[PPT] - Sampling and reconstruction CS 4620 Lecture 3 Lecture 3 Cornell PowerPoint Presentation

SLIDE 1

Lecture 3 • Cornell CS4620 Fall 2019

Sampling and reconstruction

CS 4620 Lecture 3

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SLIDE 2

Lecture 3 • Cornell CS4620 Fall 2019

Sampled representations

How to store and compute with continuous functions?
Common scheme for representation: samples

write down the function’s values at many points

[FvDFH fig.14.14b / Wolberg] 2

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Lecture 3 • Cornell CS4620 Fall 2019

Reconstruction

Making samples back into a continuous function

for output (need realizable method) for analysis or processing (need mathematical method) amounts to “guessing” what the function did in between

[FvDFH fig.14.14b / Wolberg] 3

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Lecture 3 • Cornell CS4620 Fall 2019

Filtering

Processing done on a function

can be executed in continuous form (e.g. analog circuit) but can also be executed using sampled representation

Simple example: smoothing by averaging

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Lecture 3 • Cornell CS4620 Fall 2019

Roots of sampling

Nyquist 1928; Shannon 1949

famous results in information theory

1940s: fjrst practical uses in telecommunications
1960s: fjrst digital audio systems
1970s: commercialization of digital audio
1982: introduction of the Compact Disc

the fjrst high-profjle consumer application

This is why all the terminology has a communications or audio

“fmavor”

early applications are 1D; for us 2D (images) is important

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Lecture 3 • Cornell CS4620 Fall 2019

Sampling in digital audio

Recording: sound to analog to samples to disc
Playback: disc to samples to analog to sound again

how can we be sure we are fjlling in the gaps correctly?

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Lecture 3 • Cornell CS4620 Fall 2019

Undersampling

What if we “missed” things between the samples?
Simple example: undersampling a sine wave

unsurprising result: information is lost surprising result: indistinguishable from lower frequency also was always indistinguishable from higher frequencies aliasing: signals “traveling in disguise” as other frequencies

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Lecture 3 • Cornell CS4620 Fall 2019

Undersampling

What if we “missed” things between the samples?
Simple example: undersampling a sine wave

unsurprising result: information is lost surprising result: indistinguishable from lower frequency also was always indistinguishable from higher frequencies aliasing: signals “traveling in disguise” as other frequencies

7

SLIDE 9

Lecture 3 • Cornell CS4620 Fall 2019

Undersampling

What if we “missed” things between the samples?
Simple example: undersampling a sine wave

unsurprising result: information is lost surprising result: indistinguishable from lower frequency also was always indistinguishable from higher frequencies aliasing: signals “traveling in disguise” as other frequencies

7

SLIDE 10

Lecture 3 • Cornell CS4620 Fall 2019

Undersampling

What if we “missed” things between the samples?
Simple example: undersampling a sine wave

unsurprising result: information is lost surprising result: indistinguishable from lower frequency also was always indistinguishable from higher frequencies aliasing: signals “traveling in disguise” as other frequencies

7

SLIDE 11

Lecture 3 • Cornell CS4620 Fall 2019

Undersampling

What if we “missed” things between the samples?
Simple example: undersampling a sine wave

unsurprising result: information is lost surprising result: indistinguishable from lower frequency also was always indistinguishable from higher frequencies aliasing: signals “traveling in disguise” as other frequencies

7

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Lecture 3 • Cornell CS4620 Fall 2019

Undersampling

What if we “missed” things between the samples?
Simple example: undersampling a sine wave

unsurprising result: information is lost surprising result: indistinguishable from lower frequency also was always indistinguishable from higher frequencies aliasing: signals “traveling in disguise” as other frequencies

7

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Lecture 3 • Cornell CS4620 Fall 2019

Preventing aliasing

Introduce lowpass fjlters:

remove high frequencies leaving only safe, low frequencies choose lowest frequency in reconstruction (disambiguate)

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Lecture 3 • Cornell CS4620 Fall 2019

Preventing aliasing

Introduce lowpass fjlters:

remove high frequencies leaving only safe, low frequencies choose lowest frequency in reconstruction (disambiguate)

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Lecture 3 • Cornell CS4620 Fall 2019

Linear fjltering: a key idea

Transformations on signals; e.g.:

bass/treble controls on stereo blurring/sharpening operations in image editing smoothing/noise reduction in tracking

