Overview
Making shaded images with procedures Making a more elegant language Making textures with noise functions
Making bitmaps
[bitmap-from-procedure procedure width height] Returns a bitmap width pixels by height pixels Obtains colors for pixels by calling procedure
Passes procedure a point (vector) object with the coordinates
Procedure returns a color object (color image) or a single number (grayscale image)
Making a grayscale ramp
We can use dot products to make a grayscale ramp
The dot-product of two vectors is a single number
It increases as The first vector grows in the
direction of the second, or alternatively,
The second grows in the direction of the first
i.e. it’s symmetrical Each vector gets bigger
By manipulating the direction of the vector in the dot-product, we can get different directions for the ramp
[bitmap-from-procedure[p → [dot p [vector 0.5 0.5]]]256256]
Changing the vector direction
[bitmap-from-procedure[p → [dot p [vector 1 0]]]256256]
[bitmap-from-procedure[p → [dot p [vector 0 1]]]256256]
Clipping
Why is this one different? The display only has the
ability to produce amounts of light in the range 0-255
[bitmap-from-procedure[p → [dot p [vector 1 1]]]256256]
Constructive laziness
It’s bad to keep typing the same arguments over and over again Slows you down Frustrates you Leads to errors
So make a new procedure that Just takes the parameter
we care about and Fills in the rest for us
[define show [procedure →
[bitmap-from-procedure procedure 256 256]]]
Now we can just say: [show [p → [dot p [vector 1 1]]]]
Making it (slightly) more efficient
This version calls the vector procedure for each pixel
Wasteful Arguments are always the
same Same vector is always
returned (more or less) We only need to call it once
So we can call it once and save it in a local variable
[show [p → [dot p [vector 1 1]]]]
[show [with vec = [vector 1 1] [p → [dot p vec]]]
Making another pattern
The magnitude procedure gives the length of the vector passed to it
So using it as the painting function for a bitmap gives us
A radial pattern That’s black at (0,0) And grows white farther away
Like a grayscale ramp, but circular
[show magnitude]
Question
What’s the difference between:
magnitude
and:[p → [magnitude p]]
Nothing; they behave the same Both take a vector as input And return its magnitude
as output
[show magnitude][show [p → [magnitude p]]]
Shifting it over
How do we shift it so that the black part is in the center of the image?
We shift the vector we’re taking the magnitude of We use − to shift vectors
What vector do we subtract off? It’s a 256×256 image So the center is (128, 128)
[show [p → [magnitude [− p [vector 128 128]]]]]
Making it brighter
Now it’s too dark How do we fix it?
Brighten by multiplying Multiplying by 2 Doubles the brightness Although now it clips
[show [p → [× 2 [magnitude [− p [vector 128 128]]]]]]
White on black
Okay, but what if we want it to be white in the center and black on the outside? How do we fix it?
Just subtract The brightness computed
brightness From 255 (the maximum
brightness)
[show [p → [− 255 [× 2 [magnitude [− p [vector 128 128]]]]]]]
Overview
Making shaded images with procedures Making a more elegant language Making textures with noise functions
Thinking about it differently
We started with a simple pattern (magnitude)
We shifted it over We brightened it We inverted it
But the code doesn’t make that clear Can we make the code
reflect our intentions more clearly?
[show [p → [− 255 [× 2 [magnitude [− p [vector 128 128]]]]]]]
Making the code clearer
We’d like to be able to write the code like this Take the magnitude image Shift it by [vector 128 128] Brighten it And invert it
Can we make Meta work this way?
[show[invert [brighten 2 [shift [vector 128 128] magnitude]]]]
Procedures that operate on whole patterns
Let’s call A pattern a procedure that
takes a point and returns a brightness
A pattern operator a procedure that takes a pattern and returns a new pattern
If we had pattern operators For shifting, brightening, and
inverting We could write the code much
more clearly
[shift vector pattern] Makes a new pattern that’s
pattern shifted over by vector [brighten level pattern]
Makes a new pattern that’s like pattern but with its brightness multiplied by level
[invert pattern] Makes a new pattern that’s
the same as pattern, but with black exchanged for white
Programming with pattern operators
Now we can take the magnitude pattern Since magnitude takes
a point as input And returns a number
So we can use it as a pattern
[show magnitude]
Programming with pattern operators
Now we can take the magnitude pattern
Shift it
[show [shift [vector 128 128] magnitude]]
Programming with pattern operators
Now we can take the magnitude pattern
Shift it Brighten it
[show [brighten 2 [shift [vector 128 128] magnitude]]]
Programming with pattern operators
Now we can take the magnitude pattern
Shift it Brighten it And invert it
Now we don’t even need to know that patterns are really procedures
[show[invert [brighten 2 [shift [vector 128 128] magnitude]]]]
Writing brighten Okay, now let’s write brighten
Remember we want to be able to say something like:
[brighten 2 [p → [magnitude p]]]
And get back something that behaves like:
[p → [× 2 [magnitude p]]]
► [define brighten ???]
