ELSEVIER Image and Vision Computing 14 (1996) 297-304

Short wmmunication

Alternative strategies for irregular pyramid construction

Horace H.S. Ip*, Stephen W.C. Lam

Image Computing Group, Department of Computer Science, City Polytechnic of Hong Kong, Tat Chee Avenue, Kowloon. Hong Kong

Received 19 October 1994; revised 10 July 1995

Abstract

In this paper, several alternative strategies for constructing irregular pyramids are proposed, namely, specified-rate sampling, prioritized sampling and progressive edge formation. When compared with the original scheme, these alternative strategies have shown to yield respectively pyramids in which the decimation ratio can be controlled by the application; with reduced height and pyramids which contains less nodes and edges in constructing the constituent graphs.

Keywords: Specified-rate sampling; Prioritized sampling; Progressive edge formation

1. Introduction

Pyramids are data structures that provide successively condensed information of an image. The advantages of the pyramid have been stated as reducing the compu-

tation cost of various image operations, converting global image features into local features and permitting local interactions between features that are far apart in the original image [l]. Gaussian [1] and Laplacian pyramids [2] are useful for multi-resolution image analy- sis and coding. More recently, a fuzzy pyramid has been applied in segmenting mammograms [3]. Traditionally, pyramid architecture construction is based on a rigid local sampling scheme. As stated in [4], processing in an image pyramid is parallel but not strictly local. The reason is that the next level in an image pyramid is obtained by computing the weighted averages only around addresses of the form (j x q, k x q) j, k = 0, 1, 2 . . where j, k are for indexing each cell on a two dimension array (i.e. a layer of the pyramid). q is an offset for locating the local neighbors of an individual cell. Hence, to construct a pyramid layer, we have to know the absolute positions of each cell of the lower layer. This information is global as all the cells are related to the origin of the array.

In [4], a new class of image pyramids, called stochastic image pyramids is proposed. Under this scheme, the decimation process only makes use of locally available

* Email: cship@?cphkvx.bitnet

0262~8856/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDZ 0262-8856(95)01050-5

information. A large decimation ratio is obtained using this pyramid construction scheme, and so the pyramid can be constructed in log(image_size) time [4]. The quad- tree structure obtained in a traditional pyramid structure is shift dependent; using the graph theoretic approach, though this limitation is removed. This pyramid struc- ture has been applied in synthesizing texture [S]. Similar pyramid structures have been proposed for the segmen- tation of two-dimensional grey-level images [6,7] based on grey-level homogeneity and segmenting structural texture [8]. Also, the graph-theoretic properties of dual irregular pyramid have been investigated [9]. In this paper, we propose alternative ways for improving the irregular pyramid construction process.

In the next section, an overview of the original pyra- mid construction algorithm is presented. When employ- ing sampling scheme, the sampling rate (i.e. number of survivors divided by the total number of nodes, which is the inverse of decimation ratio) is fixed. Also, due to the configuration of the graph, the sampling rate may be different in different regions of the graph. Hence, we investigate a scheme in which the sampling rate is speci- fied by the application. Furthermore, as the survivor selection scheme is not biased towards fast decimation, a prioritized sampling scheme that can reduce the pyramid height is proposed. The prioritized sampling approach aims at reducing the height of the constructed irregular pyramid. It gives priority to those nodes which have a larger number of neighboring nodes (i.e. larger degree) for being selected as survivors. In cases of applying the

298 H.H.S. Ip. S. W.C. Lam/Image and Vision Computing 14 (1996) 297-304

pyramid to segmentation or labeling, a reduction in height of the pyramid should increase the speed of the process by reducing the amount of time needed for pyramid con- struction.’ Also, a new edge formation scheme is designed which can reduce the number of edges of the graphs that needed to be processed in an irregular pyra- mid. When constructing an irregular pyramid for tasks like labeling or segmentation, non-survivors are required to link to the most similar survivor. Hence the degrees of the nodes is a critical factor for the computation load of each node during the pyramid construction process. Therefore, reducing the number of edges required to build a pyramid is crucial for increasing the speed of pyr- amid construction. Finally, we apply the last two strate- gies to labeling and segmentation. The results are shown in a later section.