Key properties

linearity: fjlter(f + g) = fjlter(f) + fjlter(g) shift invariance: behavior invariant to shifting the input

delaying an audio signal
sliding an image around
Can be modeled mathematically by convolution

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Lecture 3 • Cornell CS4620 Fall 2019

Convolution warm-up

basic idea: defjne a new function by averaging over a sliding window
a simple example to start off: smoothing

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Lecture 3 • Cornell CS4620 Fall 2019

Convolution warm-up

basic idea: defjne a new function by averaging over a sliding window
a simple example to start off: smoothing

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Lecture 3 • Cornell CS4620 Fall 2019

Convolution warm-up

basic idea: defjne a new function by averaging over a sliding window
a simple example to start off: smoothing

10

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Lecture 3 • Cornell CS4620 Fall 2019

Convolution warm-up

basic idea: defjne a new function by averaging over a sliding window
a simple example to start off: smoothing

10

SLIDE 20

Lecture 3 • Cornell CS4620 Fall 2019

Convolution warm-up

basic idea: defjne a new function by averaging over a sliding window
a simple example to start off: smoothing

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Lecture 3 • Cornell CS4620 Fall 2019

Convolution warm-up

Same moving average operation, expressed mathematically:

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c[i] = 1 2r + 1

i+r

X

j=i−r

a[j].

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Lecture 3 • Cornell CS4620 Fall 2019

Discrete convolution

Simple averaging:

every sample gets the same weight

Convolution: same idea but with weighted average

each sample gets its own weight (normally zero far away)

This is all convolution is: it is a moving weighted average

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c[i] = 1 2r + 1

i+r

X

j=i−r

a[j].

Σ

(a ? b)[i] = X

j

a[j]b[i − j].

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Lecture 3 • Cornell CS4620 Fall 2019

Filters

Sequence of weights b is called a filter
Filter is nonzero over its region of support

usually centered on zero: support radius r

Filter is normalized so that it sums to 1.0

this makes for a weighted average, not just any 

ld weighted sum
Most fjlters are symmetric about 0

since for images we usually want to treat  left and right the same a box fjlter

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Lecture 3 • Cornell CS4620 Fall 2019

Convolution and fjltering

Can express sliding average as convolution with a box filter
bbox = […, 0, 1, 1, 1, 1, 1, 0, …]/5

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Lecture 3 • Cornell CS4620 Fall 2019

1 1 0.6

1 5

/

1 5

/

1 5

/

.2 .2 .2 .2 .2 .2 .2 .2 a[j] b[i – j] a[j]b[i – j] b[i – j] a[j]b[i – j] (a b)[i]

× = × = = ×

Σ Σ Σ

6 –7 j i i – j i – j i – j

Example: box and step

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Lecture 3 • Cornell CS4620 Fall 2019

1 1 0.6

1 5

/

1 5

/

1 5

/

.2 .2 .2 .2 .2 .2 .2 .2 a[j] b[i – j] a[j]b[i – j] b[i – j] a[j]b[i – j] (a b)[i]

× = × = = ×

Σ Σ Σ

6 –7 j i i – j i – j i – j

Example: box and step

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Lecture 3 • Cornell CS4620 Fall 2019

1 1 0.6

1 5

/

1 5

/

1 5

/

.2 .2 .2 .2 .2 .2 .2 .2 a[j] b[i – j] a[j]b[i – j] b[i – j] a[j]b[i – j] (a b)[i]

× = × = = ×

Σ Σ Σ

6 –7 j i i – j i – j i – j

Example: box and step

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Lecture 3 • Cornell CS4620 Fall 2019

1 1 0.6

1 5

/

1 5

/

1 5

/

.2 .2 .2 .2 .2 .2 .2 .2 a[j] b[i – j] a[j]b[i – j] b[i – j] a[j]b[i – j] (a b)[i]

× = × = = ×

Σ Σ Σ

6 –7 j i i – j i – j i – j

Example: box and step

15

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Lecture 3 • Cornell CS4620 Fall 2019

1 1 0.6

1 5

/

1 5

/

1 5

/

.2 .2 .2 .2 .2 .2 .2 .2 a[j] b[i – j] a[j]b[i – j] b[i – j] a[j]b[i – j] (a b)[i]

× = × = = ×

Σ Σ Σ

6 –7 j i i – j i – j i – j

Example: box and step

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Lecture 3 • Cornell CS4620 Fall 2019