Writing brighten
Okay, now let’s write brighten
Brighten must Take a brightness level and a
pattern as arguments
► [define brighten [level pattern → ???]]
Writing brighten
Okay, now let’s write brighten
Brighten must Take a brightness level and a
pattern as arguments Return a new procedure that
Takes a point as an argument
► [define brighten [level pattern → [p → ???]]]
Writing brighten
Okay, now let’s write brighten
Brighten must Take a brightness level and a
pattern as arguments Return a new procedure that
Takes a point as an argument And computes the correct
brightness
► [define brighten [level pattern → [p → [× level [pattern p]]]]]
Higher-order procedures
Brighten is called a “higher-order” procedure It takes a procedure as
input And returns a new
procedure as a result
Procedures that make new procedures are perfectly acceptable And very useful
[define brighten [level pattern → [p → [× level [pattern p]]]]]
[brighten 2 magnitude]= [p →
[× 2 [magnitude p]]]
[brighten 3 magntidue]= [p → [× 3 [magnitude p]]]
[brighten 3 [brighten 2 magnitude]]
= [p → [× 3 [ [p → [× 2 [magnitude p]]] p]]
≈ [p → [× 3 [× 2 [magnitude p]]]]
Writing invert
Okay, now let’s write invert
Invert must Take a a pattern
► [define invert [pattern → ???]]
Writing invert
Okay, now let’s write invert
Invert must Take a a pattern Return a new procedure that
Takes a point as an argument
► [define invert [pattern → [p → ???]]]
Writing invert
Okay, now let’s write invert
Invert must Take a a pattern Return a new procedure that
Takes a point as an argument And computes the correct
brightness
► [define invert [pattern → [p → [− 255 [pattern p]]]]]
Writing shift
Patterns are procedures
So shift is a higher order procedure
It takes a procedure as an argument
And returns a new procedure as its result
► [define shift ???]
Writing shift
Patterns are procedures
So shift is a higher order procedure
It takes a procedure as an argument
And returns a new procedure as its result
Shift must Take an offset and a pattern as
arguments
► [define shift [offset pattern → ???]]
Writing shift
Patterns are procedures
So shift is a higher order procedure
It takes a procedure as an argument
And returns a new procedure as its result
Shift must Take an offset and a pattern as
arguments Return a new procedure that
Takes a point as an argument
► [define shift [offset pattern → [p → ???]]]
Writing shift
Patterns are procedures
So shift is a higher order procedure
It takes a procedure as an argument
And returns a new procedure as its result
Shift must Take an offset and a pattern as
arguments Return a new procedure that
Takes a point as an argument And computes the correct
brightness
► [define shift [offset pattern → [p → [pattern [− p offset]]]]]
Repeated patterns
[mod a b] For positive numbers,
returns the remainder when dividing a by b
When a is negative, returns b minus the remainder
This turns out to be just the right thing to make a repeated texture
[define replicate[width height pattern → [p → [pattern [point [mod p.X width]
[mod p.Y height]]]]]]
[show [replicate 64 64 magnitude]]
Repeated patterns
[brighten 8 [replicate 64 64 [shift [point 32 32] magnitude]]]]
[replicate 64 64 [invert magnitude]]
And now, in color …
[define colorize[r-pattern g-pattern b-pattern → [p → [color [r-pattern p] [g-pattern p] [b-pattern p]]]]]
Takes three grayscale patterns and combines them into a single color pattern
Groovy, man
[colorize [brighten 8 [replicate 64 64 [shift [point 32 32] magnitude]]] [replicate 64 64 [invert magnitude]]
[p → 50]]]
Groovy, man
[colorize [brighten 8 [replicate 64 64 [shift [point 32 32] magnitude]]] [replicate 64 64 [p → [dot p [point 1 1]]]]
[p → 50]]]
Overview
Making shaded images with procedures Making a more elegant language Making textures with noise functions
Spatial frequencies
Pictures can be thought of as having harmonic structure
Like sound Picture can be thought of as
many different frequencies combined