2. Irregular pyramid construction

Following the convention used in [7], the formalism of graph theory is employed to describe the pyramid construction algorithm. The cells on level I of the pyra- mid are taken as the vertices of an undirected graph G[Z]. The edges of the graph describe the adjacency relations between cells at level 1. Thus, G[Z] = (V[Z], E[Z]), where V[Z] is the set of vertices and E[Z] is the set of edges. The graph G[O] is defined by the 8-connected square sampl- ing grid on level 0. The pyramid is constructed by a sampling or decimation process. Each level is con- structed from the level below it by selecting a subset of the vertices.

The derivation of G[Z + l] from the G[Z] begins by defin- ing the vertices of the new graph. Since Y [Z + l] c V [I], pyramid construction is a decimation process, i.e. only a subset of vertices V[Z] are retained. As stated in [7’l, a probabilistic decimation algorithm is employed which performs graph contraction that satisfies the following constraints:

(a) Any non-survivor cell at level Z has at least one survivor in the neighborhood, thus it can be allocated to a parent by a local decision.

(b) Two adjacent cells on level Z cannot both survive, and thus the number of vertices must decrease rapidly from level to level.

Let cell co on level Z have r neighbors, i.e.

{Cl 7 c27 . . . , c,} is the set of neighboring cells of co and cl with i E {1,2, .., r} represents individual neighboring cell. The decimation algorithm employs three variables

’ Multi-resolution scene interpretation favors increasing the height of a pyramid. The reason is that it is easy to find a suitable resolution

for extracting‘ objects. However, for applications like segmentation or

labeling, the height of the pyramid should be reduced for speeding up

the process.

for every cell: two binary state variables p and q, and a random variable x uniformly distributed between [O,l]. ‘The survivors are chosen by an iterative local process. Letk=O, 1, . . . by the iteration index. A cell co survives if, at the end of the algorithm, its pa(k) state variable has the value 1 (pi(k) re p resents the state variable p of ci at iteration k). Initially, the state variables p of all the cells are set to 0. Every iteration has two steps. First q,(k) is updated based on the states pi(k - 1) of neighboring cells:

qo(k)=l ifpi(k-l)=O V’i=O,..,r

qo(k) = 0 otherwise (1)

In other words, go(k) becomes 1 if and only if there is no survivor in the neighborhood of co. Note that the neigh- borhood includes the cell itself. The initial conditions always yield qo( 1) = 1. Then PO(k) is computed on the updated values of qi(k):

pa(k) = 1 if qo(k)xo(k)

= max i=o,..,r (qi(k)xi(k)) > 0

PO(k) = po(k - 1) otherwise (2)

To become a survivor the outcome of the random variable x drawn by the cell must be the local maximum among the outcomes drawn by the neighbors. Note that only those neighbors which do not already have a sur- vivor adjacent to them (i.e. qi(k) = 1) are taken into account. The local maximum property assures that con- straint (b) is always satisfied. The state of a survivor is not reversible. Once a cell is labeled po(k - 1) = 1, at subsequent iterations the product qo(k)xo(k) in Eq. (2) is always 0 by definition of qo(k) in Eq. (1). Thus, the second condition in Eq. (2), which preserves the current state, is used. It has been shown [4] that, after a finite number of iterations (at most five, for the experiments reported there), the algorithm reaches a final global configuration in which the survivors satisfy constraints (a) as well. An example of the survivor extraction process is shown in Fig. 1.

As stated in [5], the edge formation scheme includes the following steps:

Assume node i has been selected as a survivor:

1. All the vertices of the graph sharing an edge with node i are analysed. If a non-survivor is not claimed by any of the other survivors it is incorporated into node i. If more than one surviving node competes for a non-survivor, the non-survivor allocates itself to the survivor with the largest outcome x. When applying irregular pyramid construction to labeling or segmentation, a non-survivor is incorporated into the survivor with the closest grey level value.