Convolution and fjltering

Convolution applies with any sequence of weights
Example: bell curve (gaussian-like) […, 1, 4, 6, 4, 1, …]/16

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Lecture 3 • Cornell CS4620 Fall 2019

Convolution and fjltering

Convolution applies with any sequence of weights
Example: bell curve (gaussian-like) […, 1, 4, 6, 4, 1, …]/16

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Lecture 3 • Cornell CS4620 Fall 2019

Discrete convolution

Notation:
Convolution is a multiplication-like operation

commutative associative distributes over addition scalars factor out identity: unit impulse e = […, 0, 0, 1, 0, 0, …]

Conceptually no distinction between fjlter and signal

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Lecture 3 • Cornell CS4620 Fall 2019

Discrete fjltering in 2D

Same equation, one more index

now the fjlter is a rectangle you slide around over a grid of numbers

Commonly applied to images

blurring (using box, using gaussian, …) sharpening (impulse minus blur)

Usefulness of associativity
ften apply several fjlters one after another: (((a * b1) * b2) * b3)

this is equivalent to applying one fjlter: a * (b1 * b2 * b3)

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Lecture 3 • Cornell CS4620 Fall 2019

And in pseudocode…

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Lecture 3 • Cornell CS4620 Fall 2019

riginal | box blur

sharpened | gaussian blur

[Philip Greenspun] 20

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Lecture 3 • Cornell CS4620 Fall 2019

riginal | box blur

sharpened | gaussian blur

[Philip Greenspun] 20

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Lecture 3 • Cornell CS4620 Fall 2019

Optimization: separable fjlters

basic alg. is O(r2): large fjlters get expensive fast!
defjnition: a2(x,y) is separable if it can be written as:

this is a useful property for fjlters because it allows factoring:

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Lecture 3 • Cornell CS4620 Fall 2019

Separable fjltering

first, convolve with this

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= *

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Lecture 3 • Cornell CS4620 Fall 2019

Separable fjltering

first, convolve with this second, convolve with this

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= *

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Lecture 3 • Cornell CS4620 Fall 2019

Continuous convolution: warm-up

Can apply sliding-window average to a continuous function just as

well

utput is continuous

integration replaces summation

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SLIDE 41

Lecture 3 • Cornell CS4620 Fall 2019

Continuous convolution: warm-up

Can apply sliding-window average to a continuous function just as

well

utput is continuous

integration replaces summation

23

SLIDE 42

Lecture 3 • Cornell CS4620 Fall 2019

Continuous convolution: warm-up

Can apply sliding-window average to a continuous function just as

well

utput is continuous

integration replaces summation

23

SLIDE 43

Lecture 3 • Cornell CS4620 Fall 2019

Continuous convolution: warm-up

Can apply sliding-window average to a continuous function just as

well

utput is continuous

integration replaces summation

23

SLIDE 44

Lecture 3 • Cornell CS4620 Fall 2019

Continuous convolution: warm-up

Can apply sliding-window average to a continuous function just as

well

utput is continuous

integration replaces summation

23

SLIDE 45

Lecture 3 • Cornell CS4620 Fall 2019

Continuous convolution

Sliding average expressed mathematically:

note difference in normalization (only for box)

Convolution just introduces weights

weighting is now by a function weighted integral is like weighted average again bounds are set by support of f(x)

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SLIDE 46

Lecture 3 • Cornell CS4620 Fall 2019

One more convolution

Continuous–discrete convolution

input: a sequence and a continuous function

utput: a continuous function

used for reconstruction and resampling

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SLIDE 47

Lecture 3 • Cornell CS4620 Fall 2019

Continuous-discrete convolution

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Lecture 3 • Cornell CS4620 Fall 2019

Continuous-discrete convolution

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Lecture 3 • Cornell CS4620 Fall 2019

Continuous-discrete convolution

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Lecture 3 • Cornell CS4620 Fall 2019

Continuous-discrete convolution

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Lecture 3 • Cornell CS4620 Fall 2019

Continuous-discrete convolution

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SLIDE 52

Lecture 3 • Cornell CS4620 Fall 2019

And in pseudocode…

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function reconstruct(sequence a, filter f, real x) s = 0 r = f.radius for i = dx re to bx + rc do s = s + a[i]f(x i) return s

