Recursive filtering techniques are often used to reduce the computational complexity of a repeated operation such as filtering. If an image filter is applied to each location in an image, a (horizontally) recursive formulation of the filtering operation expresses the result at location (x +1, y) in terms of the previously computed result at location (x, y). A box convolution filter, B, which has coefficients equal to one inside a rectangular win- dow, and zero elsewhere is given by: w-1h-1 B(r, y, w, h) = ΣΣ+ ,y +) i=0 j=0 where I(r, y) is the pixel intensity of image I at (r, y). We can speed up the computation of arbitrary sized box filters using recursion as described above. In this problem, you will derive the procedure to do this. (a) The function J at location (x,y) is defined to be the sum of the pixel values above and to the left of (x,y), inclusive: J(r, y) = EI(i, j) i=0 j=0 Formulate a recursion to compute J(r, y). Assume that I(r, y) 0 if r <0 or y < 0. Hint: It may be useful to consider an intermediate image to simplify the recursion. (b) Given J(r, y) computed from an input image, the value of an arbitrary sized box filter (BJ) applied anywhere on the original image can be computed using four references to J(r, y). BJ(r,y, w, h) = aJ(?,?) + bJ(?, ?) + cJ(?, ?) + dJ(?, ?) h (x, y) Figure 2: Visualization of box filter computation. Find the values of a, b, c, d and the ?'s to make this formula correct. Specifically, your answer should be the above equation with appropriate values for the above unknowns. Hint: It may be useful to visualize this process as shown in Fig. 2
Recursive filtering techniques are often used to reduce the computational complexity of a repeated operation such as filtering. If an image filter is applied to each location in an image, a (horizontally) recursive formulation of the filtering operation expresses the result at location (x +1, y) in terms of the previously computed result at location (x, y). A box convolution filter, B, which has coefficients equal to one inside a rectangular win- dow, and zero elsewhere is given by: w-1h-1 B(r, y, w, h) = ΣΣ+ ,y +) i=0 j=0 where I(r, y) is the pixel intensity of image I at (r, y). We can speed up the computation of arbitrary sized box filters using recursion as described above. In this problem, you will derive the procedure to do this. (a) The function J at location (x,y) is defined to be the sum of the pixel values above and to the left of (x,y), inclusive: J(r, y) = EI(i, j) i=0 j=0 Formulate a recursion to compute J(r, y). Assume that I(r, y) 0 if r <0 or y < 0. Hint: It may be useful to consider an intermediate image to simplify the recursion. (b) Given J(r, y) computed from an input image, the value of an arbitrary sized box filter (BJ) applied anywhere on the original image can be computed using four references to J(r, y). BJ(r,y, w, h) = aJ(?,?) + bJ(?, ?) + cJ(?, ?) + dJ(?, ?) h (x, y) Figure 2: Visualization of box filter computation. Find the values of a, b, c, d and the ?'s to make this formula correct. Specifically, your answer should be the above equation with appropriate values for the above unknowns. Hint: It may be useful to visualize this process as shown in Fig. 2
Computer Networking: A Top-Down Approach (7th Edition)
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Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
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Transcribed Image Text:Recursive filtering techniques are often used to reduce the computational complexity of
a repeated operation such as filtering. If an image filter is applied to each location in an
image, a (horizontally) recursive formulation of the filtering operation expresses the result
at location (x +1, y) in terms of the previously computed result at location (x, y).
A box convolution filter, B, which has coefficients equal to one inside a rectangular win-
dow, and zero elsewhere is given by:
w-1h-1
B(r, y,w, h) =
ΣΣΤ+ i,y + )
i=0 j=0
where I(r, y) is the pixel intensity of image I at (x, y). We can speed up the computation
of arbitrary sized box filters using recursion as described above. In this problem, you will
derive the procedure to do this.
(a) The function J at location (x,y) is defined to be the sum of the pixel values above
and to the left of (x,y), inclusive:
J(r, y) =
- ΣΣ14.0
i=0 j=0
Formulate a recursion to compute J(r, y). Assume that I(r, y) = 0 if r <0 or y < 0.
Hint: It may be useful to consider an intermediate image to simplify the recursion.
(b) Given J(r, y) computed from an input image, the value of an arbitrary sized box filter
(BJ) applied anywhere on the original image can be computed using four references
to J(r, y).
BJ(r, y, w, h) = aJ(?,?) + bJ(?,?)+ cJ(?, ?) + dJ(?, ?)
(х, у);
w
Figure 2: Visualization of box filter computation.
Find the values of a, b, c, d and the ?'s to make this formula correct.
Specifically, your answer should be the above equation with appropriate values for the
above unknowns. Hint: It may be useful to visualize this process as shown in Fig. 2
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