4.3 System Implementation
4.3.2 Normalization techniques
4.3.2.1 Proposed method for rule based hand extraction
Consider a binary image f defined over a gridBof size (N+1)×(M+1).Bis composed of two comple- mentary regions R and R representing the hand (object) and the background respectively. Thus,
R=
(x,y)
(x,y)∈B and f (x,y)=1
(4.4) and the complementary region R is given by
R=B\R (4.5)
4.3 System Implementation
R
Finger region
Circle enclosing
maximum neighbourhood region.
Wrist crease.
Forearm region Binary image
R
Estimated palm centroid
Object pixels within the circle:Palm region
(xc, yc) f
Rf ingers
Rpalm
Rf orearm
Figure 4.9: Pictorial representation of the regions composing the binary image f . R denotes the hand region and R denotes the background region.
The boundaryδR of the hand region is defined by the set of pixels in R that are adjacent to at least one pixel in the region R. It is represented as
δR=
(x,y)
(x,y)∈R and (x,y) is adjacent to a pixel in R
(4.6) The hand region R can be partitioned in to three subregions. They are (a) Rf ingers(fingers), (b) Rpalm(palm) and (c) Rf orearm(forearm). Hence
R=Rf ingers∪Rpalm∪Rf orearm (4.7)
such that
Rf ingers∩Rpalm=∅ Rf ingers∩Rf orearm=∅ Rpalm∩Rf orearm=∅.
(4.8)
Figure 4.9 illustrates these elementary regions comprising the hand object R. Based on the anatomy, the palm and the forearm can be considered as continuous smooth regions. The forearm extends outside the palm and its width is less than that of the palm region. Conversely, the region containing the fingers is discontinuous under abduction. Also, the width of a finger is much smaller than that of the palm and the forearm. Therefore, the geometrical variations in the width and the continuity of these subregions in the hand image are used as cues for detection.
(a) Computation of width
The variation in the width along the longest axis of the hand image is calculated from the distance map obtained using the Euclidean distance transform (EDT). The EDT gives the minimum distance of an object
pixel to any pixel on the boundary setδR. The Euclidean distance between a boundary pixel (xb,yb)∈δR and an object pixel (x,y)∈R is defined as
d(xb,yb),(x,y) = q
(x−xb)2+(y−yb)2 (4.9)
The value of the EDT, D(x,y)for the object pixel (x,y) is computed as D(x,y) = min
(xb,yb)∈δRd(xb,yb),(x,y) (4.10)
The values of D(x,y)at different (x,y) are used to detect the subregions of R.
The straightforward implementation of EDT defined through (4.9) and (4.10) is computationally expen- sive. Therefore, the conventional approach to fast EDT based on the Voronoi decomposition of the image proposed in [177] is employed. A study on the several other algorithms proposed for reducing the compu- tational complexity of EDT is discussed in [178].
(b) Verification of region continuity
The continuity of the subregions after detection is verified through connected component labelling pre- ceded by morphological erosion. The erosion operation with a small structuring element is performed to disconnect the weakly connected object pixels. The structuring element considered is a disk operator with radius 3. The resultant is verified to be a continuous region if there is only one connected component. If there is more than one connected component, the detected region is verified as discontinuous.
The geometrical measurements along the finger region vary with the users and they get altered due to geometric distortions. However, the measures across the palm and the forearm can be generalized and their ratios are robust to geometric distortions. The palm is an intactly acquired part that connects the fingers and the forearm. Since Rpalmlies as an interface between Rf ingersand Rf orearm, the separation of palm facilitates the straightforward detection of the other two regions. Hence, the anthropometry of palm is utilized for detecting the regions in the hand image.
4.3.2.1.1 Anthropometry based palm detection The parameters of the hand considered for palm detection are the hand length, palm length and the palm width as illustrated in Figure 4.10(a). The anthropometric studies in [179–181] present the statistics of the above mentioned hand parameters. From these studies, we infer that the minimum value of the ratio of palm length (Lpalm) to palm width (Wpalm) is approximately 1.322 and its maximum value is 1.43. Similar observations were made from our photometric experiments. Figure 4.10(b)
4.3 System Implementation
Hand length
Forearm region
Palm width
Minimum forearm width Palm length Lpalm
Wpalm
(a)
1.325 1.35 1.37 1.39 1.41 1.43 1.45 10
15 20 25 30
Values of
Number of occurences
Lpalm:W
palm
(b)
Figure 4.10: (a) Hand geometry and (b) Histogram of the experimental values of palm length (Lpalm) to palm width (Wpalm) ratio calculated for 140 image samples taken from 23 persons.
