Practice Midterm, EE 264 Spring 2018
1) Below left you see a Fourier series of frequencies and below
on the right an empty frequency plot. Please, fill in the empty
frequency plot with the Fourier series!
Note: One frequency is twice as high as the other one. They both
have the same amplitude.
2) What would be the result if you were to convolve the image
below and the kernel below?
Answer: The image will be shifted one pixel to the left.
3) What operation could you use to produce the image (c) from
the images (a) and (b)?
(a) (b) (c)
Answer: a – b = c
Note: You need to subtract b from a. If you instead add a and b
you make the shading gradient worse. If you instead to perform some
type of multiplication or division with the images, you could
manage to remove the background shading gradient in the light
squares, but you would corrupt the black squares, of the
checkerboard pattern.
4) An image histogram lets us visualize what gray level values
are present in an image, and how many pixels there are in the image
with a specific one of these gray levels. What does the histogram
of the synthetic image of squares (below on the left) look like?
Please, draw out the histogram in the empty histogram plot!
(Hint: As you can see, there are four gray levels present in
this image. You can assume that this is an 8 bit image, meaning
gray values go from 0 black to 255 white)
5) When we take a photograph with a camera (or image with a
microscope) the image is effectively a convolution of every point
in the object with the optical transfer function of the camera.
This transfer function is airy function, which is the Fourier
transform of the circular aperture of the camera. The effect is a
blurring the finest details in the object. Below you see an example
of an image convolution that works in the same way. The image on
the left is convolved with the Gaussian kernel in the middle which
produces the image on the right.
Now, what would happen if we convolved the image with the same
Gaussian again?
· Answer: The image would become even more blurred
Note: The convolution is a linear operation. The convolution of
a Gaussian with itself is a wider Gaussian. Thus, the convolution
of an image followed by another convolution of the resulting image
with the same Gaussian, will produce more blur. Identically, the
convolution of the original image with the Gaussian convolved by
itself will produce this more blurred image.
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EE 264: Image Processing and Reconstruction Peyman Milanfar UCSC
EE Dept. 7
Histogram Processing:
Graylevel
9
15
7
5 Histogram:
• Distribution of gray-levels can be judged by measuring a
Histogram
EE 264: Image Processing and Reconstruction Peyman Milanfar UCSC
EE Dept. 8
For B-bit image,
• Initialize 2B counters with 0 • Loop over all pixels x,y •
When encountering gray level f(x,y)=i, increment the counter number
i
• With proper normalization, the histogram can be interpreted as
an estimate of the probability density function (pdf) of the
underlying random variable (the graylevel) •You can also use fewer,
larger bins to trade off amplitude
Histogram Processing:
4
EE 264: Image Processing and Reconstruction
Peyman Milanfar
UCSC EE Dept.
7
Histog ram Proce ssin g:
Graylevel
9
15
7
5
Histogram:
•Distributi on of gray -levels can be judged
by meas uring a Histogram
EE 264: Image Processing and Reconstruction
Peyman Milanfar
UCSC EE Dept.
8
For B-bit image,
• Initialize 2
B
counters with 0
• Loop ov er all pixels x,y
• When encountering gra y level f(x,y)=i, increment
the counter number i
• With proper normalizat ion, the histogram can be
interpret ed as an estimate of the probability densi ty
function (pdf) of the underly ing random variable (the
graylevel)
•You can also use fewer, larger bins to trade off
amplitude
His togram Proc essing:
14
EE 264: Image Processing and Reconstruction Peyman Milanfar UCSC
EE Dept.
Choosing kernel width
• Rule of thumb: set filter half-width to about 3 σ
EE 264: Image Processing and Reconstruction Peyman Milanfar UCSC
EE Dept.
Example: Smoothing with a Gaussian
14
EE 264: Image Processing and Reconstruction
Peyman Milanfar
UCSC EE Dept.
Choos ing kernel width
•Rule of thum b: set filter half -width to about
3 σ
EE 264: Image Processing and Reconstruction
Peyman Milanfar
UCSC EE Dept.
Examp le: Smoo thin g with a Gau ssia n
14
EE 264: Image Processing and Reconstruction Peyman Milanfar UCSC
EE Dept.
Choosing kernel width
• Rule of thumb: set filter half-width to about 3 σ
EE 264: Image Processing and Reconstruction Peyman Milanfar UCSC
EE Dept.
Example: Smoothing with a Gaussian
14
EE 264: Image Processing and Reconstruction
Peyman Milanfar
UCSC EE Dept.
Choos ing kernel width
•Rule of thum b: set filter half -width to about
3 σ
EE 264: Image Processing and Reconstruction
Peyman Milanfar
UCSC EE Dept.
Examp le: Smoo thin g with a Gau ssia n
14
EE 264: Image Processing and Reconstruction Peyman Milanfar UCSC
EE Dept.
Choosing kernel width
• Rule of thumb: set filter half-width to about 3 σ
EE 264: Image Processing and Reconstruction Peyman Milanfar UCSC
EE Dept.
Example: Smoothing with a Gaussian
14
EE 264: Image Processing and Reconstruction
Peyman Milanfar
UCSC EE Dept.
Choos ing kernel width
•Rule of thum b: set filter half -width to about
3 σ
EE 264: Image Processing and Reconstruction
Peyman Milanfar
UCSC EE Dept.
Examp le: Smoo thin g with a Gau ssia n
9
EE 264: Image Processing and Reconstruction Peyman Milanfar UCSC
EE Dept.
Practice with linear filters
0 0 0
1 0 0
0 0 0
Original
?
Source: D. Lowe
EE 264: Image Processing and Reconstruction Peyman Milanfar UCSC
EE Dept.
Practice with linear filters
0 0 0
1 0 0
0 0 0
Original Shifted left By 1 pixel
Source: D. Lowe
9
EE 264: Image Processing and Reconstruction
Peyman Milanfar
UCSC EE Dept.
Prac tice with linear filters
0 0 0
1 0 0
0 0 0
Original
?
Source: D. Lowe
EE 264: Image Processing and Reconstruction
Peyman Milanfar
UCSC EE Dept.
Prac tice with linear filters
0 0 0
1 0 0
0 0 0
Original
Shifted left
By 1 pixel
Source: D. Lowe
