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Module 5
Module - 5 (Image Enhancement in Spatial
Domain and Image Segmentation)
Basic gray level transformation functions - Log
transformations, Power-Law transformations,
Contrast stretching. Histogram equalization. Basics
of spatial filtering - Smoothing spatial filter- Linear
and nonlinear filters, and Sharpening spatial filters-
Gradient and Laplacian. Fundamentals of Image
Segmentation. Thresholding - Basics of Intensity
Thresholding and Global Thresholding. Region
based Approach - Region Growing, Region Splitting
and Merging. Edge Detection - Edge Operators-
Sobel and Prewitt.
Spatial Domain vs. Transform Domain
• Spatial domain
image plane itself, directly process the intensity values of the image plane
• Transform domain
process the transform coefficients, not directly process the intensity values
of the image plane
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Spatial Domain Process
g ( x, y ) T [ f ( x, y )])
f ( x, y ) : input image
g ( x, y ) : output image
T : an operator on f defined over
a neighborhood of point ( x, y )
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Spatial Domain Process
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Spatial Domain Process
Intensity transformation function
s T (r )
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Contrast Stretching
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Contrast Stretching
Some Basic Intensity Transformation Functions
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Image Negatives
Image negatives
s L 1 r
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Example: Image Negatives
Small
lesion
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Log Transformations
Log Transformations
s c log(1 r )
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Example: Log Transformations
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Power-Law (Gamma) Transformations
s cr
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Example: Gamma Transformations
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Example: Gamma Transformations Cathode ray tube
(CRT) devices have an
intensity-to-voltage
response that is a
power function, with
exponents varying
from approximately
1.8 to 2.5
sr 1/2.5
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Example: Gamma Transformations
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Example: Gamma Transformations
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Histogram Processing
• Histogram Equalization
• Histogram Matching
• Local Histogram Processing
• Using Histogram Statistics for Image Enhancement
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Histogram Processing
Histogram h(rk ) nk
rk is the k th intensity value
nk is the number of pixels in the image with intensity rk
nk
Normalized histogram p( rk )
MN
nk : the number of pixels in the image of
size M N with intensity rk
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Histogram Processing
Histogram Processing
Histogram Processing
Histogram Processing
• Histogram
The histogram of an image is a plot of the number of
occurrences of gray levels in the image against the gray
level values.
• Histogram Equalization
Histogram equalization is a process that attempts to
improve the contrast by spreading out the gray levels in
an image using probability density function, so that they
are evenly distributed across the image.
Histogram Equalization –continuous
The intensity levels in an image may be viewed as
random variables in the interval [0, L-1].
Let pr (r ) and ps ( s) denote the probability density
function (PDF) of random variables r and s.
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Histogram Equalization – discrete
Histogram Equalization
Histogram Equalization - conditions
s T (r ) 0 r L 1
a. T(r) is a strictly monotonically increasing function
in the interval 0 r L -1;
b. 0 T (r ) L -1 for 0 r L -1.
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Histogram Equalization
Histogram Equalization
Histogram Equalization - Problem
• Perform histogram
16
Histogram - original
equalization for the
16 Histogram
following -equalized
image.
14 14
4 4 4 4 4 12 12
Pixel Count
Pixel Count
10 10
3 4 5 4 3 8 8
6 6
3 5 5 5 3 4 4
2 2
3 4 5 4 3
0 0
4 4 4 4 4 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
nk Cumulative
Grey Levels sk =(C/(MxN))x(L-1) Grey Levels
Value (C)
0 0 0 (0/25) x 7 = 0 0 0 0
Equalized Image 1 0 0 (0/25) x 7 = 0 0 1 0
6 6 6 6 6 2 0 0 (0/25) x 7 = 0 0 2 6
3 6 6 (6/25) x 7 = 1.68 2 3 0
2 6 7 6 2
4 14 20 (20/25) x 7 = 5.6 6 4 0
2 7 7 7 2
5 5 25 (25/25) x 7 = 7 7 5 0
2 6 7 6 2
6 0 25 (25/25) x 7 = 7 7 6 14
6 6 6 6 6
7 0 25 (25/25) x 7 = 7 7 7 5
Histogram Equalization - Problem
• Equalize the histogram of the following image:
Histogram Equalization - Problem
Spatial Filtering / Mask Processing
Spatial Processing - Point processing (discussed already) and
Mask processing
A spatial filter consists of a neighborhood, and a predefined
operation
Linear spatial filtering of an image of size MxN with a filter of size
mxn is given by the following expression (Normally, we deal with
filters of odd size – smallest being 3x3)
a b
g ( x, y ) w(s, t ) f ( x s, y t )
s a t b
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Spatial Filtering
a b
g ( x, y ) w(s, t ) f ( x s, y t )
s a t b
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Spatial Filtering
Spatial Filtering
Spatial Filtering
Smoothing Spatial Filters
Smoothing filters are used for blurring and for noise
reduction
Blurring is used in removal of small details and bridging of
small gaps in lines or curves
Smoothing spatial filters include linear filters and nonlinear
filters.