gives the histogram of the WLpalm
palm values obtained through our experimentation. This ratio will be utilized to approximate the palm region as an ellipse. Considering all the variations of this ratio, we take
Lpalm=1.5×Wpalm (4.11)
Based on the geometry, we approximate the palm region Rpalmas an elliptical region with
Major axis length =1.5×minor axis length (4.12)
Assuming apalmas the semi-major axis length and bpalmas the semi-minor axis length, we can write apalm= Lpalm
2 (4.13)
bpalm = Wpalm
2 (4.14)
Therefore,
apalm =1.5×bpalm (4.15)
From (4.15), it can be inferred that all the pixels constituting Rpalm will lie within the ellipse of semi-major axis length apalm. Therefore, the palm centre and the value of apalmhave to be estimated for detecting the palm region.
a) Computing the palm centre Given that the boundary of Rpalmis an ellipse, its centre is known to have the maximum distance to the nearest boundary. Therefore, the centre of Rpalmis computed using the EDT in (4.10).
The pixels (x,y) with EDT values D(x,y)greater than a thresholdζare the points belonging to the neighborhood
of the centre of the palm. This neighborhood is defined as C=n
(x,y)∈R|D(x,y) > ζo
(4.16) The centre (xc,yc) is defined as the palm centroid and given by
(xc,yc)=j Xm
,j Ym
(4.17) where
X = 1
|C| X
(xi,yi)∈C
xi , Y = 1
|C| X
(xi,yi)∈C
yi,
|C|is the cardinality of C and⌊⌉denotes rounding off to the nearest integer.
The thresholdζis selected as max(D(x,y))−τ. The offsetτis considered to compensate for the inaccuracies due to the viewing angles. For small values ofτ, the centroid may not correspond to the exact palm centre and large values ofτwill tend to deviate the centroid from the palm region. The optimal value ofτis experimentally chosen as 2.
b) Computing the semi-major axis length From the geometry, it can be understood that the nearest boundary points from the palm centroid correspond to the end points of the minor axis. Hence, the EDT value at (xc,yc) is the length of the semi-minor axis and therefore,
bpalm= D(xc,yc) (4.18)
From (4.15), it follows that the length of the semi-major axis can be given as
apalm=1.5×D(xc,yc) (4.19)
c) Detecting the palm In order to ensure proper detection of the palm, the finger regions (Rf ingers) are sheared from the segmented object through the morphological opening operation. The structuring element is a disk with radius drempirically chosen as
dr= bpalm
1.5 (4.20)
The resultant is considered as the residual and will be referred as the oddment. The oddment is generally composed of the palm region and may or may not contain the forearm. This implies A ⊆ R. Therefore, the
4.3 System Implementation
Input Segmented image ‘f’
Separation of forearm from ‘f’
Oddment ‘A’
Detected palm ˆ
fingers forearm
R = R R
Morphological
opening palm from ‘R’
Abstraction of
Region containing pixels with Dx, y!"T
Forearm detection Output:Hand
Computed Distance
transformation
(xc,yc)andbpalm
Figure 4.11: Illustration of the rule based region detection and separation of the hand from the acquired posture image f . The intensity of the background pixels is assigned a 0 and the object pixels are assigned the maximum intensity value 1.
oddment A can be defined as A=Rpalm∪Rf orearm
For R with no forearm region, Rf orearm = ∅ and A = Rpalm. Rpalm is a part of A that is approximated as an elliptic region. Thus,
Rpalm =
(xo,yo)
(xo,yo)∈A and xo−xc
apalm
!2
+ yo−yc bpalm
!2
61
(4.21)
d) Detection of forearm The forearm is detected through the abstraction of the palm region Rpalm from the posture image R. The abstraction separates the forearm and the finger regions, such that R is modified as
Rˆ =R\Rpalm=Rf ingers∪Rf orearm (4.22)
As in the case of palm detection, the finger region is removed from ˆR through the morphological opening operation. The structuring element is a disk with its radius calculated from (4.20). The resultant is a forearm region and has the following characteristics:
(i) The resultant Rf orearm⊆ A and the region enclosing Rf orearmis continuous.
(ii) The width of the wrist crease is considered as the minimum width of the forearm region. From the anthropometric measures in [180], the minimum value of the ratio of the palm width to wrist breadth is obtained as 1.29 and the maximum value is computed as 1.55. Using this statistics, the empirical value
for the width of the forearm should satisfy the relation Wf orearm> 2bpalm
1.29 (4.23)
e) Identifying the finger region Having detected the palm and the forearm, the remaining section of the hand image R will contain the finger region if it satisfies the following conditions,
• Rf ingers*A.
• The region enclosing Rf ingersis marked by irregular boundary, if more than one finger is abducted.
• The width of a finger (maximum EDT value in this section) is much less than that of the palm and the forearm. Experimentally,
Wf inger≤ bpalm
2 (4.24)
A procedural illustration of the proposed rule-based method for detecting the hand region from the input image is shown in Figure 4.11. After detecting the hand region, the pixels belonging to the forearm Rf orearm are assigned with zero values, thus including Rf orearmin the background.