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SLIDE 53

Lecture 3 • Cornell CS4620 Fall 2019

Resampling

Changing the sample rate

in images, this is enlarging and reducing

Creating more samples:

increasing the sample rate “upsampling” “enlarging”

Ending up with fewer samples:

decreasing the sample rate “downsampling” “reducing”

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SLIDE 54

Lecture 3 • Cornell CS4620 Fall 2019

Resampling

Reconstruction creates a continuous function

forget its origins, go ahead and sample it

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SLIDE 55

Lecture 3 • Cornell CS4620 Fall 2019

Resampling

Reconstruction creates a continuous function

forget its origins, go ahead and sample it

29

SLIDE 56

Lecture 3 • Cornell CS4620 Fall 2019

Resampling

Reconstruction creates a continuous function

forget its origins, go ahead and sample it

29

SLIDE 57

Lecture 3 • Cornell CS4620 Fall 2019

Resampling

Reconstruction creates a continuous function

forget its origins, go ahead and sample it

29

SLIDE 58

Lecture 3 • Cornell CS4620 Fall 2019

Resampling

Reconstruction creates a continuous function

forget its origins, go ahead and sample it

29

SLIDE 59

Lecture 3 • Cornell CS4620 Fall 2019

And in pseudocode…

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function resample(sequence a, filter f, real x0, real ∆x, int N) allocate output sequence b of length N for j = 0 to N − 1 do b[j] = reconstruct(a, f, x + j∆x) return b

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1 x0 x0 + (N – 1)Δx Δx

SLIDE 60

Lecture 3 • Cornell CS4620 Fall 2019

Defjning the source interval

Convenient: defjne samples using an interval to be resampled, and N
Exactly how to fjll an interval with N samples? Desiderata:

sample pattern should be centered in the interval (prevents shifting of content as you change resolution) sample patterns of adjacent intervals should tile (makes it meaningful to break up a signal into pieces) N all the way down to 1 should work

Solution: think of breaking into N subintervals and centering a

sample in each one

sample i goes at

31

xl + i + 0.5 N (xr − xl)

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1 x0 x4 xl xr x0 + (N – 1)Δx

x0 = xl + 1

2∆x

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xl + (N − 1

2)∆x

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e.g. fjrst sample at e.g. last sample at

SLIDE 61

Lecture 3 • Cornell CS4620 Fall 2019

And in pseudocode…

32

1 x0 x4 xl xr x0 + (N – 1)Δx

function resample(sequence a, filter f, real xl, real xr, int N) allocate output sequence b of length N for j = 0 to N − 1 do b[j] = reconstruct(a, f, xl + (j + 1

2)(xr − xl)/N)

return b

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Note that this expands into a double loop

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Lecture 3 • Cornell CS4620 Fall 2019

Cont.–disc. convolution in 2D

same convolution—just two variables now

loop over nearby pixels,   average using fjlter weight looks like discrete fjlter,  but offsets are not integers  and fjlter is continuous remember placement of fjlter  relative to grid is variable

33

pixel locations support of reconstruction fjlter sample point for reconstruction

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Lecture 3 • Cornell CS4620 Fall 2019

Cont.–disc. convolution in 2D

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Example showing  fjlter weights used  to compute a  reconstructed value  at a single point.

SLIDE 64

Lecture 3 • Cornell CS4620 Fall 2019

And in pseudocode…

35

function reconstruct(sequence a, filter f, real x, real y) s = 0 for j = dy f.rye to by + f.ryc do for i = dx f.rxe to bx + f.rxc do s = s + a[i, j]f(x i, y j) return s

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SLIDE 65

Lecture 3 • Cornell CS4620 Fall 2019

Defjning the source rectangle

exactly the same way we defjned the source interval in 1D
except now there is

an interval (l, r) in x  and an interval (b, t)  in y

intervals in x and y

constitute a rectangle

sample points are

positioned with the  same considerations  (at centers of a grid 

f sub-rectangles)

36

(l, b) (r, t)

SLIDE 66

Lecture 3 • Cornell CS4620 Fall 2019

And in pseudocode…

37

Note that this expands into a quadruple loop

function resample(image a, filter f, rectangle r, int w, int h) allocate output image b of size (w, h) for j = 0 to h − 1 do y = r.b + (j + 1