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Spatial Smoothing Linear Filters
The general implementation for filtering an M N image
with a weighted averaging filter of size m n is given
a b
w(s, t ) f ( x s, y t )
g ( x, y ) s a t b
a b
w(s, t )
s a t b
where m 2a 1, n 2b 1.
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Two Smoothing Averaging Filter Masks
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Example: Gross Representation of Objects
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Order-statistic (Nonlinear) Filters
— Nonlinear
— Based on ordering (ranking) the pixels contained in the
filter mask
— Replacing the value of the center pixel with the value
determined by the ranking result
E.g., median filter, max filter, min filter
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Order-statistic (Nonlinear) Filters
Order-statistic (Nonlinear) Filters
Example: Use of Median Filtering for Noise Reduction
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Sharpening Spatial Filters
► Foundation
► Laplacian Operator
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Sharpening Spatial Filters: Foundation
► The first-order derivative of a one-dimensional function f(x)
is the difference
f
f ( x 1) f ( x)
x
► The second-order derivative of f(x) as the difference – it is
the difference between the successive second order
derivatives
2 f
f ( x 1) f ( x 1) 2 f ( x)
x 2
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Sharpening Spatial Filters:
Foundation
Sharpening Spatial Filters: Foundation
Sharpening Spatial Filters: Foundation
Sharpening Spatial Filters: Laplace Operator
The second-order isotropic derivative operator is the
Laplacian for a function (image) f(x,y)
2
f 2
f
f 2 2
2
x y
2 f
f ( x 1, y ) f ( x 1, y ) 2 f ( x, y )
x 2
2 f
f ( x, y 1) f ( x, y 1) 2 f ( x, y)
y 2
2 f f ( x 1, y ) f ( x 1, y ) f ( x, y 1) f ( x, y 1)
- 4 f ( x, y)
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Sharpening Spatial Filters: Laplace Operator
2 f f ( x 1, y ) f ( x 1, y ) f ( x, y 1) f ( x, y 1)
- 4 f ( x, y)
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Sharpening Spatial Filters: Laplace Operator
Image sharpening using the Laplacian filter:
g ( x, y ) f ( x, y ) c 2 f ( x, y )
where,
f ( x, y ) is input image,
g ( x, y ) is sharpenend images,
c -1 if 2 f ( x, y ) corresponding to Fig. 3.37(a) or (b)
and c 1 if either of the other two filters is used.
Note: Sharpening is obtained by adding the Laplacian image to the Original
image. If the definition has a negative center coefficient , it will be a
subtraction. Otherwise, it will be an addition.
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Example:
Combining
Spatial
Enhancement
Methods
Goal:
Enhance the
image by
sharpening it and
by bringing out
more of the
skeletal detail
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Example:
Combining
Spatial
Enhancement
Methods
Goal:
Enhance the
image by
sharpening it
and by bringing
out more of the
skeletal detail
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Image Segmentation
• Image Segmentation is the process of dividing an image
into different regions based on the characteristics of
pixels to identify objects or boundaries to simplify an
image and more efficiently analyze it. Segmentation
impacts a number of domains, from the filmmaking
industry to the field of medicine.
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Image Segmentation - Fundamentals
• Let R represent the entire spatial region occupied by an image. Image segmentation
is a process that partitions R into n sub-regions, R1, R2, …, Rn.
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Image Segmentation
Image Segmentation
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Image Segmentation
Approaches in Image Segmentation
• Similarity approach: This approach is based on
detecting similarity between image pixels to form a
segment, based on a threshold. ML algorithms like
clustering are based on this type of approach to
segment an image.
• Discontinuity approach: This approach relies on the
discontinuity of pixel intensity values of the image.
Line, Point, and Edge Detection techniques use this
type of approach for obtaining intermediate
segmentation results which can be later processed to
obtain the final segmented image.
Background
• First-order derivative
f
f '( x) f ( x 1) f ( x)
x
• Second-order derivative
2 f
f ( x 1) f ( x 1) 2 f ( x)
x 2
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Characteristics of First and Second Order Derivatives
• First-order derivatives generally produce thicker edges in
image
• Second-order derivatives have a stronger response to fine
detail, such as thin lines, isolated points, and noise
• Second-order derivatives produce a double-edge response
at ramp and step transition in intensity
• The sign of the second derivative can be used to determine
whether a transition into an edge is from light to dark or
dark to light
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Detection of Isolated Points
• The Laplacian
2
f 2
f
f ( x, y ) 2 2
2
x y
f ( x 1, y ) f ( x 1, y ) f ( x, y 1) f ( x, y 1)
4 f ( x, y )
9
1 if | R( x, y ) | T R wk zk
g ( x, y ) k 1
0 otherwise
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Line Detection
• Second derivatives to result in a stronger response and to produce thinner lines
than first derivatives
• Double-line effect of the second derivative must be handled properly
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Detecting Line in Specified Directions
• Let R1, R2, R3, and R4 denote the responses of the masks in Fig. 10.6. If, at a given
point in the image, |Rk|>|Rj|, for all j≠k, that point is said to be more likely
associated with a line in the direction of mask k.