2. Node i tries to expand from the vertices already incor- porated into it. The expansion is stopped whenever a survivor or an already-allocated non-survivor is met.

H.H.S. Ip. S. W.C. Lam/Image and Vision Computing 14 (1996) 297-304 299

l 0.000 a0 bl b0 b3 c5 c3

-

0000.0

al a2 b2 cl CO ~2

000000

d0 dl el e2 c4 f0

000.0.

d2 d3 e3 e4 e5 fl

(I.4

a

d e f

(c)

Fig. 1. Example of an irregular tessellation hierarchy. (a) The eight- connected square lattice as a graph; (b) arbitrary partition of the graph (0: survivors retained for the next level; 0: non-survivors): (c) graph of the next level.

For convenience, we call the set of nodes occupied by a survivor, including the survivor itself, ‘territory.’ The edges of the pyramid at layer I + 1 can now be defined based on the adjacency of the territories at layer 1. If two territories are neighbors at layer I, the two nodes at layer 1+ 1 corresponding to their center nodes (i.e. the two expanding survivors at layer 1) are linked together by an edge. An example of the edge formation process is shown in Fig. 1. After this edge formation process, the son-to-father linkages can also be defined. As survivors at layer 1 are duplicated to layer I+ 1, they are simply linked to their corresponding vertices at layer I + 1. For those non-survivors, they are linked to the vertices corresponding to the survivors into which they are incor- porated during the survivor expanding process carried out at layer 1. When applying irregular pyramid construc- tion to labeling or segmentation, the feature value of a father node is the weighted average of the son nodes. The weight is set to the area represented by a son node.

A large decimation ratio is obtained using this pyra- mid construction scheme, and so the pyramid can be constructed in log(image_size) time. The quadtree struc- ture obtained in a traditional pyramid structure is shift dependent but using the graph theoretic approach this limitation is removed. Despite these advantages, further improvements in terms of construction efficiency can be obtained if the scheme is modified in some ways. The following sections describe these modifications.

3. Specified rate sampling

When employing the above sampling scheme, the sampling rate (i.e. number of survivors divided by total number of nodes, which is the inverse of decimation ratio) is fixed. Also, due to the configuration of the graph, the sampling rate may be different in different regions of the graph. This is because different regions of the graph consist of nodes having different degrees. Hence, regions where the nodes have a smaller degree on average can- accommodate more survivors than other regions. The advantages of specified rate sampling are that (a) it allows the decimation ratio to be specified by the user, and (b) it removes the bias in sampling rate across different regions of the graph. To implement specified rate sampling, a node is selected as a survivor based on a fixed probability. The iterative survivor selec- tion scheme is replaced by the following single step procedure. Initially, the same as the original algorithm, the state variables p of all the cells are set to 0. Then

p. = 1 if x0 < in, where w < 1 is the sampling rate (3)

A cell co survives if, after this step, its p. state variable has the value 1. Note that the iteration indicator k is omitted because this is not an iterative process. Also, the state variable q is not used. The sampling rate w can easily be verified as nw is the expected number of nodes extracted as survivors given that the graph consists of n nodes. The decimation ratio is 1 /w, which is equal to the average number of cells of an individual support. The pyramid can be constructed in logi,w. (image-size) time.

The edge formation scheme of this sampling scheme is the same as that of the original scheme. However, the expanding cells may collide in more than two steps. Assume a survivor occupies a disk like region with radius r (i.e. expanding at the same rate in all directions), then the area occupied by a node is rr2. As the average number of cells of an individual support is l/w, we have 7r~’ = l/w. The average number of steps in which the cells may collide can be estimated as ~wT. It is obvious that as we try to do sampling at a specified rate, both constraint (a) and (b) may be violated. That is, any non-survivor can have no survivor in the neighborhood, and two neighboring nodes can be selected as survivors.

As mentioned before, if different regions of the graph consist of nodes having different degrees, using the original scheme the sampling rates of these regions are different. To illustrate this, the graph is divided into two regions. One corresponds to the boundary region and the other to the non-boundary region.

