑥, 𝑌𝑖 < 𝑑𝑋 𝑥, 𝑌𝑗 }
9
GEODESIC SKELETON BY ZONES
OF INFLUENCE
 The boundaries between
the various zones of
influence give the geodesic
skeleton by Zones of
influence of 𝑌 in 𝑋.
𝐼𝑍𝑋 𝑌 =
𝑔𝑧𝑋 (𝑌𝑖 )
𝑖
𝑆𝐾𝐼𝑍𝑋 𝑌 = 𝑋/𝐼𝑍𝑋 (𝑌)
10
MINIMA AND MAXIMA
The set of points in the
function 𝑓 can be seen as
topographic surface 𝑆, The
lighter the gray value of the
function at the point 𝑥 the
higher the altitude of the
corresponding point on the
surface
11
MINIMA AND MAXIMA
 An ascending path is a sequence {𝑠1 , 𝑠2 , . . } On the surface
such that:
∀𝑠𝑖 𝑥𝑖 , 𝑓 𝑥𝑖 , 𝑠𝑗 𝑥𝑗 , 𝑓 𝑥𝑗
𝑖 ≥ 𝑗 ⟺ 𝑓 𝑥𝑖 ≥ 𝑓(𝑥𝑗 )
 A point 𝑠 belongs to a minimum if there is a no ascending
path starting from 𝑠. It can be considered as a sink of the
topographic surface (see next slide). The set 𝑀 of all the
minima of 𝑓 is made of various connected components
𝑀𝑖 (𝑓)
12
ASCENDING PATH
13
NON-ASCENDING PATH
14
THE WATERSHED
TRANSFORMATION
If we look at the image 𝑓 as a topographic surface, imagine
that we pierce each 𝑀𝑖 (𝑓) of the topographic surface 𝑆
and then we plunge this surface into a lake, the water
entering through the holes floods the surface and if two or
more floods coming from different minima attempt to
merge, we avoid this event by building a dam on the
points of the surface where the floods would merge.
At the end of the process only these dams will emerge
and this is what define the watershed of the function 𝑓
15
THE WATERSHED
TRANSFORMATION
16
THE WATERSHED
TRANSFORMATION
http://cmm.ensmp.fr/~beucher/lpe1.gif
17
BUILDING THE WATERSHED
Suppose the flood of the surface has reached the section
𝑍𝑖 (𝑓), when it continue and reach 𝑍𝑖+1 𝑓 the flooding is
performed in the zones of influence 𝐼𝑍𝑍𝑖+1
𝑓
𝑋𝑖 𝑓 .
The components of 𝑍𝑖+1 𝑓 which are not reached by the
flood are the minima at this level and must be added to the
flooded area
18
BUILDING THE WATERSHED
19
BUILDING THE WATERSHED
If we define 𝑊𝑖 𝑓 as the catchment basins of 𝑓 at level 𝑖
and 𝑀𝑖 (𝑓) as the minima of 𝑓 at height 𝑖 + 1 then:
𝑊𝑖+1 𝑓 = 𝐼𝑍𝑍𝑖+1
𝑓
𝑋𝑖 𝑓
𝑀𝑖+1 = 𝑍𝑖+1 (𝑓)/𝑅𝑍𝑖+1
∪ 𝑀𝑖+1 𝑓
𝑓
(𝑍𝑖 𝑓 )
• The initiation of this iterative algorithm is 𝑊−1 𝑓 =⊘
• In the end the watershed line is 𝐷𝐿 𝑓 = 𝑊 𝑐 𝑁 𝑓 when
𝑁 = max(𝑓)
 Visual illustration
20
OVER-SEGMENTATION
PROBLEM
Unfortunately, most times the real watershed transform of the
gradient present many catchment basins, Each one corresponds
to a minimum of the gradient that is produced by small
variations, mainly due to noise.
21
OVER-SEGMENTATION:
SOLUTION
The over-segmentation
could be reduced by
appropriate filtering, but
the best results is obtained
by marking the patterns to
be segmented before
preforming the watershed
transformation of the
gradient.