2)(r.t − r.b)/h

for i = 0 to w − 1 do x = r.l + (i + 1

2)(r.r − r.l)/w

b[i, j] = reconstruct(a, f, x, y) return b

<latexit sha1_base64="J4cXop21yLJpCB828rpkvKUs=">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</latexit>

SLIDE 67

Lecture 3 • Cornell CS4620 Fall 2019

Separable fjlters for resampling

just as in fjltering, separable fjlters are useful

separability in this context is a statement about a continuous fjlter, rather than a discrete one:

with a separable fjlter the region of support is rectangular
all widely used resampling fjlters

are separable

there are good reasons best explained   with frequency domain arguments (it’s easiest to design separable fjlters   to suppress grid artifacts)

38

(x, y)

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[i, j]

<latexit sha1_base64="K+lfoJOS9nSwJe0ZcE70FEFEi0g=">AB7HicbZDLSsNAFIZPvNZ6q7p0M1gEF1KSNmB3Fty4rGDaQhrKZDpx04mYWYilNBncONCEbc+kDvfwkcwTap4+2Hg4z/ncM78fsyZ0qb5Ziwtr6yurZc2yptb2zu7lb39joSahDIh7Jno8V5UxQRzPNaS+WFIc+p1/cjGvd2+pVCwS13oaUy/EI8ECRrDOLMdlpzfeoFI1a2Yu9BesBVTP3yFXe1B57Q8jkoRUaMKxUq5lxtpLsdSMcDor9xNFY0wmeETdDAUOqfLS/NgZOs6cIQoimT2hUe5+n0hxqNQ09LPOEOux+l2bm/V3EQHTS9lIk40FaRYFCQc6QjNf46GTFKi+TQDTCTLbkVkjCUmOsunIdg1+tNu4EKaFif0PwKoVOvWY2aeWVXW3aRBpTgEI7gBCw4gxZcQhscIMDgDh7g0RDGvfFkPBetS8Zi5gB+yHj5ABTVj8k=</latexit>

|y − j|

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|x − i|

<latexit sha1_base64="SbWzCKrM1Pz5uOEGkhVFWFTj2WI=">AB7HicbZDLSsNAFIZP6q3W9Wlm8EiuLEkbcDuLhxWcG0QhvKZDph04mYWYilrTP4MaFIm59IHe+hY9gmlTx9sPAx3/O4Zz5vYgzpU3zSgsLa+srhXSxubW9s75d29tgpjSahDQh7Kaw8rypmgjma0+tIUhx4nHa8fm83rmhUrFQXOlJRN0ADwXzGcE6tZzp7Qmb9sVs2pmQn/BWkDl7B0ytfrl194gJHFAhSYcK9W1zEi7CZaEU5npV6saITJGA9pN0WBA6rcJDt2ho5SZ4D8UKZPaJS53ycSHCg1Cby0M8B6pH7X5uZ/tW6s/YabMBHFmgqSL/JjnSI5j9HAyYp0XySAiaSpbciMsISE53mU8pCsGu1hl1HOdStT2h8hdCuVa161by0K07TwOKcACHcAwWnEITLqAFDhBgcAcP8GgI4954Mp7z1oKxmNmHzJePgCNQ5AY</latexit>

SLIDE 68

Lecture 3 • Cornell CS4620 Fall 2019

Optimized separable resampling

for larger fjlters, separability provides an opportunity for an
ptimization
resample in two passes, one resampling each row and one

resampling each column

intermediate storage required: product of one dimension of source

and the other dimension of destination

39

SLIDE 69

Lecture 3 • Cornell CS4620 Fall 2019

two-stage resampling using a  separable fjlter

[Philip Greenspun] 40

O(rNsrcMdst)

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O(rNdstMdst)

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O(r2NdstMdst)

<latexit sha1_base64="VQEX6BGf+q8wLAL3pXoDs6pW7Q0=">ACB3icbZDLSsNAFIZP6r3eqi4FGSyCbkqSFuxOwY0b2Ct0MYymU516OTCzIlYQndufBU3LhRx6yu48y18BNOkSr38MPDx/+cwM78bSqHRN+N3Nj4xOTU9Ex+dm5+YbGwtHymg0gxXmOBDNS5SzWXwuc1FCj5eag49VzJ6253b5DXr7nSIvBPsRdyx6OXvugIRjGxWoW1o01YR+2mshvMG5r7B+M8FarUDRLZiryF6whFHc+INVxq/DWbAcs8riPTFKtG5YZohNThYJ3s83I81Dyr0kjcS9KnHtROn/+iTjcRpk06gkuMjSd3RjZh6Wvc8N5n0KF7p39nA/C9rRNipOrHwi5z7KLOpEkGJBKaQtFGcoewlQpkTyVsKuqKIMk+ryaQkV265WyiSDsvUF1e8SzuySVS6ZJ5XibiVrA6ZhFdZhEyzYhl3Yh2OoAYNbuIdHeDLujAfj2XjJRnPGcGcFfsh4/QGxZsA</latexit>

SLIDE 70

Lecture 3 • Cornell CS4620 Fall 2019

A gallery of fjlters

Box fjlter

Simple and cheap

Tent fjlter

Linear interpolation

Gaussian fjlter

Very smooth antialiasing fjlter

B-spline cubic

Very smooth

Catmull-rom cubic

Interpolating

Mitchell-Netravali cubic

Good for image upsampling

41

SLIDE 71

Lecture 3 • Cornell CS4620 Fall 2019

Box fjlter

42

SLIDE 72

Lecture 3 • Cornell CS4620 Fall 2019

Tent fjlter

43

SLIDE 73

Lecture 3 • Cornell CS4620 Fall 2019

Gaussian fjlter

44

SLIDE 74

Lecture 3 • Cornell CS4620 Fall 2019

B-Spline cubic

45

fB(x) = 1 6      −3(1 − |x|)3 + 3(1 − |x|)2 + 3(1 − |x|) + 1 −1 ≤ x ≤ 1, (2 − |x|)3 1 ≤ |x| ≤ 2,

therwise.

SLIDE 75

Lecture 3 • Cornell CS4620 Fall 2019

Catmull-Rom cubic

46

fC(x) = 1 2      −3(1 − |x|)3 + 4(1 − |x|)2 + (1 − |x|) −1 ≤ x ≤ 1, (2 − |x|)3 − (2 − |x|)2 1 ≤ |x| ≤ 2,

therwise.

SLIDE 76

Lecture 3 • Cornell CS4620 Fall 2019

Michell-Netravali cubic

47

fM(x) = 1 3fB(x) + 2 3fC(x) = 1 18      −21(1 − |x|)3 + 27(1 − |x|)2 + 9(1 − |x|) + 1 −1 ≤ x ≤ 1, 7(2 − |x|)3 − 6(2 − |x|)2 1 ≤ |x| ≤ 2,

therwise.

SLIDE 77

Lecture 3 • Cornell CS4620 Fall 2019

Lanczos

48

1

2
1

1 2 1

3
2
1

1 2 3

fL2(x) = ( sinc(x) sinc(x/2) |x| < 2

therwise

fL3(x) = ( sinc(x) sinc(x/3) |x| < 3

therwise

sinc(x) = sin(πx) πx

SLIDE 78

Lecture 3 • Cornell CS4620 Fall 2019

Effects of reconstruction fjlters

For some fjlters, the reconstruction process winds up implementing a

simple algorithm

Box fjlter (radius 0.5): nearest neighbor sampling

box always catches exactly one input point it is the input point nearest the output point so output[i, j] = input[round(x(i)), round(y(j))]  x(i) computes the position of the output coordinate i on the input grid

Tent fjlter (radius 1): linear interpolation

tent catches exactly 2 input points weights are a and (1 – a) result is straight-line interpolation from one point to the next

49

SLIDE 79

Lecture 3 • Cornell CS4620 Fall 2019

Properties of fjlters

Degree of continuity
Impulse response
Interpolating or no
Ringing, or overshoot

interpolating filter used for reconstruction

50

SLIDE 80

Lecture 3 • Cornell CS4620 Fall 2019

Ringing, overshoot, ripples

Overshoot

caused by  negative fjlter  values

Ripples

constant in,   non-const. out ripple free when:

51

SLIDE 81

Lecture 3 • Cornell CS4620 Fall 2019

Constructing 2D fjlters

Separable fjlters (most common approach)

52

SLIDE 82

Lecture 3 • Cornell CS4620 Fall 2019

Yucky details

What about near the edge?

the fjlter window falls off the edge of the image need to extrapolate methods:

clip fjlter (black)
wrap around
copy edge
refmect across edge
vary fjlter near edge

53 [Philip Greenspun]

SLIDE 83

Lecture 3 • Cornell CS4620 Fall 2019

Yucky details

What about near the edge?