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Edge Detection
• Edges are pixels where the brightness function changes
abruptly
• Edge models
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Fundamental steps in edge detection
The four steps of edge detection
• (1) Smoothing: suppress as much noise as possible, without destroying the true
edges.
• (2) Enhancement: apply a filter to enhance the quality of the edges in the image
(sharpening).
• (3) Detection: determine which edge pixels should be discarded as noise and which
should be retained (usually, thresholding provides the criterion used for detection).
• (4) Localization: determine the exact location of an edge (sub-pixel resolution might
be required for some applications, that is, estimate the location of an edge to better
than the spacing between pixels). Edge thinning and linking are usually required in
this step.
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Basic Edge Detection by Using First-Order Derivative
f
g x x
f grad ( f )
g y f
y
The magnitude of f
M ( x, y ) mag(f ) g x 2 g y 2
The direction of f
gx
( x, y ) tan
1
g y
The direction of the edge
- 90
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Basic Edge Detection by Using First-Order Derivative
f
g x x
Edge normal: f grad ( f )
g y f
y
Edge unit normal: f / mag(f )
In practice,sometimes the magnitude is approximated by
f f f f
mag(f )= + or mag(f )=max | |,| |
x y x y
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Thresholding for Segmentation
• Thresholding is a very popular segmentation
technique, used for separating an object from its
background.
• The process of thresholding involves, comparing
each pixel value of the image (pixel intensity) to
specified threshold. This divides all the pixels of the
input image into groups:
• Pixels having intensity value lower than threshold.
• Pixels having intensity value greater than threshold.
• Specific values are assigned to these groups
depending on the type of segmentation used.
Thresholding
1 if f ( x, y ) T (object point)
g ( x, y )
0 if f ( x, y ) T (background point)
T : global thresholding
Multiple thresholding
a if f ( x, y) T2
g ( x, y ) b if T1 f ( x, y ) T2
c if f ( x, y) T1
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Basic Global Thresholding
1. Select an initial estimate for the global threshold, T.
2. Segment the image using T. It will produce two groups of pixels: G1 consisting of all pixels with
intensity values > T and G2 consisting of pixels with values T.
3. Compute the average intensityvalues m1 and m2 for the pixels in G1 and G2, respectively.
4. Compute a new threshold value.
1
T m1 m2
5. Repeat Steps 2 through24 until the difference between values of T in successive iterations is
smaller than a predefined parameter .
T
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Using Image Smoothing to Improve Global Thresholding
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Multiple Thresholds
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Variable Thresholding: Image Partitioning
• Subdivide an image into nonoverlapping rectangles
• The rectangles are chosen small enough so that the illumination of each is
approximately uniform.
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Region-Based Segmentation
Region Growing
• Region growing is a procedure that groups pixels or subregions
into larger regions.
• The simplest of these approaches is pixel aggregation, which
starts with a set of “seed” points and from these grows regions by
appending to each seed points those neighboring pixels that have
similar properties (such as gray level, texture, color, shape).
• Region growing based techniques are better than the edge-based
techniques in noisy images where edges are difficult to detect.
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Region-Based Segmentation
Example: Region Growing based on 8-connectivity
f ( x, y ) : input image array
S ( x, y ): seed array containing 1s (seeds) and 0s
Q( x, y ): predicate
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Region Growing based on 8-connectivity
1. Find all connected components in S ( x, y ) and erode each
connected components to one pixel; label all such pixels
found as 1. All other pixels in S are labeled 0.
2. Form an image fQ such that, at a pair of coordinates (x,y),
let fQ ( x, y ) 1 if the Q is satisfied otherwise f Q ( x, y ) 0.
3. Let g be an image formed by appending to each seed point
in S all the 1-value points in fQ that are 8-connected to that
seed point.
4. Label each connencted component in g with a different region
label. This is the segmented image obtained by region growing.
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TRUE if the absolute difference of the intensities
Q between the seed and the pixel at (x,y) is T
FALSE
otherwise
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Region Growing
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4-connectivity
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8-connectivity
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Region Splitting and Merging
R : entire image Ri :entire image Q: predicate
1. For any region Ri , If Q( Ri ) = FALSE,
we divide the image Ri into quadrants.
2. When no further splitting is possible,
merge any adjacent regions R j and Rk
for which Q( R j Rk ) = TRUE.
3. Stop when no further merging is possible.
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Image Segmentation – NPTEL Videos
Image Segmentation Fundamentals
https://www.youtube.com/watch?v=3qJej6wgezA&t=5s
Thresholding and Region growing
https://www.youtube.com/watch?v=vaS6rS8ZpkU&t=328s
https://www.youtube.com/watch?v=CD4KyEHfVx4
