DIP Lecture 7-9.pdf

Digital Image Processing
Week # 3
Lecture # 7-9
1

The Basics
• Spatial domain techniques work directly on the pixels as opposed to, for example, frequency
domain, which work in a transformed domain (for instance Fourier)
• Spatial domain processing is more efficient, needs less computaitons
• g(x,y)=T[f(x,y)]
Neighborhood operation:
• For the pixels at the corners of the image, normal practice is to omit the missing neighbors or
to pad zeros.
• Smallest possible neighborhood is of size 1x1 i.e. the pixel itself.

Intensity Transformation: Contrast Stretching
• The transformation produces higher contrast than the original by
darkening the intensity levels below m and brightening the levels above m
• Thresholding: Produces a binary image. Pixels above a threshold are
designated as ‘1’ and the others are declared ‘0’

Images Negative and Identity Image (Linear Operations)
• The negative of an image with intensity levels in the range [0,L-1] is
obtained: s=L-1-r -> linear operation
• This processing is used to get an image negative
• Particularly useful in enhancing the white or gray detail embedded in dark
regions of an image
• For an 8 bpp (bits per pixel) image L-1=255
• Identity image: s=r

Log Transformation (Dynamic Range Compression)
• The log transformation corresponds to the operation: s = c log (1+r)
• ‘c’ is a constant
• The transformation maps a narrow range of low intensity inputs into a wider
range of output levels and vice versa for the higher values of input.
• The logarithmic transformation compresses the dynamic range of images with
large variations in pixel values

• The most common application for the dynamic range compression is for the display of
the Fourier Transform.
• The maximum magnitude value of its Fourier Transform is 7.9x10^6, and the second
largest value is approximately 10 times smaller. If we simply linearly scale this image,
we obtain the middle figure.
• Due to the large dynamic range, we can only recognize the largest value in the center
of the image. All remain values appear as black on the screen. If we instead apply the
logarithmic operator to the Fourier image, we obtain the third image.
• Here, smaller pixel values are enhanced and therefore the image shows significantly
more details.
• The logarithmic operator enhances the low intensity pixel values, while compressing
high intensity values into a relatively small pixel range. Hence, if an image contains
some important high intensity information, applying the logarithmic operator might
lead to loss of information.

• Thus, a logarithmic transform is appropriate when we want to enhance the low
pixel values at the expense of loss of information in the high pixel values
• Subject was photographed in front of a bright background. The dynamic range of
the film material is too small, so that the gray levels on the subject's face are
clustered in a small pixel value range. A logarithmic transform spreads them over a
wider range, while the higher values are compressed

• Applying logarithmic transform is less appropriate, because most of its
details are contained in the high pixel values.
• This image shows that a lot of information is lost during the transform

Power-Law (Gamma) Transformation
• Power-law transformations have the basic form:
•

• Two important observations:
• (1) For fractional values of gamma: A narrow range of dark input values
are mapped into a wider range of output values. Opposite is true for
higher input intensities. Enhances the darker pixels
• (2) For gamma>1: Narrow range of high input intensities are mapped to a
wider range of output values. Opposite is true for the lower input
intensities. Enhances the brighter pixels.
• An Illustrated Example:

Contrast Manipulation: Another Application of Gamma
Correction

Piece-wise Linear Transformation Function
• A complementary approach to the methods discussed in the previous three sections is to
use piecewise linear functions. The principal advantage of piecewise linear functions over
the types of functions we have discussed thus far is that the form of piecewise functions can
be arbitrarily complex. In fact, as we will see shortly, a practical implementation of some
important transformations can be formulated only as piecewise functions. The principal
disadvantage of piecewise functions is that their specification requires considerably more
user input.
• Contrast stretching Low-contrast images can result from poor illumination, lack of dynamic
range in the imaging sensor, or even wrong setting of a lens aperture during image
acquisition. The idea behind contrast stretching is to increase the dynamic range of the gray
levels in the image being processed.
• The locations of points (r1, s1) and (r2, s2) control the
shape of the transformation
• If r1=s1 and r2=s2, the transformation is a linear function
that produces no changes in gray levels.
• If r1=r2, s1=0 and s2=L-1, then the transformation becomes
a thresholding function and will result in a binary image.
• Intermediate values of r1, s1 and r2, s2 produce
various degrees of spread in the gray levels of the output
image