We won’t go into this in any detail, but …
Frequency corresponds roughly to size
Fine detail is high frequency Bigger structures are lower
frequeny
White noise
White noise is a random signal Every pixel (sample)
computed using a random number generator
Called “white” because it contains equal amounts of all frequencies
Not very interesting as a texture
► [show [p → [random-integer 0 255]]]
Bandpass noise
But now suppose we zoom in between the randomly chosen pixel values
And smoothly interpolate between them
The result is still a random texture, but it’s missing the very high and very low frequencies
[noise point] Interpolated noise Gaussian distribution Result is between -0.7 and 0.7
[show [p → [+ 128 [× 128 [noise [ ⁄ p 30]]]]]]
Bandpass noise at different frequences
[show [p → [+ 128 [× 128 [noise [ ⁄ p 10]]]]]]
[show [p → [+ 128 [× 128 [noise [ ⁄ p 80]]]]]]
Summing bandpass noise
You can get interesting effects by summing bandpass noise at different frequencies
[show [p → [+ 128 [× 128 [+ [noise [ ⁄ p 80]] [noise [ ⁄ p 40]] [noise [ ⁄ p 20]]]]]]]
“1/f noise”
Important kind of noise Amplitude of frequency f is 1/f Self-similar (like fractals)
Zoom in on it and it still looks like itself
Approximated using bandpass noise
Compute at different scales Sum with weights that vary
inversely with frequency
Also known as Brown noise (Brownian motion) Turbulence
[show [p → [+ 128 [× 128 [+ [noise [ ⁄ p 80]] [ ⁄ [noise [ ⁄ p 40]] 2] [ ⁄ [noise [ ⁄ p 20]] 4]]]]]]
Perlin noise
Ken Perlin (1985) Technique for
approximating 1/f noise using interpolated bandpass noise
Built into most graphics cards
[turbulence point] Computes a sum of many
calls to noise All you really need to
understand for this class
[show[p → [+ 128 [× 128 [turbulence [ ⁄ p 30]]]]]]
Annoying bug/misfeature
You have to divide p by some number before calling noise or turbulence The noise function returns 0 for all points
whose coordinates are both integers So you have to divide by something to make
sure the coordinates are usually not integers
The Art of Noise
[show[p → [× 255 [turbulence [ ⁄ p 30]]]]]
Noise clips on low end
[show[p → [abs [× 255 [turbulence [ ⁄ p 30]]]]]]
Abs produces abrupt change on low end
The Art of Noise
[show[with center = [point 128 128] [p → [× 255 [sin [+ [ ⁄ [magnitude [− p center]] 10]
[turbulence [ ⁄ p 30]]]]]]]
Noise used in input to another function (sin)
[show[p → [× 255 [sin [+ [turbulence [ ⁄ p 20]] [dot p [point 0.1 0.1]]]]]]]
Extra arguments to turbulence Turbulence works by calling noise at
different frequencies and summing
You can call turbulence with three extra arguments
Drop-off factor Amplitude of the new noise component
drops by this factor each iteration Frequency multiplier
Frequency get multiplied by this each iteration
Iteration count
Try playing with the extra arguments The default values are 2, 2, and 4
[show[p → [+ 127 [× 128 [turbulence [ ⁄ p 30] 1.5 2 8]]]]]
Extra arguments to turbulence Turbulence works by calling noise at
different frequencies and summing
You can call turbulence with three extra arguments
Drop-off factor Amplitude of the new noise component
drops by this factor each iteration Frequency multiplier
Frequency get multiplied by this each iteration
Iteration count
Try playing with the extra arguments The default values are 2, 2, and 4
[show[p → [+ 127 [× 128 [turbulence [ ⁄ p 30] 5 2 8]]]]]
The Art of Noise
[show[p → [× 256
[turbulence [× 0.001 p.X p.Y]]]]]
You can also call noise or turbulencewith a number as an argument (rather than a point)
[show[p → [× 256 [cos [× 10 [turbulence [ ⁄ p 50]]]]]]]
Using the code from today’s lecture
Meta is distributed with the code for brighten, invert, etc.
To load it and make it available, just type:[using Examples.Painting] Doing it twice won’t hurt anything You only need to do this when you start using Meta But you have to do it over again if you exit Meta and
run it again later
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