Chapter 3. Intensity Transformations and Spatial Filtering.pdf

XỬ LÝ ẢNH TRONG CƠ ĐIỆN TỬ
Machine Vision
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TRƯỜNG ĐẠI HỌC BÁCH KHOA HÀ NỘI
Giảng viên: TS. Nguyễn Thành Hùng
Đơn vị: Bộ môn Cơ điện tử, Viện Cơ khí
Hà Nội, 2021

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Chapter 3. Intensity Transformations and Spatial Filtering
Rafael C. Gonzalez, Richard E. Woods, “Digital image processing,” Pearson (2018).
❖Two principal categories of spatial processing are intensity transformations and
spatial filtering.
➢ Intensity transformations operate on single pixels of an image for tasks such
as contrast manipulation and image thresholding.
➢ Spatial filtering performs operations on the neighborhood of every pixel in an
image.
➢ Examples of spatial filtering include image smoothing and sharpening.

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1. Background
2. Some Basic Intensity Transformation Functions
3. Histogram Processing
4. Fundamentals of Spatial Filtering
5. Smoothing (Lowpass) Spatial Filters
6. Sharpening (Highpass) Spatial Filters
7. Highpass, Bandreject, and Bandpass Filters from Lowpass Filters
8. Combining Spatial Enhancement Methods

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1. Background
❖The Basics of Intensity Transformations and Spatial Filtering
➢The spatial domain processes are based on the expression
where f(x, y) is an input image, g(x, y) is the
output image, and T is an operator on f defined
over a neighborhood of point (x, y).
A 3x3 neighborhood about a point (x0, y0) in an image. The neighborhood
is moved from pixel to pixel in the image to generate an output image.

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1. Background
❖The Basics of Intensity Transformations and Spatial Filtering
➢intensity (also called a gray-level, or mapping) transformation function
Intensity transformation functions.
(a) Contrast stretching function.
(b) Thresholding function.

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1. Background

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❖Three basic types of functions
➢ linear (negative and identity transformations)
➢ logarithmic (log and inverse-log transformations)
➢ power-law (nth power and nth root
transformations)
Some basic intensity transformation functions.

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❖Image Negatives
(a) A digital mammogram. (b) Negative image obtained using Eq. (3-3) .
(Image (a) Courtesy of General Electric Medical Systems.)

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❖Log Transformations
where c is a constant and it is assumed that r  0
(a) Fourier spectrum displayed as a grayscale image. (b) Result of applying the log
transformation in Eq. (3-4) with c = 1. Both images are scaled to the range [0, 255].

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❖Power-Law (Gamma) Transformations
where c and are positive constants
Plots of the gamma equation s = cr for various values
of  (c = 1 in all cases).

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(a) Image of a human retina. (b) Image as
as it appears on a monitor with a gamma
setting of 2.5 (note the darkness). (c)
Gammacorrected image. (d) Corrected
image, as it appears on the same monitor
(compare with the original image). (Image
(a) courtesy of the National Eye Institute,
NIH)

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➢Contrast enhancement using power-law intensity transformations.
a) Magnetic resonance image (MRI) of a fractured human spine (the region of the fracture is enclosed by the circle). (b)–(d)
Results of applying the transformation in Eq. (3-5) with and and 0.3, respectively. (Original image courtesy of Dr. David R. Pickens,
Department of Radiology and Radiological Sciences, Vanderbilt University Medical Center.)

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➢Another illustration of power-law transformations.
(a) Aerial image. (b)–(d) Results of applying the transformation in Eq. (3-5) with  = 3.0, 4.0 and 5.0, respectively.
(c = 1 in all cases.) (Original image courtesy of NASA.)

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❖Piecewise Linear Transformation Functions
➢Contrast Stretching
where rmin and rmax denote
the minimum and maximum
intensity levels in the input
image, respectively
where m is the mean intensity
level in the image

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➢Intensity-Level Slicing
Figure 1: (a) This transformation function highlights range [A, B] and
reduces all other intensities to a lower level. (b) This function highlights
range [A, B] and leaves other intensities unchanged.
Figure 2: (a) Aortic angiogram. (b) Result of using a slicing transformation
of the type illustrated in Fig. 1(a) , with the range of intensities of interest
selected in the upper end of the gray scale. (c) Result of using the
transformation in Fig. 1(b) , with the selected range set near black, so that
the grays in the area of the blood vessels and kidneys were preserved.
(Original image courtesy of Dr. Thomas R. Gest, University of Michigan
Medical School.)