• Figure shows an 8-bit image with low contrast. Fig. 3.10(c) shows the result of
contrast stretching, obtained by setting (r1, s1) =(r_min, 0) and (r2, s2)=(r_max,L-1)
where r_min and r_max denote the minimum and maximum gray levels in the
image, respectively. Thus, the transformation function stretched the levels linearly
from their original range to the full range [0, L-1]. Finally, Fig. 3.10(d) shows the
result of using the thresholding function defined previously, with r1=r2=m, the
mean gray level in the image

Piece-wise Linear Transformations Contd……
• Intensity Level Slicing: The objective is to highlight a specific range of intensities.
• Applications: Enhancing masses of water in satellite images, highlighting flaws in
X-ray images.
• Two main approaches are adopted
• (1) Represent the intensities in region of interest (ROI) with one value (say white)
and the rest with the other value (say, black). Result is a binary image.
• (2) Brighten or darken the intensities in ROI and leave others unchanged

• A little twist in part (c) above, the interest lies in the original intensities of the ROI so we kept the higher
intensities as is and transformed the mid-gray intensities to black

Piece-wise Linear Transformations Contd……
• Bit-plane Slicing: To highlight the contribution made to the total image
appearance by specific bits.
• Application: Image compression
• Assuming that each pixel is represented by 8 bits, the image is composed
of eight 1-bit planes.
• Plane 1 contains the least significant bit and plane 8 contains the most
significant bit.
• Useful for analyzing the relative importance played by each bit of the
image.
• Only the higher order bits (top four) contain visually significant data. The
other bit planes contribute the more subtle details
• Plane 8 corresponds exactly with an image thresholded at gray level 127.
We map all intensities between 0-127 to 0 and map all levels between
128-255 to 1.

Image Reconstruction using Bit-planes
•
• We multiply nth significant bit by 2^(n-1) and add the resultant images.
• Note that 4 top planes are enough for visually significant reconstruction
• Adding more planes will not matter a lot

Histogram Processing
• Let rk be the kth intensity value (i.e. k=0,1,2,…,L-1) and let nk be the
number of pixels in an image with intensity rk. Then histogram of the
image is defined as h(rk)= nk
• It is normal practice to divide each component of histogram with the total
number of pixes of an image =MN
• A normalized histogram is therefore given as: p(rk)= nk /(MN); k=0,1,…,L-1
• p(rk) is an estimate of the probability of occurrence of intensity level rk in
an image. The sum of all components of a normalized histogram is equal
to 1.

• Image whose pixels tend to occupy the
entire range of intensity levels and
tend to be distributed uniformly will
have an appearance of high contrast
and will exhibit a large variety of gray
tones.
• Could there be a tool through which
we can enhance the contrast of dull
images using the concept of spreading
intensities?

Histogram Equalization
• A method through which intensities of an image are spread to the entire
range thereby increasing the contrast of the image

• r=0,1,2,…..,L-1
• Effect of histogram equalization

An Illustrative Example
Intensity # pixels
0 20
1 5
2 25
3 10
4 15
5 5
6 10
7 10
Total 100
Accumulative Sum of Pr
20/100 = 0.2
(20+5)/100 = 0.25
(20+5+25)/100 = 0.5
(20+5+25+10)/100 = 0.6
(20+5+25+10+15)/100 = 0.75
(20+5+25+10+15+5)/100 = 0.8
(20+5+25+10+15+5+10)/100 = 0.9
(20+5+25+10+15+5+10+10)/100 = 1.0
1.0

Intensity
(r)
No. of Pixels
(nj)
Acc Sum
of Pr
Output value Quantized
Output (s)
0 20 0.2 0.2x7 = 1.4 1
1 5 0.25 0.25*7 = 1.75 2
2 25 0.5 0.5*7 = 3.5 4
3 10 0.6 0.6*7 = 4.2 4
4 15 0.75 0.75*7 = 5.25 5
5 5 0.8 0.8*7 = 5.6 6
6 10 0.9 0.9*7 = 6.3 6
7 10 1.0 1.0x7 = 7 7
Total 100

Histogram Specification
• Image of Mars moon-most pixels have intensities around 0
• Although it appears that histogram equalization will enhance details in the
dark region?
• Let’s find out

• The transformation suggest the HE would result in a washed-out image.
• Note how quickly the curve rises to high intensity
• HE is not the desired fix
• We can fix the image by devising a suitable transformation which would change
the histogram in a desirable way.

• Problem? Lots of pixel near ‘0’
• We cannot change the histogram altogether as it will entirely change the image.
• We keep the original look of the image in tact i.e. majority of pixels lying in lower
intensity region but we make the transition of levels smoother in the dark region
Smoother transition of levels in dark region

DIP Lecture 7-9.pdf

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