The sampling rates of the two regions using _ the original sampling scheme are also shown in Table 1 for comparison. For instance, in the case of base dimen- sion 128 x 128, the sampling rate of the non-boundary region is 0.19 and the boundary region is 0.30. The difference between the two sampling rates is 0.11. For

300 H.H.S. Ip, S. W.C. Lam/Image and Vision Computing 14 (1996) 297-304

Table 1

Actual sampling rate of the specified rate sampling scheme. Original sampling scheme data is shown in the last row for comparison

Specified sampling

rate

0.2

0.4

0.6

0.8

Original

32 x 32

Non-boundary

0.20

0.39

0.59

0.80

0.18

Boundary

0.21

0.41

0.61

0.81

0.29

64 x 64

Non-boundary

0.20

0.40

0.60

0.80

0.19

Boundary

0.19

0.40

0.60

0.78

0.28

128 x 128

Non-boundary

0.20

0.40

0.60

0.80

0.19

Boundary

0.18

0.38

0.59

0.79

0.30

256 x 256

Non-boundary

0.20

0.40

0.60

0.80

0.19

Boundary

0.20

0.40

0.61

0.81

0.28

our specified sampling rate scheme, the difference in the sampling rates is 0.02 at maximum. Similar measures are obtained in cases of other base dimensions. Hence, the difference in the sampling rate in the boundary and non- boundary regions is eliminated to a large extent. Also, as the value of the specified sampling rate nearly equals the actual sampling rate, the sampling rate can be controlled by employing our scheme. For the original sampling scheme, the overall sampling rate of GIO] lies between

&in and &,x9 where Rmin = l/9 and R,,, = l/4, as shown in Fig. 2. Rmin occurs when all the non-survivors have just one survivor as a neighbor (except those situated near the boundary, see Fig. 2(a)), while R,, occurs when the graph attains the highest spatial density of the survivor (see Fig. 2(b)). Intuitively the minimum and maximum sampling rates for the non-boundary region also lie between Rgn and R,,,. However, in the boundary region, the minimum sampling rate Bmin and the maximum sampling rate B,,,,, are Bmin = 0 and B - l/4. Bti, refers to the case where no node at max - the boundary is selected as a survivor, and B,, happens when the boundary attains the highest spatial density of the survivor.

4. Prioritized sampling

One way in which to increase the decimation ratio is to allow some nodes to have a higher priority over others in being selected as survivors. Clearly, these nodes are those which are connected to more neighbors (i.e. larger degree) in comparison with their competitors. The reason is that, by selecting those nodes as survivors, a

Fig. 2. Situations in which the minimum and maximum sampling

rates occur. (a) Minimum sampling rate; (b) maximum sampling rate.

(a)/(b) happens when the graph attains the lowest/highest spatial

density of a survivor. W: survivors; 0: non-survivors.

larger number of nodes will become non-survivors. It follows that the number of survivors obtained will be smaller than the original selection mechanism.

To introduce priority in the sampling scheme, a ranking approach is adopted. The range [O,l] is divided into n sub-ranges. The value n is an estimated constant indicating the maximum degree of a node. A random value x’ is generated in the range l/n. The random vari- able x associated with a node is then set by

x=x’+(r- 1) x (l/n) ifr<n

X = x’ + (rr - 1)/n if ran, (4

where r is the degree of the node

Hence, for any two neighboring nodes with different degrees, the one which has a larger degree will be assigned a higher priority in being chosen as a survivor. For those neighboring nodes having the same degree, they fall in the same rank. Survivors are selected randomly from among them using the original local maximum mechanism defined in Eq. (2). As stated in [4], the probability P(i) of a node i being selected as a survivor with r neighbors using the local maximum survivor selection mechanism is calculated by

P(i) = l/(r + 1) (5)

Employing the priority sampling scheme, P(i) is calcu- lated as

P(i) = l/(r’ + 1) if i has the largest number of neighbors among its neighbors

P(i) = 0 otherwise

where r’ is the number of neighbors of i which have the same number of neighbors as i. It is obvious that, in general, r’ d r.

