ECE 525.674
Image Processing and Analysis
Prof. John Goutsias
Assignment #1
Please READ instructions at the end of assignment
1. For each input-output relationship given below, determine whether the corresponding image
processor is linear/shift invariant.
(a) g ( x , y )  f ( x , y ) f ( x  z , y ) .
(b) g( x ) 
z

f ( x, z )dz .

(c) g ( m, n )  f ( m,  1)  f ( m, 0 )  f ( m,1)
2. Given a continuous image f ( x , y )  ( x  y )3 , compute images
g1 ( x, y )  f ( x, y )  ( x  1, y  2) and g 2 ( x, y )  f ( x, y )   ( x  1, y  2) .
3. Convolve the discrete image f ( m, n )  a m n u( m, n ) with the discrete image
h( m, n ) 

 ( m  rM , n  rN ) , where u( m, n ) is the unit step sequence, defined by
r 
1, if m  0 and n  0
u (m, n)  
.
0, otherwise
4. Consider two images f ( x , y ) and g ( x , y ) that are separable, i.e., such that
f ( x , y )  f1 ( x ) f 2 ( y )
and
g ( x , y )  g1 ( x ) g 2 ( y ) .
Is their convolution a separable image? Express their convolution in terms of f1 , f 2 , g1 , and
g2 .
5. Compute the Fourier transforms of the following 2-D continuous images
(a) f ( x, y )  sin(2 ax ) cos(2 y ) .
(b) f ( x, y )  cos(2 ( ax   y )) .
6. If F ( u, v ) is the Fourier transform of the continuous image f ( x , y ) , compute the Fourier
transform of the continuous image f ( ax, by ) for any constants a and b . How do the values
of a and b affect the image f ( x , y ) ?
7. If F ( u, v ) is the Fourier transform of the continuous image f ( x , y ) , compute the Fourier
transform of the continuous image f ( ax  by , cx  dy ) when a, b, c, d are such that
ad  bc  1.
1
8. For the imaging system shown below, with a, b  0 , show that the output image g ( x , y ) is a
scaled and inverted replica of the input.
f ( x, y )
f ( ax , by )
scaling
F 
F  
g( x, y )
9. Calculate the convolution ( f  h )( m, n ) , where:
Note that the image f ( m, n ) is 0 at all pixels, except at those pixels indicated by the black
dots at which the value is 1. On the other hand, image h ( m, n ) takes value 0 at all pixels,
except at those pixels indicated by the black dots at which the values are shown within the
parentheses.
10. The Fourier transform F ( u, v ) of a discrete image f ( m, n ) is given by
F ( u, v )  3  2 cos( 2 u )  4 j sin( 2 v )  8e  j 2  ( u v ) .
Determine f ( m, n ) .
11. Consider the following discrete image f ( m, n ) , taking value 0 at all pixels, except at those
pixels indicated by the black dots at which the values (which are all real) are shown within
the parentheses.
2
(a) What can you say about F ( u, v ) without explicitly computing it?
(b) Determine F( 0,0 ) .
(c) Determine F ( u, v ) .
12. (MATLAB) For each image file image1.bmp, image2.jpg, and image3.jpg:
(a) Determine its type (binary, grayscale, or colored).
(b) Determine its size.
(c) Determine the maximum and minimum grayscale values.
(d) Determine the number of bits required to represent the raw information.
(e) Use the imwrite function of MATLAB to write image3.jpg into files of type TIFF,
BMP and PNG. How many bytes each file (including the original) occupies on the hard
disk? What do you observe?
13. (MATLAB)
(a) By using the imresize function of MATLAB, demonstrate for image4.jpg the effect of
reducing the sample grid size by a factor of 2, 4 and 8. To see the effect, the resulting
images must be displayed using the same size as of the original image. Mark the number
of pixels associated with each image.
(b) By using the grayslice function of MATLAB, quantize image2.jpg into 256, 128, 64,
32, 16, 8, 4, and 2 gray levels. At which number of gray levels you start observing
significant distortion?
14. (MATLAB)
(a) Generate a 256  256 pixel binary image that contains a disk of radius 15 located at the
center of the image. By using the fft2 function of MATLAB, compute and display the
magnitude of its Fourier transform. The resulting magnitude is subject to ringing, why?
Gently modify the disk to reduce the ringing effect.
(b) Convolve image4.jpg with a 2-D image processor whose point-spread sequence is a
512  512 pixel zero-mean Gaussian curve with standard deviation 4 (using the
fspecial function of MATLAB). What do you observe? Show that the result is the
same if the image processor is implemented in the frequency domain.
INSTRUCTIONS
1.
2.
3.
4.
Each problem carries 10 points.
You need to solve only 7 of problems 1-11.
If you solve additional problems, you will receive more points.
The images required to complete the MATLAB problems can be downloaded from
http://www.cis.jhu.edu/~goutsias/temp/ipdata.zip
5. You must include ALL results (images, explanations, and answers to questions)
associated with each problem, as well as your MATLAB code.
3
6. Your work (including the MATLAB code) should be submitted as ONE file using the
following convention: LastName-FirstName-Asgmt#.pdf, where # is the assignment
number (1, 2, etc).
4