Pixels, Numbers, and Programs Introduction · 2014. 8. 26. · Intro. Image Proc. & Python © S. Tanimoto Sampling Introduction to Image Processing and Python--Image Representation:

Intro. Image Proc. & Python © S. Tanimoto Sampling

Introduction to Image Processing and Python--Image Representation: Sampling

Sampling

Steven L. Tanimoto

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1Saturday, August 24, 13

SamplingIntro. Image Proc. & Python © S. Tanimoto

Key Issues in Image

• Image capture• Image representation• Image transformation• Image analysis• Image synthesis• Image data management

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Key Issues in Image

• Image capture• Image representation• Image transformation• Image analysis• Image synthesis• Image data management

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Digital Image Representation

When we want to capture a visual scene in a digital image, we face two fundamental questions:

1. How densely should our sample points be arranged? ("sampling")

2. How should each sample (e.g., color) be represented with number(s)? ("quantization")

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Sampling

Density of samples (sampling rate or "frequency")Placement of samples (grid systems and alignment

or "phase")

Samples could represent very small regions or larger areas. This is an issue that the designers of digital cameras must consider.

We'll focus mainly on the concept of sampling rate.

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Representing a Continuous Image

• Let’s call our original image or view to be represented our “picture.”

• The picture may be formed with a lens focusing light from a scene onto a surface.

• The picture might be a work of art created with watercolors on white artist’s paper.

• The picture is “continuous” in the sense that it has a color defined at every (infinitessimal) point within its boundaries.

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Sampling

• Sampling means making a selection of points (samples) to represent the picture.

• Usually, the samples are organized with a grid.

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Pixels, Numbers, and Programs; © S. Tanimoto --

Sampling

Example

Photo of Lincoln Sample points Digitized

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Steven Tanimoto 4/3/08if ((x+10) mod 20)+((y+10) mod 20) = 0 then 255 else s1(x,y)/2Is the formula used to produce the dots on the darkened version of the original.


SamplingIntro. Image Proc. & Python © S. Tanimoto 9



Sampling Rate

• How many samples per unit (distance or area for pictures, time for sound)

• For pictures, we will typically use distance, but the units might vary.

• Direct units: inches, centimeters• Indirect units: “width of the picture”• (If we are re-sampling a digital image) “old

pixel widths”.

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How Many Samples is Enough?

• An important question because it relates to:• Perceived Image quality• Faithfulness in representing details• Cost of various kinds: • amount of memory required• time to download images from camera or

web.• time to process the image.

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Oscillating Patterns

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English Camp on San Juan Island, near Roche Harbor, WA.

S. Tanimoto



Oscillating Patterns

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• We use a picket fence as an example of an oscillating pattern because it’s a clear case for teaching and learning – almost a caricature.

• In reality, there may be high-frequency oscillating patterns subtly embedded in the image.

• For example, any sharp edge in a scene contains high-frequency wavy patterns, but they may be invisible except at the edge due to interference elsewhere.



Consider the Picket Fence

To represent this "oscillation", a series of samples from left to right will need a sample on each picket and in each space between pickets.

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Consider the Picket Fence

To represent this "oscillation", a series of samples from left to right will need a sample on each picket and in each space between pickets.

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Nyquist Criterion

• Suppose that the most rapidly changing pattern changes at a rate of k cycles per unit distance.

• The Nyquist criterion says that we’ll need at least 2k samples per unit distance to represent this pattern.

• That’s a minimum of 2 samples per cycle.• (Also known as Shannon sampling or

Nyquist-Shannon sampling)

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Aliasing

If the sampling rate is lower than the Nyquist rate, we won't have the true number of pickets showing. The oscillating pattern will seem to have a different (lower) frequency than the correct frequency. This is known as aliasing.

In graphics, the blocky character of diagonal or curved lines is also called aliasing.

Both forms of aliasing are due to insufficient resolution (i.e., sampling at too low a rate).

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Aliased Pickets

original high-res image downsampled using 1 to 10 in each direction (horizontally and vertically)

s1(10*floor(x/10), 10*floor(y/10))

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PixelMath example of aliasing:Moire effect with an artificial image

• Start with sin(x) --not visible in PixelMath• 128 * (1 + sin(x)) --visible but boring• Multiply x by 0.5 to widen the stripes (actually

waves)• Add a small component that is a nonlinear

function of y to get some curvature. y2 / 1000.• Final PixelMath formula to synthesize a pattern of

curved stripes: 128 * (1+ sin(x*0.5+y*y/1000))

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source pattern of stripes

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PixelMath example

Resample at every other pixel. Transformation formula: Source1(x*2, y*2) Aliasing is not a problem at this rate.

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S1(x*2, y*2)

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PixelMath example

• Resample with a rate that is too low to capture the highest frequency oscillation.

• Take every 10th pixel in each direction• Transformation formula: Source1(x*10, y*10)

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destination (shrunk)

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destination (blowup)

The aliasing here is so bad that some of the curves bend in a completely different direction.

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One Dimensional Case

Square wave and sine wave. Shown are 4 cycles of each. Here y is a function of x.

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1D Case (cont.)

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Sums of Sinusoids

• An arbitrary function y = f(x) can be expressed as a sum of sinusoids.∗

• * The functions must be “smooth” and bounded.

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Adding Sinusoids

http://www.udel.edu/idsardi/sinewave/sinewave.html

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magenta = red + blue



The “Nyquist Rate”

• Let g(x) be a signal to be sampled.• Let fh be the highest frequency component

in g(x)• The Nyquist rate fr = 2 fh is a lower bound

on the sampling frequency needed to accurately represent g(x).

• We actually must sample at a frequency higher (at least slightly higher) than fr to capture g(x)

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The “Nyquist Frequency”

• Let fs be a chosen sampling frequency.• The frequency fN = fs / 2 is the Nyquist

frequency.• The Nyquist frequency is an upper bound on

the frequency of any oscillation in g(x) that is to be accurately represented in the sampling result.

• Note that fr < 2 fN

• Thus, fr / 2 < fN

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The Shannon-Nyquist Theorem

• Let g(x) be a continuously varying function whose highest frequency component, at fB satisfies 2 fB < fs, the chosen sampling frequency.

• Then the samples obtained by sampling g(x) using sampling frequency fs uniquely determine g(x).

• That is, it’s theoretically possible to exactly reconstruct g(x) from the samples.

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Nyquist freq., rate, etc.

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Assumptions

• g(x) truly is bandlimited to fB.• The samples are perfectly accurate.• The reconstruction is perfect.In practice, none of these is really true.However, the theorem does guide us to the

correct sampling rates for pretty darn accurate (though not perfect) representations.

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Conclusion

To avoid aliasing, use a sampling rate that allows more than two samples (pixel) per cycle of the highest-frequency oscillation in the image.

Or, prior to sampling, remove (using filtering) components at frequencies more than half the sampling frequency.

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Pixels, Numbers, and Programs Introduction · 2014. 8. 26. · Intro. Image Proc. & Python © S. Tanimoto Sampling Introduction to Image Processing and Python--Image Representation: