ConstantWorking Space Algorithms for Image Procesing - PowerPoint PPT Presentation

Title: ConstantWorking Space Algorithms for Image Procesing


1
Constant-Working Space Algorithmsfor Image
Procesing
Colloquium on Emerging Trends in Visual
Computing Paris, November 2008

• Tetsuo Asano
• School of Information Science
• JAIST
• Joint work with
• L. Buzer, S. Bereg, and D. Kirkpatrick

2
What is a constant-working-space algorithm?

• Input data are given on a read-only array.
• Only constant number of working storage cells are
  available.
• Each working storage cell is of O(log n) bits,
  and hence the total working storage size is also
  O(log n).
• A class of problems solved in this framework is
  known as log-space in complexity theory.
• Want to develop efficient algorithms in this
  computational model.

3
Why constant-working-space?

• Working space is limited in applications to
  software embedded in hardware
• e.g. scanner, digital camera, etc.
• In some applications we want to keep input data
  unchanged data on a read-only array.
• Recursion
• working space ? recursion depth
• algorithmic techniques such as controlled
  recursion.

4
What is known?

• Median-finding Given a read-only array of n
  numbers, find their median using constant number
  of variables. Munro-Raman, Theoretical
  Computer Science, 1996.
• st-connectivity in graph Given an undirected
  graph using a read-only array, determine whether
  two arbitrarily given vertices belong to the same
  connected component.
• Reingold, STOC, 2005.
• st-connectivity in image Given a read-only array
  of a binary image, determine whether two
  arbitrarily specified pixels of the same value
  belong to the same connected component.
• Malgouyresa-Moreb, Theoretical Computer
  Science, 2002.

5
st-connectivity in maze
t
s
Is s connected to t?
checked by right-hand rule
6
st-connectivity in maze
removed boundary cells
t
s
t
Is s connected to t?
7
st-connectivity in maze
t
s
t
Is s connected to t?
cannot be checked by right-hand rule
8
st-connectivity in a graph
s
t
Is there any path from s to t?
A graph is given using a read-only array.
9
st-connectivity in a graph
s
t
Is there any path from s to t?
A graph is given using a read-only array.
10
st-connectivity in a binary image
0000000011000111000 0011100000111100011 0111111000
000111111 0110011110001110000 0111110011100111111
t
s
Is there any rectilinear path of white pixels (of
values 1) interconnecting s to t?
11
st-connectivity in a binary image
0000000011000111000 0011100000111100011 0111111000
000111111 0110011110001110000 0111110011100111111
t
s
Is there any rectilinear path of white pixels (of
values 1) interconnecting s to t?
12
Median-finding
1
13
3
5
6
9
11
13
20
25
27
31
35
39
17
sorted seq 3, 5, 6, 9, 11, 13, 17, 20, 25, 27,
31, 35, 39
median
Linear-time algorithm based on prune and
search, Blum, Floyd, Pratt, Rivest, Tarjan, 1972.
How fast can we find the median in the
computational model? O(n) time? O(n log
n) time? W(n2) time?
13
(No Transcript)
14
Algorithm A3
Partition the array into n1/3 blocks instead of
n1/2 blocks. Compute a block median using
Algorithm A2.
Algorithm A3 finds the median in O(n4/3 log2 n)
time using only constant working space.
Algorithm Ak
Partition the array into n1/k blocks. Compute a
block median using Algorithm Ak-1.
Algorithm Ak finds the median in O(n(k1)/k
logk-1 n) time using O(k) working space.
15
Applications to Image Processing
Thresholding Intensity Images Given an
intensity image G, find an appropriate threshold
for G to generate a good-looking binary image.
threshold100
too low
threshold150
intensity image
too high
each threshold is to be evaluated by the
resulting binary image.
The input image should be treated as a read-only
array.
16
intensity image
17
intensity image
threshold 50 100
150 200
18
Ohtsu's Optimal Thresholding
Divide a given histogram into two classes S0, S1
at a threshold t
class S0
class S1
Choose a threshold t so that their difference is
maximized by considering balance of their areas.
19

