chapter 2 digital image fundamentals - bgudip111/wiki.files/dip3e_chapter02_art.pdf · digital...
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
The Cornea is a tough, transparent tissue g , pthat covers the anterior surface of the eye.The Sclera is an opaque membrane that encloses the remainder of the optic globeThe Choroid contrarians the blood vessels,The Choroid contrarians the blood vessels,which are the major source of nutrition tothe eye.The lens is made up of concentric layer of fibers cells and is supported by fibers that pp yattach to the ciliary body.The Retina lines the inside of the wall’s entire posterior portion. Its surface includetwo classes of light receptors: cones and rods.The cones lies on the fovea and highly sensitive to color (6-7 million in each eye).The rods (75-150 million), which are distributed over the retinal surface and serves to give general
© 1992–2008 R. C. Gonzalez & R. E. Woods
overall picture of the field of view and are sensitive to low level illumination.
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
The human eye can adapt to enormous range of light intensity levels – on the order of 1010 . The subjectivebrightness (the intensity perceived by the eye) of the eye is a logarithmic function.Ph t i i i i th i i f th d ll litPhotopic vision is the vision of the eye under well-lit conditions. In humans and many other animals, photopic vision allows color perception, mediated by cone cells.Scotopic vision is the vision of the eye under low light conditions
© 1992–2008 R. C. Gonzalez & R. E. Woods
conditions.
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Simple Image Operationp g pWe assume the image pixels as elements of a set and use the set operation to manipulate images
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Simple Image OperationNegating a binary image is done simply by exchanging the white pixels “1” by back pixels “0” and vise versa. For gray scale images this is done by subtracting 255 from the intensity of each pixel; i.e., Ci=255- Ci
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Logical Operations on Imagesg p gWe assume binary values for pixels of the image (B/W) and treat the image as a binary buffer.
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Image Operationg pMost of image operation take into account the adjacent pixels when computing the resulting values (for pixels).Example: The value of a pixel, in the resulting image, is taken as the average of of its adjacent neighbors.
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Geometric Spatial TransformationpThese operations modify the spatial relationship between pixels in an image.In digital images Geometric Transformations involvesinvolves1. Spatial Transformation of coordinates.2. Intensity interpolations.The geometric coordinate could be expressed as (xp, yp) = T{(up, vp) }, where(up, up) are the coordinate of the pixel p before applying the transformation T and (xp, yp) are thecoordinate of the pi el p before appl ing Tcoordinate of the pixel p before applying TIt is often written as vector operations
Tvuyx pppp ],[),(
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
TranslationTranslationAlter the position of apoint or an object in 2D space.p j p
)( vu
),( dyvdxu ),(),(),( , dyvdxuvuTyx dydx
),( vu
11001
),( vu
dydx
yx
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
R t tiRotation• Rotate a pixel P(x, y) around
the origin by an angle α.Th lt i l iti i• The results pixel position is P’(x’,y’)
)sin()cos( vux
),( vu
α))cos()sin(),sin()cos((),(
)cos()sin(
vuvuvuRvuy
),( yx
)i ()(
vu
yx
)cos()sin()sin()cos(
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Rotation around the center of the image by 250
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Image Scale• Scale a region of pixels by a
factor• Each pixel in this region P(u v) is
Image Scale),( yx
),( yx ss• Each pixel in this region P(u, v) is
“scaled” to a new position is P’(x,y)
)(sux
The scale of include additional translation when no point of the
),( vu
y
x
suysux
)()( svsuyx translation when no point of the scaled region is in the origin
uS
Sx x
00
),(),( yx svsuyx
© 1992–2008 R. C. Gonzalez & R. E. Woods
vSy y0
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
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Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
S2 2S2,2
S2,1
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
I ShImage Sheer• Scale a region of pixels by a factor
(Shx,Shy).• Each pixel in this region P(u v) is
(x’, y’)
• Each pixel in this region P(u, v) is “Sheered” to a new position is P’(x, y)
(x y) (x, y)
hy
hx
suvysvux
)()( ),(),( hyhx suvsvuyx
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
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Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Horizontal SheerHorizontal Sheer
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
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Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
H C di tHomogenous Coordinates
Let us consider applying several transformation T0, T1,…, Tk on the same image.
This requires k matrix multiplication for each pixel}}...}}{{...{{ 011
v
uTTTT kk
uu
If we can perform the following, where T= T0 T1… Tk , we are required to perform one time k matrix multiplication and one matrix multiplication for each vertex
v
uT
vu
TTTT kk 011...
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Example:Let us consider the following transformation
u
u
vu
TSRTS
20107)30sin()30cos(50101
4,25,7303,52,1
v41050)30cos()30sin(31020
It is impossible to multiply these matrices. p p y
To overcome this limitation we use a homogeneous coordinate (x, y, w), where w =1. This also update all the spatial transformation to 3×3 matrices.