the fjlter window falls off the edge of the image need to extrapolate methods:

clip fjlter (black)
wrap around
copy edge
refmect across edge
vary fjlter near edge

53 [Philip Greenspun]

SLIDE 84

Lecture 3 • Cornell CS4620 Fall 2019

Yucky details

What about near the edge?

the fjlter window falls off the edge of the image need to extrapolate methods:

clip fjlter (black)
wrap around
copy edge
refmect across edge
vary fjlter near edge

53 [Philip Greenspun]

SLIDE 85

Lecture 3 • Cornell CS4620 Fall 2019

Yucky details

What about near the edge?

the fjlter window falls off the edge of the image need to extrapolate methods:

clip fjlter (black)
wrap around
copy edge
refmect across edge
vary fjlter near edge

53 [Philip Greenspun]

SLIDE 86

Lecture 3 • Cornell CS4620 Fall 2019

Yucky details

What about near the edge?

the fjlter window falls off the edge of the image need to extrapolate methods:

clip fjlter (black)
wrap around
copy edge
refmect across edge
vary fjlter near edge

53 [Philip Greenspun]

SLIDE 87

Lecture 3 • Cornell CS4620 Fall 2019

Yucky details

What about near the edge?

the fjlter window falls off the edge of the image need to extrapolate methods:

clip fjlter (black)
wrap around
copy edge
refmect across edge
vary fjlter near edge

53 [Philip Greenspun]

SLIDE 88

Lecture 3 • Cornell CS4620 Fall 2019

Yucky details

What about near the edge?

the fjlter window falls off the edge of the image need to extrapolate methods:

clip fjlter (black)
wrap around
copy edge
refmect across edge
vary fjlter near edge

53 [Philip Greenspun]

SLIDE 89

Lecture 3 • Cornell CS4620 Fall 2019

Yucky details

What about near the edge?

the fjlter window falls off the edge of the image need to extrapolate methods:

clip fjlter (black)
wrap around
copy edge
refmect across edge
vary fjlter near edge

53 [Philip Greenspun]

SLIDE 90

Lecture 3 • Cornell CS4620 Fall 2019

Yucky details

What about near the edge?

the fjlter window falls off the edge of the image need to extrapolate methods:

clip fjlter (black)
wrap around
copy edge
refmect across edge
vary fjlter near edge

53 [Philip Greenspun]

SLIDE 91

Lecture 3 • Cornell CS4620 Fall 2019

Yucky details

What about near the edge?

the fjlter window falls off the edge of the image need to extrapolate methods:

clip fjlter (black)
wrap around
copy edge
refmect across edge
vary fjlter near edge

53 [Philip Greenspun]

SLIDE 92

Lecture 3 • Cornell CS4620 Fall 2019

Yucky details

What about near the edge?

the fjlter window falls off the edge of the image need to extrapolate methods:

clip fjlter (black)
wrap around
copy edge
refmect across edge
vary fjlter near edge

53 [Philip Greenspun]

SLIDE 93

Lecture 3 • Cornell CS4620 Fall 2019

Yucky details

What about near the edge?

the fjlter window falls off the edge of the image need to extrapolate methods:

clip fjlter (black)
wrap around
copy edge
refmect across edge
vary fjlter near edge

53 [Philip Greenspun]

SLIDE 94

Lecture 3 • Cornell CS4620 Fall 2019

Yucky details

What about near the edge?

the fjlter window falls off the edge of the image need to extrapolate methods:

clip fjlter (black)
wrap around
copy edge
refmect across edge
vary fjlter near edge

53 [Philip Greenspun]

SLIDE 95

Lecture 3 • Cornell CS4620 Fall 2019

Reducing and enlarging

Very common operation

devices have differing resolutions applications have different memory/quality tradeoffs

Also very commonly done poorly
Simple approach: drop/replicate pixels
Correct approach: use resampling