n(Si) size of Class i (number of pixels) m(Si)
average intensity level of Class i mT average
intensity level of the whole image
Choose a threshold t that minimizes V(t) as an
optimal threshold (Note an optimal threshold can
be computed in linear time)
20
Observation Let n be the number of pixels in a
given image and L be the number of intensity
levels. Given such an image, we can find in O(n
L) time using O(L) space in addition to the
image array an optimal threshold that maximizes
the intercluster distance.
If O(L) working space is not available we rely on
binary search for an optimal threshold.
intercluster distance max
all these values can be computed using
constant working space.
21
Observation Let n be the number of pixels in a
given image and L be the number of intensity
levels. Given such an image, we can find in O(n
log L) time using only constant working space in
addition to the image array an optimal threshold
that maximizes the intercluster distance.
Remark The binary search works only when
intensity levels are given by integers 1..L.
What happens if all intensity levels are real
numbers?
Median intensity level instead of Ohtsu's
threshold. O(n1e) time algorithm using O(1/e)
working space.
22
Experimental results
23
Histogram Computation
Problem Let G be a monochrome image containing n
pixels, and L be the number of intensity levels.
Given such an image, report the frequency of each
intensity level. histogram computation
Easy solution If O(L) working storage is
available, then it is done in O(n)
time. What about the case of constant working
space? O(k) working storage
gt O(nL/k) time using O(k) working space
inefficient if L gtgt n.
24
Constant working space algorithm
for histogram computation
Let Q be a priority queue keeping O(k) values
together with their frequencies in the decreasing
order of their keys.
T positive infinity do for each pixel (x,
y) in a given image if its the
intensity level p(x, y) is less than T
then put it into Q report the content of
the queue Q T the smallest key value in
Q while(Q is not empty)
This algorithm runs in O(n2/k log k) time using
O(k) space.
25
Connected Components Counting
What is a connected component?
M an nxn binary image (matrix) M(x, y) 0
or 1 Two 1-pixels are connected if there is a
rectilinear sequence of 1-pixels between
them. A connected component is a maximal set of
1-pixels any two of which are connected.
26
Theorem There is a constant working space
algorithm for connected components labeling in
linear time which put labels onto the input image
without using any other working storage.
linear-time in-place algorithm for connected
components labeling
linear-time in-place algorithm for counting
connected components
What about the case when input image is given as
a read-only array?
27
Key Idea for Counting
Basic assumption Component boundary is directed
so that the interior always lies to the
left of the boundary External boundary
counter-clockwise order Internal boundary
clockwise order
28
Key Idea for Counting
Basic assumption Component boundary is directed
so that the interior always lies to the
left of the boundary External boundary
counter-clockwise order Internal boundary
clockwise order
Canonical edge for each boundary the
lowest downward edge
29
Each boundary (internal/external) has a unique
canonical edge, which is the lowest downward
edge on the boundary.
30
Checking orientation
So easy to check the orientation!!
31
Algorithm for Connected Components
Counting counter 0. Perform Raster Scan
whenever we find a candidate downward edge e.
Apply Boundary_trace_from(e). if it returns 1
then increment the count Report the count.
0 1 ? 0
Boundary_trace_from(e) es e. do e
next edge of e on the same
boundary. if e is downward and lower
than es then return 0. while(e ! es)
return 1.
32
Theorem The algorithm Connected-Components-Counti
ng correctly computes the number of connected
components in a given nxn binary image in O(n4)
time using only constant amount of working
storage.
worst-case example
33
How to improve the time complexity
bidirectional search whenever we find a
candidate edge e, search on the boundary in two
opposite directions for any lower edge.
Now, it is an easy case.
34
How to improve the time complexity
bidirectional search whenever we find a
candidate edge e, search on the boundary in two
opposite directions for any lower edge.
worst-case example which requires O(n2log n)
35
Enumeration of all components together with their
areas
Given a binary image G, enumerate all connected
components together with their areas (number of
pixels) each exactly once.
input
output component 1 21 component 2 3
component 3 7 component 4 15
Question Can we obtain such a list in a
read-only model?
36
Enumeration of all components together with their
areas
Given a binary image G, enumerate all connected
components together with their areas (number of
pixels) each exactly once.
There is a constant-working-space algorithm for
enumerating all components together with their
areas for a read-only array for a binary image
which runs in O(n2 log n) time.
37
Pattern Matching
Given two binary images G1 and G2, determine
whether a pattern in G1 appears in G2.
pattern G1
given image G2
38
1
1
1
0
0
1
1
0
holes, islands, island hole canonical edges,
candidates
detecting an internal boundary
search within the external boundary
search within the internal boundary
39
search within the external boundary
search OUTSIDE the external boundary
detection of an internal boundary and then
search within it
search within the external boundary
40
detection of an internal boundary and then
search within it
completing the search on the external boundary,
and then returning to the previous boundary
continue the search detecting a new boundary
41
Conclusions and Future Works

• Extend the algorithm for geometric problems such
  as planar subdivisions or graphs.
• Establish some non-linear lower bound on the time
  complexity for constant-working-space algorithms
• element uniqueness, sorting, largest
  gap?
• Devise more algorithmic paradigms for
  constant-working-space algorithms

42
Thank you!
Kenrokuen Garden, Kanazawa, Japan