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Spatial Transformationp
• Scaling an image by mapping each pixel to a target.
• In the resulting image• In the resulting image– The number of pixels
increasesS i l “ t ”– Some pixels are “empty”
– How to determine their values?
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
After Scaling
Original
After Scaling
A better Result
Original Image
After Scaling withNearest Pixel
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Spatial Transformation - Forward Scanningp g
Scale +
In this approach missing pixel are clearly visible. To fill the missing pixel,
ld li tRotate one could replicate pixels. However, the quality of the resultingis image
Mapping the image of the square pixel covers all thesquare pixel covers all the pixels in the target image.The value of a pixel is determined by the mapped squares that cover it
© 1992–2008 R. C. Gonzalez & R. E. Woods
squares that cover it.
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Spatial Transformation - Backward ScanningIn this approach we fill the target image pixel by pixel. For each pixel we compute it source position. The value of the target pixel is computed by interpolating the pixels around the source position.
Spatial Transformation Backward Scanning
p
T-1
This approach does not need to take into account the type of transformationInverse
Transformation
the type of transformation to determine the adjacent pixels of the source position
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Interpolation
• According to WikipediaIn the mathematical subfield of numerical analysis interpolation is aIn the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.
• In Image Processing• In Image ProcessingIt is a techniques to assign values to pixels in an images based on their adjacent pixels, or adjacent samples that my not be correlate with pixel location in the image
© 1992–2008 R. C. Gonzalez & R. E. Woods
location in the image.
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Value InterpolationValue Interpolation
• Let us assume we have several f0 f f
f3
samples of a function f– f0, f1, f2,…,fn
N t (Z O d )
0 f1 f
2
3fk
f0
f3• Nearest (Zero Order)
•
f0 f
1 f2
3fk
f f• Bilinear (First Order)
f0 f
1 f2
f3
fk
f f
© 1992–2008 R. C. Gonzalez & R. E. Woods
• Bicubic (Third Order) f0 f
1 f2
f3
fk
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Bili I t l ti
Length = 1t1D
bai tCCtC )1(
Bilinear Interpolation
Ca Cb
tCi
bai )(
2D Length = 12DCtl Ctr
wCt
ngth
= 1
trtlt
wCCwCwCCwC
)1()1(
Len
tbwh
brblb
hCChCwCCwC
)1()1(
h Cwh
© 1992–2008 R. C. Gonzalez & R. E. Woods
Cbl CbrCb
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Q d i d C bi I t l tiQuadric and Cubic Interpolation
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Q d i d C bi I t l ti
The value of the sample position f(x+dx, y+dy) is determined based
Quadric and Cubic Interpolation
)()(),(),(2
1
2
1jdyCidxCjyixfdyydxxf aa
i j
( y y)on the following equation.
||202||1,4||8||5||1||0,1||)3(||)2(
)( 23
23
1 1
xxaxaxaxaxxaxa
xCa
i j
||2,0 x
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
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Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
1D Quadric InterpolationXi-1 X Xi
f3
fGeneral Quadric Equation
1D Quadric Interpolation
iXi+1
fkcbxaxy 2
Using the three points to determine the equation parameters
p
cbxaxy
cbxaxy
iii
iii
211
21
q p
After determining the parameters of the quadric function, we plug in xp to
cbxaxy iii
ii
112
1
determine its value, yp
Similarly we can derive the 1D cubic interpolation of a point xp But we need
© 1992–2008 R. C. Gonzalez & R. E. Woods
Similarly we can derive the 1D cubic interpolation of a point xp, But we need four points.
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
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Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
(x,y)yx yCxCyxf )(*)(),(
2D Cubic Interpolation
yyyyy
xxxxx
dycybyayC
dxcxbxaxC
23
23
)(
)(
3,12,11,10,1
3,02,01,00,0
aaaaaaaa
dbcbbbabdacabaaa
yxyxyxyx
yxyxyxyx
1,31,31,30,3
3,22,21,20,2
,,,,
aaaaaaaa
ddcdbdaddcccbcac
yxyxyxyx
yxyxyxyx
yyyy
© 1992–2008 R. C. Gonzalez & R. E. Woods
We need to determine the value of the 16 parameters to compute the value of (xp,xp) But we need four points.
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Approximating Cubic InterpolationApproximating Cubic Interpolation
© 1992–2008 R. C. Gonzalez & R. E. Woods
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Digital Image Processing, 3rd ed.Digital Image Processing, 3rd ed.
www.ImageProcessingPlace.com
Gonzalez & Woods
Chapter 2Digital Image Fundamentals
Chapter 2Digital Image Fundamentals
Visual Examples
BiLinear
Visual Examples
Original Imageg g
BiCubicNearest
BiCubic
© 1992–2008 R. C. Gonzalez & R. E. Woods