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➢Bit-Plane Slicing
Bit-planes of an 8-bit image.

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(a) An 8-bit gray-scale
image of size pixels. (b)
through (i) Bit planes 8
through 1, respectively,
where plane 1 contains
the least significant bit.
Each bit plane is a binary
image. Figure (a) is an
SEM image of a
trophozoite that causes a
disease called giardiasis.
(Courtesy of Dr. Stan
Erlandsen, U.S. Center
for Disease Control and
Prevention.)

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Image reconstructed from bit planes: (a) 8 and 7; (b) 8, 7, and 6; (c) 8, 7, 6, and 5.

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1. Background

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❖Histogram
➢The unnormalized histogram:
where rk is the k-th intensity level of an L-level
digital image f(x, y); nk is the number of pixels
in f with intensity rk and the subdivisions of the
intensity scale are called histogram bins.

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❖Histogram
➢The normalized histogram:
where M and N are the number of image rows and columns, respectively.
෍
𝑘=1
𝐿−1
𝑝 𝑟𝑘 = 1

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❖Histogram
Four image types and their corresponding histograms. (a) dark; (b) light; (c) low contrast; (d) high contrast.
The horizontal axis of the histograms are values of rk and the vertical axis are values of p(rk) .

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❖Histogram Equalization
➢The probability of occurrence of intensity level in a digital image is approximated
by
where MN is the total number of pixels in the image, and nk denotes the number of pixels that have
intensity rk.
a histogram equalization or histogram linearization transformation
➢The discrete form of the transformation function is

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➢Example: Illustration of the mechanics of histogram equalization.
• Suppose that a 3-bit image (L = 3) of size 64x64 pixels (MN = 4096) has the intensity distribution
in Table

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We round them to their nearest integer values in the range [0, 7]:
The values of the equalized histogram.

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Histogram equalization. (a) Original histogram. (b) Transformation function. (c) Equalized histogram.

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➢Algorithm for Histogram Equalization

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Histogram-
equalized images
Source images Histogram Histogram-
equalized images
Source images Histogram

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(a) Image from Phoenix Lander. (b) Result of
histogram equalization. (c) Histogram of image
(a). (d) Histogram of image (b). (Original image
courtesy of NASA.)

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1. Background

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❖The Mechanics of Linear Spatial Filtering
➢Spatial filter kernel: filter kernel, kernel, mask,
template, and window
➢Linear spatial filtering

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❖Spatial Correlation
and Convolution
➢1-D illustration

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❖Spatial Correlation and Convolution
➢2-D illustration
➢Correlation
➢Convolution

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❖Spatial Correlation and Convolution

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1. Background

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➢Smoothing (also called averaging) spatial filters are used to reduce sharp
transitions in intensity.
➢Application: noise reduction, reduce aliasing, reduce irrelevant detail in an image,
smoothing the false contours, …
➢Linear spatial filtering
➢Nonlinear smoothing filters

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❖Box Filter Kernels

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❖Box Filter Kernels
➢Example: Lowpass filtering with a box
kernel
(a) Test pattern of size 1024x1024 pixels. (b)-(d)
Results of lowpass filtering with box kernels of
sizes 3x3, 11x11, and 21x21 respectively.

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❖Lowpass Gaussian Filter Kernels
➢Gaussian kernels of the form
Distances from the center for
various sizes of square kernels.

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(a) Sampling a Gaussian function to obtain a discrete Gaussian kernel.
The values shown are for K = 1 and  = 1. (b) Resulting kernel.

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➢Example: Lowpass filtering with a Gaussian kernel
(a)A test pattern of size 1024x1024. (b) Result of lowpass filtering the pattern with a Gaussian kernel of
size 21x21, with standard deviations  = 3.5. (c) Result of using a kernel of size 43x43, with  = 7. We
used K = 1 in all cases.

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➢Example: Lowpass filtering with a Gaussian kernel
(a) Result of filtering using a Gaussian kernels of size43x43, with  = 7. (b) Result of using
a kernel of 85x85, with the same value of . (c) Difference image.

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➢Example: Comparison of Gaussian and box filter smoothing characteristics.

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➢Example: Using lowpass filtering and thresholding for region extraction.

