Downsampling, upsampling, and filtering are used in a pyramid architecture to get multiple resolution levels for images and shown in lecture slides At each stage of the pyramid, a filtered input signal is downsampled by a factor of 2 to provide the input to the next level of the pyramid, That signal is then processed to try to predict the input signal, and the residual difference between the input and the prediction can be used for analysis and compression. This process can be analyzed for a one dimensional signal. The signals at level j are defined as follows: • ??[?] is the input to level j. It is filtered by ℎ[?] to produce ??[?] • ??−1[?′] is the output of level j and the input to the next level, ( j-1). It is created by downsampling ??[?] by a factor of 2. • ??[?] is created by upsampling ??−1[?′] by a factor of 2 • ??[?] is created by filtering ??[?] by ℎ[?] and then multiplying by 2. • ??[?] is subtracted from ??[?] to compute the residual difference for the jth level. a. Show that ??[?] = 0.5 ∙ (1 + ???(??)) ∙ ??[?]. (This relationship does not depend on the specific input values or specific filter values.) b. Can ??[?] have energy at frequencies that have zero energy in ??[?]? Why or why not c. If ℎ[?] were an ideal low pass filter that introduced no delay, describe all the signals ( ??[?], ??−1[?′], ??[?], ??[?]) in terms the input signal, ??[?].
Downsampling, upsampling, and filtering are used in a pyramid architecture to get multiple resolution levels for images and shown in lecture slides At each stage of the pyramid, a filtered input signal is downsampled by a factor of 2 to provide the input to the next level of the pyramid, That signal is then processed to try to predict the input signal, and the residual difference between the input and the prediction can be used for analysis and compression. This process can be analyzed for a one dimensional signal. The signals at level j are defined as follows: • ??[?] is the input to level j. It is filtered by ℎ[?] to produce ??[?] • ??−1[?′] is the output of level j and the input to the next level, ( j-1). It is created by downsampling ??[?] by a factor of 2. • ??[?] is created by upsampling ??−1[?′] by a factor of 2 • ??[?] is created by filtering ??[?] by ℎ[?] and then multiplying by 2. • ??[?] is subtracted from ??[?] to compute the residual difference for the jth level. a. Show that ??[?] = 0.5 ∙ (1 + ???(??)) ∙ ??[?]. (This relationship does not depend on the specific input values or specific filter values.) b. Can ??[?] have energy at frequencies that have zero energy in ??[?]? Why or why not c. If ℎ[?] were an ideal low pass filter that introduced no delay, describe all the signals ( ??[?], ??−1[?′], ??[?], ??[?]) in terms the input signal, ??[?].
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
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Chapter1: Computer Networks And The Internet
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Downsampling, upsampling, and filtering are used in a pyramid
architecture to get multiple resolution levels for images and shown in lecture slides At each
stage of the pyramid, a filtered input signal is downsampled by a factor of 2 to provide the
input to the next level of the pyramid, That signal is then processed to try to predict the input
signal, and the residual difference between the input and the prediction can be used for
analysis and compression. This process can be analyzed for a one dimensional signal. The
signals at level j are defined as follows:
• ??[?] is the input to level j. It is filtered by ℎ[?] to produce ??[?]
• ??−1[?′] is the output of level j and the input to the next level, ( j-1). It is created by
downsampling ??[?] by a factor of 2.
• ??[?] is created by upsampling ??−1[?′] by a factor of 2
• ??[?] is created by filtering ??[?] by ℎ[?] and then multiplying by 2.
• ??[?] is subtracted from ??[?] to compute the residual difference for the jth level.
a. Show that ??[?] = 0.5 ∙ (1 + ???(??)) ∙ ??[?]. (This relationship does not depend on the
specific input values or specific filter values.)
b. Can ??[?] have energy at frequencies that have zero energy in ??[?]? Why or why not
c. If ℎ[?] were an ideal low pass filter that introduced no delay, describe all the signals
( ??[?], ??−1[?′], ??[?], ??[?]) in terms the input signal, ??[?].
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