Table 2 shows the effect of employing this prioritized scheme for pyramid construction. Compared with the original sampling scheme, this scheme achieved a reduc- tion of the pyramid height.

4. Progressive edge formation

In this section we introduce a new edge formation scheme that can lead to a smaller number of edges used

Tab

le

2

Ave

rage

he

ight

of

the

py

ram

id

gene

rate

d by

th

e or

igin

al

and

prio

ritiz

ed

sam

plin

g sc

hem

es.

The

da

ta

show

n in

the

ta

ble

are

obta

ined

by

ta

king

th

e av

erag

e of

con

stru

ctin

g 10

0 py

ram

ids

Bas

e si

ze

32 x

32

Bas

e si

ze

64 x

64

Bas

e si

ze

128

x 12

8 B

ase

size

256

x

256

Ori

gina

l Pr

iori

tized

sam

plin

g

sche

me

Dif

fere

nce

Ori

gina

l Pr

iori

tized

sam

plin

g

sche

me

Dif

fere

nce

Ori

gina

l Pr

iori

tized

sam

plin

g

sche

me

Dif

fere

nce

Ori

gina

l Pr

iori

tized

sam

plin

g

sche

me

Dif

fere

nce

Ave

rage

heig

ht

Stan

dard

devi

atio

n

6.87

6.

02

0.85

7.

97

6.98

0.

99

9.47

8.

02

1.45

10

.6

8.8

1.8

0.62

0.

14

0.97

0.

19

1.12

0.

24

1.15

0.

37

302 H.H.S. Ip. S. W.C. Lam/Image and Vision Computing 14 (1996) 297-304

Table 3 Total number of edges of the original irregular pyramid construction process and the new graph formation scheme. X represents the number of edges of the whole pyramid excluding the base of the pyramid (i.e. G[O]). The number of edges shown in this table is obtained by taking the average of 100 pyramids

Base size 32 x 32 Base size 64 x 64 Base size 128 x 128 Base size 256 x 256

Original New % Original New % Original New % Original New % scheme difference scheme difference scheme difference scheme difference

X 139.6 634.8 14.2 3047.7 2600.0 14.7 12320.2 10623.5 13.8 49378.5 42521.1 13.9

to build an irregular pyramid. To reduce the number of edges required for building the pyramid, we propose a progressive edge formation scheme for which, in each processing cycle, each survivor tries to link to one of its neighboring survivors. The definition of a neighborhood among the survivors follows the original algorithm, which has been stated previously. In general, a survivor m links to another survivor n through an edge if m cannot find a path that connects it to n through a previously connected neighboring survivor.

Let s{f$]} = {s~,s: , . .sf,} be the set of neighbor- ing survivors of a survivor ci[Z], where s i - x > s&x> . . . > si - x. The neighborhood relation is defined as in the original algorithm, and has been stated pre- viously. Let a tf b denotes the edge joining nodes a and b. The edge formation scheme is a local repetitive process which, in each processing cycle, executes the following procedure. Each node ci[Z] examines the set of neighboring survivors starting from S; to sk. During the examination, either cases (i) or (ii) can happen:

(i) No edge is formed between cilZ] and ~6. (ii) An edge is already formed between ci[Z] and ~6.

Note that case (ii) refers to the situation in which an edge is formed by another survivor. The following rules govern the edge formation process: (a) Fpr case (i), generate ci[Z] +-+ sk if there does not exist

sb * sI, where q < p.

(b) For case (ii), take no action.

Rule (a) links the node to a neighboring survivor if it cannot arrive at that survivor through a previously linked neighboring survivor. The random variable x of each neigboring survivor is employed for defining the sequence of linking. Rule (b) deals with the situation in which the node discovers that an edge has already formed between it and a survivor. When compared with the original edge formation algorithm, the number of edges generated is reduced. This is because rules (a) and (b) eliminate some of the edges formed by the origi- nal algorithm. The edges formed using this scheme still represent the region adjacency relationship of the survivors. The region adjacency relationship represented by the eliminated edges are represented indirectly through other edges. Finally, since an edge a * b is

eliminated only if node a is connected to node b through a previously connected neighboring survivor, no isolated node will appear.