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❖Order-Statistic (Nonlinear) Filters
➢Median filter: replaces the value of the center pixel by the median of the intensity
values in the neighborhood of that pixel
→Effective in the presence of impulse noise (salt-and-pepper noise)
→The 50th percentile of a ranked set of numbers
➢Max filter:
→Finding the brightest points in an image or for eroding dark areas adjacent to light
regions
→The 100th percentile filter

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➢Min filter:
→used for the opposite purpose
→The 0th percentile filter

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➢Example: Median filtering

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1. Background

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❖Foundation
➢First-order derivative
➢Second-order derivative

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❖Image Sharpening—the Laplacian
➢Laplacian

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➢Laplacian kernel

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➢The basic way in which the Laplacian is used for image sharpening:
▪ c = 1 if the center element of the Laplacian kernel is positive
▪ c = -1 if the center element of the Laplacian kernel is negative

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➢Example: Image sharpening using the Laplacian
a) Blurred image of the North Pole of the moon. (b) Laplacian image obtained using the kernel in Fig.
3.51(a). (c) Image sharpened using Eq. (3-63) with c = -1. (d) Image sharpened using the same procedure,
but with the kernel in Fig. 3.51(b). (Original image courtesy of NASA.)

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❖Unsharp Masking and Highboost Filtering
➢Unsharp masking
▪ Blur the original image
▪ Subtract the blurred image from the original (the resulting difference is called the mask)
▪ Add the mask to the original
blurred image
▪ When k = 1 → unsharp masking
▪ When k > 1 → highboost filtering
▪ When 0  k < 1 → reduces the contribution of the unsharp mask

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1-D illustration of the mechanics of unsharp masking. (a) Original signal. (b) Blurred signal with original
shown dashed for reference. (c) Unsharp mask. (d) Sharpened signal, obtained by adding (c) to (a).

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(a) Unretouched “soft-tone” digital image of size 469x600 pixels. (b) Image blurred using a 31x31 Gaussian lowpass
filter with  = 5. (c) Mask. (d) Result of unsharp masking using Eq. (3-65) with k = 1. (e) and (f) Results of highboost
filtering with k = 2 and k = 3 respectively.

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❖Image Sharpening—the Gradient
➢The gradient of an image f at coordinates (x, y)
➢The magnitude (length) of vector f

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➢Roberts cross-gradient operators

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➢Sobel operators

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➢Filter masks
(a) A 3x3 region of an image, where the zs are intensity values. (b)–(c) Roberts cross-gradient operators.
(d)–(e) Sobel operators. All the kernel coefficients sum to zero, as expected of a derivative operator.

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➢Example: Using the gradient for edge enhancement.
(a) Image of a contact lens (note defects on the boundary at 4 and 5 o’clock).
(b) Sobel gradient. (Original image courtesy of Perceptics Corporation.)

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1. Background

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❖Transfer functions of ideal 1-D filters
Transfer functions of ideal 1-D
filters in the frequency domain
(u denotes frequency). (a)
Lowpass filter. (b) Highpass
filter. (c) Bandreject filter. (d)
Bandpass filter. (As before, we
show only positive frequencies
for simplicity.)

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A zone plate image of size 597x597 pixels
(a) A 1-D spatial lowpass filter function. (b) 2-D kernel
obtained by rotating the 1-D profile about its center.

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(a) Zone plate image filtered with a separable lowpass kernel. (b) Image
filtered with the isotropic lowpass kernel in Fig. 3.60(b).

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Spatial filtering of the zone
plate image. (a) Lowpass
result; this is the same as Fig.
3.61(b) . (b) Highpass result.
(c) Image (b) with intensities
scaled. (d) Bandreject result.
(e) Bandpass result. (f) Image
(e) with intensities scaled.

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1. Background

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➢ Laplacian is superior for enhancing fine detail.
➢ The gradient has a stronger response in areas of significant intensity transitions (ramps and steps).
(a) Image of whole body bone scan. (b) Laplacian of (a). (c) Sharpened image obtained by adding (a) and (b).
(d) Sobel gradient of image (a). (Original image courtesy of G.E. Medical Systems.)

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(e) Sobel image smoothed with a 3x3 box filter. (f) Mask image formed by the product of (b) and (e). (g) Sharpened image
obtained by the adding images (a) and (f). (h) Final result obtained by applying a power-law transformation to (g). Compare
images (g) and (h) with (a). (Original image courtesy of G.E. Medical Systems.)

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