22
OVER-SEGMENTATION:
SOLUTION
FIRST: we mark each blob
of protein of the original
image (by extracting the
minima of the image
function)
23
OVER-SEGMENTATION:
SOLUTION
SECOND: by applying the
watershed on the initial
image we can mark the
background with
connected marker
surrounding the blobs
 We define these two
steps as marker set 𝑀
24
HOMOTOPY MODIFICATION
The first two steps of this
algorithm can be done by
modifying the gradient
function to a new wery similar
function 𝑔′ , the difference
between the two is that in 𝑔′
the initial minima are replaced
by the set 𝑀, this modification
is called homotopy
modification
25
OVER-SEGMENTATION:
SOLUTION
Now we look at the final
result of the marking as a
topographic surface, but in
the flooding process
instead of piercing the
minima, we only make
holes through the
components of the marker
set that we produced
The initial image marked with the
set 𝑀
26
OVER-SEGMENTATION:
SOLUTION
This way the flooding will
produce as many
catchment basins as there
are markers in 𝑀, this way
the watershed lines of the
contours of the objects
will be on the crest lines
of this topographic
surface
27
OVER-SEGMENTATION:
SOLUTION
The algorithm for this solution is as follows:
𝑊𝑖 (𝑔) – section at level 𝑖 of the new catchment basins of 𝑔
Then:
𝑊𝑖+1 (𝑔) = 𝐼𝑍𝑍𝑖+1 ∪𝑀 (𝑊𝑖 𝑔 )
Initialization:
𝑊−1 𝑔 = 𝑀
28
OVERLAPING GRAINS
 In some cases we have an
image with overlapping
figures, that we need to
segment, in order to do that
we need to point out the
overlapping regions.
 For example the figure here is
a TEM (transmission electron
microscopy) image of grains
of silver nitrate scattered on a
photographic plate.
29
OVERLAPING GRAINS
To point out the
overlapping regions we
first threshold the initial
image to a binary image
with only two gray
values
30
REMINDER: DISTANCE
FUNCTION
the distance function of an image assigns for each pixel a
number that is the Euclidean distance between that pixel
and the nearest nonzero pixel.
For example: suppose we have this image matrix0 0 0
0 1 0
0 0 1
Then the distance matrix will be1.4142 1 1.4142
1
0
1
1.4142 1
0
31
OVERLAPING GRAINS
By calculation the maxima
of the distance function of
the binary image we can
provide the markers of
the grains
32
OVERLAPING GRAINS
The markers of the
overlapping regions are
obtained by executing the
watershed transformation
of the inverted distance
function −𝑑 𝑥 , it will
produce divide lines which
will cut the overlapping
grains, that way we can
mark them.
33
OVERLAPING GRAINS
Finally after marking the
background and
calculation the gradient
function we run the
homotopy modification
and the watershed
construction are
preformed
34
THE SEGMENTATION
PARADIGM
The segmentation process is divided into two steps:
I. Finding the markers and
the segmentation.
II. Performing a markercontrolled watershed
with these two
elements
35
WATERSHED TRANSFOTMATION
PROCESS
Source: A gray scale
image
Step 1: Use the Gradient
Magnitude as the
Segmentation Function The gradient is high at
the borders of the objects
and low (mostly) inside
the objects.
FROM - WWW.MATHWORKS.COM
Step 2: Mark the
foreground objects
36
WATERSHED TRANSFOTMATION
PROCESS
Step 3: computing the
opening-byreconstruction of the
image
Step 4: Following the
opening with a closing
can remove the dark
spots and stem marks.
FROM - WWW.MATHWORKS.COM
Step 5: Calculate the
regional maxima to
obtain good
foreground markers.
37
WATERSHED TRANSFOTMATION
PROCESS
Step 6: Superimpose the
foreground marker
image on the original
image, Notice that the
foreground markers in
some objects go right up
to the objects' edge
Step 7: cleaning the
edges of the marker
blobs and then
shrinking them a bit
FROM - WWW.MATHWORKS.COM
Step 8: Compute
Background Markers,
Starting with
thresholding operation
38
WATERSHED TRANSFOTMATION
PROCESS
Step 9: Compute Background
Markers, using the watershed
transform of the distance
transform and then looking
for the watershed ridge lines
of the result
Step 10: Visualize the Result,
one of the techniques is to
superimpose the foreground
markers, background markers,
and segmented object
boundaries.