Table 3 shows the reduction in the number of edges required for building an irregular pyramid for various base sizes. The reduction in the number of edges above layer 0 is above 10% for irregular pyramids generated from lattices with sizes 32 x 32, 64 x 64, 128 x 128 and 256 x 256.

5. Applications

Finally, we apply the priority sampling and progressive edge formation schemes to labeling and segmentation. When applying these schemes to labeling and segmenta- tion, the pixels are categorized into classes [7]. Two nodes are of the same class if their grey level difference is below a threshold T. For labeling, the threshold T is set to 1. In other words, pixels of the same grey level are grouped into one class. Two neighboring nodes can be selected as survivors if they are of different classes. Also, we have to define clearly what r means in Eq. (4). Only the number of neighbors of the same class of the node, contributes to the degree of Y of the node. A 128 q 128 ‘checker board’ image is used as a test image for labeling.

For segmentation, we employ the non-symmetric class membership criterion scheme used in [7] in con- structing the pyramid. When applying the progressive edge formation scheme in constructing the pyramid for segmentation, the sequence of linking based on random variable x stated in Section 5 is changed. With

s{ci[ll) = {s!7si~ . . sk} representing the set of neighbor- ing survivors of a survivor ci[Z], let ci[Z] -f and s; -f represent the feature value (i.e. the average grey value) of cJZ] and s& respectively. We define ~6, sf where p < q

if S(S~ *fci[Z] -f) < S(S~ -f, ci[Z] sf), where S is the abso- lute difference function. The rationale of using this linking sequence is that a node should be at least linked to its most similar neighbor. Fig. 3 shows the results of applying these schemes to segment a 128 x 128 grey level image. As shown in the figure, the segmentation results using the two schemes are similar to the results using the original scheme. The pyramid height and number of edges used in pyramid construction for labeling and

H.H.S. Ip, S. W.C. Lam/Image and Vision Computing 14 (1996) 297-304

Table 5

303

Number of edges in constructing the pyramid excluding level 0 for labeling and segmentation when using the original and progressive edge formation schemes. The last row shows the reduction in percentage

(4 (b)

(4 W Fig. 3. Image segmentation of an g-bit 128 x 128 grey-level image (a) using an irregular pyramid with (b) the original scheme, (c) priority sampling scheme and (d) progressive edge formation scheme. T 53. Number of segments in (b), (c) and (d) = 13, 13 and 10, respectively.

segmentation are shown in Tables 4 and 5, respectively. Compared with the original irregular construction scheme, the priority sampling scheme reduces the height of the pyramid, while the progressive edge formation scheme reduces the number of edges used in constructing the pyramid.

6. Discussion

This paper presents several alternative strategies for constructing irregular pyramids: specified rate sampling prioritized sampling and a progressive edge formation scheme. For specified sampling, the decimation ratio can be controlled by the application. The prioritized

Table 4 Height of the pyramid constructed for labeling and segmentation when using the original and priority sampling schemes. The last row shows the reduction in percentage

Labeling Segmentation

Original scheme 22 17 Priority sampling scheme 8 12 % reduction 27.2 29.4

Labeling Segmentation

Original scheme 14482 31000 Progressive edge formation scheme 13185 23385 % reduction 9.0 24.6

sampling approach aims at reducing the height of the constructed irregular pyramid. It gives priority to those nodes which have a larger number of neighboring nodes (i.e. larger degree) for being selected as survivors. Results of our experiment confirm that the height of a pyramid can be reduced using this prioritized scheme when com- pared to the original scheme.

Finally, we introduce a new edge formation scheme. When constructing an irregular pyramid for tasks like labeling or segmentation, non-survivors are required to link to the most similar survivor. Hence the degrees of the nodes is a critical factor for the computation load of each node during the pyramid construction process. As our edge formation scheme reduces the number of edges required to build the pyramid, it also reduces the average computation load of the nodes during pyramid construction.

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Ill

PI

[31

141

[51

bl

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