54

SLIDE 96

Lecture 3 • Cornell CS4620 Fall 2019

1000 pixel width

[Philip Greenspun] 55

SLIDE 97

Lecture 3 • Cornell CS4620 Fall 2019

250 pixel width by dropping pixels gaussian fjlter

[Philip Greenspun] 56

SLIDE 98

Lecture 3 • Cornell CS4620 Fall 2019

4000 pixel width box reconstruction fjlter bicubic reconstruction fjlter

[Philip Greenspun] 57

SLIDE 99

Lecture 3 • Cornell CS4620 Fall 2019

Types of artifacts

Garden variety

what we saw in this natural image fjne features become jagged or sparkle

Moiré patterns

58

SLIDE 100

Lecture 3 • Cornell CS4620 Fall 2019

600ppi scan of a color halftone image

[Hearn & Baker cover] 59

SLIDE 101

Lecture 3 • Cornell CS4620 Fall 2019

downsampling a high resolution scan by dropping pixels gaussian fjlter

[Hearn & Baker cover] 60

SLIDE 102

Lecture 3 • Cornell CS4620 Fall 2019

Types of artifacts

Garden variety

what we saw in this natural image fjne features become jagged or sparkle

Moiré patterns

caused by repetitive patterns in input produce large-scale artifacts; highly visible

These artifacts are aliasing just like in the audio example earlier
How do I know what fjlter is best at preventing aliasing?

practical answer: experience theoretical answer: there is another layer of cool math behind all this

based on Fourier transforms
provides much insight into aliasing, fjltering, sampling, and reconstruction

61

Sampling and reconstruction

CS 4620 Lecture 3

Sampled representations

write down the function’s values at many points

Reconstruction

for output (need realizable method) for analysis or processing (need mathematical method) amounts to “guessing” what the function did in between

Filtering

can be executed in continuous form (e.g. analog circuit) but can also be executed using sampled representation

Roots of sampling

famous results in information theory

the fjrst high-profjle consumer application

“fmavor”

early applications are 1D; for us 2D (images) is important

Sampling in digital audio

how can we be sure we are fjlling in the gaps correctly?

Undersampling

unsurprising result: information is lost surprising result: indistinguishable from lower frequency also was always indistinguishable from higher frequencies aliasing: signals “traveling in disguise” as other frequencies

Undersampling

unsurprising result: information is lost surprising result: indistinguishable from lower frequency also was always indistinguishable from higher frequencies aliasing: signals “traveling in disguise” as other frequencies

Undersampling

unsurprising result: information is lost surprising result: indistinguishable from lower frequency also was always indistinguishable from higher frequencies aliasing: signals “traveling in disguise” as other frequencies

Undersampling

unsurprising result: information is lost surprising result: indistinguishable from lower frequency also was always indistinguishable from higher frequencies aliasing: signals “traveling in disguise” as other frequencies

Undersampling

unsurprising result: information is lost surprising result: indistinguishable from lower frequency also was always indistinguishable from higher frequencies aliasing: signals “traveling in disguise” as other frequencies

Undersampling

unsurprising result: information is lost surprising result: indistinguishable from lower frequency also was always indistinguishable from higher frequencies aliasing: signals “traveling in disguise” as other frequencies

Preventing aliasing

remove high frequencies leaving only safe, low frequencies choose lowest frequency in reconstruction (disambiguate)

Preventing aliasing

remove high frequencies leaving only safe, low frequencies choose lowest frequency in reconstruction (disambiguate)

Linear fjltering: a key idea

bass/treble controls on stereo blurring/sharpening operations in image editing smoothing/noise reduction in tracking

linearity: fjlter(f + g) = fjlter(f) + fjlter(g) shift invariance: behavior invariant to shifting the input

Convolution warm-up

Convolution warm-up

Convolution warm-up

Convolution warm-up

Convolution warm-up

Convolution warm-up

c[i] = 1 2r + 1

i+r

X

j=i−r

a[j].

Discrete convolution

every sample gets the same weight

each sample gets its own weight (normally zero far away)

c[i] = 1 2r + 1

i+r

X

j=i−r

a[j].

Σ

(a ? b)[i] = X

j

a[j]b[i − j].

Filters

usually centered on zero: support radius r

this makes for a weighted average, not just any

since for images we usually want to treat left and right the same a box fjlter

Convolution and fjltering

× = × = = ×

Σ Σ Σ

Example: box and step

× = × = = ×

Σ Σ Σ

Example: box and step

× = × = = ×

Σ Σ Σ

Example: box and step

× = × = = ×

Σ Σ Σ

Example: box and step

× = × = = ×

Σ Σ Σ

Example: box and step

Convolution and fjltering

Convolution and fjltering

Discrete convolution

this makes for a weighted average, not just any 

since for images we usually want to treat  left and right the same a box fjlter