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ECE 6364 Spring 2009 HW 3 Due 2/
Problem 1. Fundamentals of Digital Image Processing - Jain: Problem 2.4 a,b,c,d
Problem 2. Fundamentals of Digital Image Processing - Jain: Problem 4.
Problem 3. Fundamentals of Digital Image Processing - Jain: Problem 4.
Note the typo in book: the reconstruction filter has bandwidths of Hz x y ⎟
Problem 4. Fundamentals of Digital Image Processing Jain Problem 4.
Problem 5. Recall, given a continuous image g ( x , y ), an ideal point sample of g ( x , y )is simply g ( mTx , nTy ) where Tx
and T (^) y are the sample spacing in the x and y directions. Mathematically, we can form g ( mTx , nTy )as
g mTx nTy ∫ g x y x mTx y nTy dx dy
+∞
−∞
( , )= (^) ∫ ( , )δ( − , − ) , but this is simply the same as evaluating g ( x , y )at the
point ( x , y )= ( mTx , nTy ).
Consider a CCD. Let us set up our coordinate-system so that ( x,y )=(0,0) is at the very center of the upper-left cell of the CCD. Then the CCD forms a discrete image by integrating the photons from some continuous image f ( x , y )that is projected onto the
CCD by the lens. The ( m , n ) th sample f (^) d ( m , n )from the CCD in
formed by integrating photons from within the continuous region of f ( x , y )that corresponds to the ( m , n ) th cell of the CCD.
We can express this with the equation
f mn f γ β d γ d β
n Ty
n Ty
m Tx
m Tx
d ∫ ∫
−
−
( 0. 5 )
( 0. 5 )
( 0. 5 )
( 0. 5 )
Let us define a window function ⎩
⎨
⎧ ≤ ≤
else
xy x Tx and y Ty wT (^) xTy 0 1 0. 5 0. 5 , ( , )
(a) Draw a 2-D graph of w (^) , ( x , y ) Tx Ty , labeling pertinent points on the x and y axes.
Now, consider a second continuous function g ( x , y ), related to f ( x , y )as
= (^) ∫ γβ γ− β− γ β
+∞
−∞
g ( x , y ) ∫ f ( , ) wTx , T (^) y ( x , y ) d d.
Notice: (1) the integrand f ( γ, β) wTx , T (^) y (γ− x ,β− y )is a product of two functions, each of which is a function of
independent variables ( γ, β), and the integral forms g ( x , y )by integrating the product over all values of ( γ, β).
x
y
(0,0) 1.5Tx
1.5Ty
x
y
(0,0) 1.5T1.5Txx
1.5T1.5Tyy
(2) wT (^) x , T (^) y (γ − x ,β− y )is a function of ( γ ,β)formed by shifting wT (^) x , Ty (γ,β) by an amount ( x , y )in the
( γ, β)directions.
(3) f (^) d ( m , n )= g ( mTx , nTy ); i.e. the CCD-samples are equal to point-samples of the continuous image g ( x , y ).
(4) The window function w (^) , ( x , y ) Tx Ty is symmetric about the origin, so that
wT (^) x , T (^) y ( γ − x ,β− y )= wTx , Ty ( x −γ, y −β)by which
g ( x , y )= (^) ∫ f (γ,β) wTx , T (^) y ( x −γ, y −β) d γ d β= f ( x , y )** wTx , Ty ( x , y )
+∞
−∞
By taking the continuous Fourier transform of both sides of the above equation, we obtain G ( u , v )= F ( u , v ) WTx , T (^) y ( u , v )
(b) To show that g ( x , y )is a low-pass version of f ( x , y ), compute the Fourier transform W (^) , ( u , v ) Tx Ty of
w (^) , ( x , y ) Tx Ty
. Graph W (^) , ( u , v ) T (^) xTy . Label pertinent points on the u , v axes of the graph. Find the particular
values of u , v where W (^) Tx , Ty ( u , v )= 0. Compute and label the amplitude WT (^) x , Ty ( 0 , 0 ) as a function of T (^) x , Ty.
(c) Find the -3 dB frequencies u (^) − 3 dB and v (^) − 3 dB for W (^) , ( u , v ) T (^) xTy as a function of T (^) x and T (^) y ; i.e. where
( 0 , 0 ) ( , 0 ) ( 0 , ) 2
1 WT (^) x , Ty = WTx , Ty u − 3 dB = WTx , Ty v − 3 dB
Conclusions: You should now be able to see that (1) The CCD-samples f (^) d ( m , n )are equivalent to ideal point-samples of a continuous image g ( x , y )where
g ( x , y )is the low-pass-filtered version f ( x , y )** wTx , Ty ( x , y )of the continuous image f ( x , y )that is
projected onto the CCD by the lens. (2) The -3 dB cutoff frequencies of the lowpass filter w (^) Tx , Ty ( x , y )are important from the viewpoint of the
sampling theorem. These frequencies, as you computed in 4c above, are sufficiently low as to preclude aliasing when a CCD forms a discrete image { f (^) d ( m , n )}whose samples are equivalent to point samples of
f ( x , y )** wTx , Ty ( x , y ), a lowpass filtered f ( x , y ).
(3) If you consider W (^) , ( u , v ) T (^) xTy as approximating an ideal lowpass filter whose cutoff frequency is equal to
the -3 dB cutoff freq of W (^) , ( u , v ) T (^) xTy , then the CCD sample spacing is sufficiently small as to recover the
lowpass image g ( x , y )from the point samples { f (^) d ( m , n )}. That is, you cannot recover f ( x , y )from { f (^) d ( m , n )},
but you can recover g ( x , y )where ( , ) ( , )** , ( x , y ) TxTy g xy = f xy w from { f (^) d ( m , n )}.
(4) A CCD has a sort of self-regulating lowpass filter W (^) , ( u , v ) Tx Ty that is proportional to the size of the CCD
cells. If the cells are larger, then the -3 dB cutoff frequencies of W (^) , ( u , v ) T (^) xTy are lower, making the image
g ( x , y )that can be recovered from { f (^) d ( m , n )}a more lowpass filtered version of f ( x , y )than would be the
case if the cells of the CCD were smaller.