FROM - WWW.MATHWORKS.COM
39
WATERSHED TRANSFOTMATION
PROCESS – ADVANCE OPTIONS
*Another useful visualization
technique is to display the
label matrix as a color image
We can use transparency to
superimpose this pseudocolor label matrix on top of
the original intensity image.
FROM - WWW.MATHWORKS.COM
40
ROAD SEGMENTATION
In this study they use the
watershed algorithm among
others to extract vehicle
position on the road and
possible obstacles ahead.
The algorithms have been
tested on a small database
representing different
driving situations.
41
ROAD SEGMENTATION
The original road image
The morphological
gradient image
42
ROAD SEGMENTATION
Due to noise and
inhomogeneities in the
gradient image, the
watershed will produce a
lot of minima which leads
to over-segmentation of
the image
43
ROAD SEGMENTATION
We can enhance the
watershed on the
gradient image by
modifying the gradient
function by defining
new markers which will
be imposed as the new
minima.
44
ROAD SEGMENTATION
The difference between watershed on simple gradient and
watershed on the gradient after modifying using the
regularized gradient
45
ROAD SEGMENTATION
Then by selecting the
catchment basin located at
the front of the vehicle we
can extract a coarse marker
of the road.
After smoothing this marker
we define it as 𝑀1
46
ROAD SEGMENTATION
Then we build an outer
marker to mark the region
of the image which do not
belong to the road
This marker is defined by
𝑀2
47
ROAD SEGMENTATION
Using 𝑀1 and 𝑀2 we
modify the gradient
which now contain
only two minima and
the divide lines are
the contours of the
road
48
ROAD SEGMENTATION
To obtain the road markers we do a
simplification on the image using its
gradient, the result is an image made
of catchment basins tiles of constant
gray values- this image is called the
mosaic-image.
The gradient of this image will be
null everywhere except on the divide
lines where it will be equal to the
absolute difference of the gray-tone
values of the to catchment basins.
49
ROAD SEGMENTATION
Watershed of the mosaic-image points out only the regions
surrounded by higher contrast edges, and we can still extract a
marker for the road
50
ROAD SEGMENTATION
The result – Road
borders,
corresponding to the
watershed of the
modified gradient
image
51
LANE BY LANE ROAD
SEGMENTATION
Original
image
Mosaicimage
Watershed
of mosaicimage
Lanes
Final result
markers
enhancement
52
POTENTIAL OBSTACLES
DETECTION
The second part of this
study was identifying
obstacles on the road, but
this detection is useless
without a good definition of
the nature of the obstacles,
the problem in this part was
distinguishing a dangerous
obstacle from a light
variation in intensity due,
for instance to a shading.
Black marker- the edges of the road
White marker- obstacles-free zone
53
POTENTIAL OBSTACLES
DETECTION
Difficulties in this segmentation:
false detection due to the
shadows, because they are
considered as obstacles, this
can be solved if given
complementary information by
telemetry or stereovision
54
VISUAL EXAMPLES
 Illustration of watershed road segmentation:
https://www.youtube.com/watch?v=Tibi6a_aeeE
 Road Detection Using Region Growing and
Segmentation:
https://www.youtube.com/watch?v=ADdkfE_J4a0
55
REFERENCES
THE WATERSHED TRANSFORMATION APPLIED TO IMAGE
SEGMENTATION – S.Beucher
ROAD SEGMENTATION BY WATERSHEDS ALGORITHEMS –
S.Beucher, M.Billodeau and X.Yu
USE OF WATERSHEDS IN CONTOUR DETECTION- S.Beucher
and C.Lantuejoul
MATHWORKS.COM
WIKIPEDIA
56
57
TOPOGRAPHIC MAP
58