Computer Science 121
Scientific ComputingWinter 2014Chapter 14
Images
Background: Images
● Signal (sound, last chapter) is a single (one-dimensional) quantity that varies over time.
● Image (picture) can be thought of as a two-dimensional quantity that varies over space.
● Otherwise, the computations for sounds and images are about the same!
14.1 Black-and-White Images
14.1 Black-and-White Images
14.1 Black-and-White Images
14.1 Black-and-White Images
● Eventually we get down to a single point (pixel, “picture element”) whose color we need to represent.
● For black-and-white images, two basic options– Halftone: Black (0) or white (1); having lots of
points makes a region look gray– Grayscale: Some set of values (2N, N typically = 8)
representing shades of gray.
14.1 Black-and-White Images● To turn a b/w image into a grid of pixels, imagine
running a stylus horizontally along the image:
14.1 Black-and-White Images● This gives us a one-dimensional function showing
gray level at each point along the horizontal axis:
14.1 Black-and-White Images● Doing this repeatedly with lots of evenly-spaced scan
lines gives us a grayscale map of the image:
● Height = grayscale value
● Spacing between lines =
spacing within line
14.2 Color● Light comes in difference wavelengths (frequencies):
14.2 Color● Like a sound, energy from a given light source (e.g.,
star) can be described by its wavelength (frequency) spectrum : the amount of each wavelength present in the source.
Spectrum of vowel “ee”Frequency (Hz)
Ene
rgy
(dB
)
Spectrum of star ICR3287
Wavelength (nm)
Inte
nsity
14.2 Color● Light striking our eyes is focused by a lens and
projected onto the retina.● The retina contains photoreceptors (sensors)
called cones that respond to different wavelengths of light: red, green, and blue.
● So retina can be thought of as a transducer that inputs light and outputs a three-dimensional value for each point in the image: [R, G, B], where R, G, and B each fall in the interval (0,1).
● So we can represent any color image as three “grayscale” images:
14.2 Color
=
,R
,G B
14.3 Digital Sampling of Images● Same questions arise for sampling images as for
sound:–Sampling Frequency (how often to sample)–Quantization (# of bits per sample)
● With images, “how often” means “how many times per linear unit (inch, cm) – a.k.a. resolution
● Focus on quantization–Each sample is an index into a color map–With too few bits, we lose gradual shading
Color Maps
● With 8 bits per color, 3 colors per pixel, we get 24 bits per pixel: 224 ≈ 100,000,000 distinct colors
● We can actually get away with far fewer, using color maps● Each pixel has a value that tells us what row in the color map to use● Color map rows are R, G, B values:
Color Maps>> colormap
ans =
0 0 0.5625 0 0 0.6250 ... 0.4375 1.0000 0.5625 0.5000 1.0000 0.5000 ...
0.5625 0 00.5000
0 0>> size(colormap)
ans = 64 3
Color MapsMatlab provides us with some default color maps:>> drawMandelbrot([-2.5 2.5], [-2 2])
Color Maps
>> colormap(hot)
Color Maps
>> colormap(flag)
Quantization ProblemsWith too few bits, we lose gradual shading:
5 bits
1 bit
2 bits
3 bits
4 bits
6 bits
7 bits
8 bits
14.4 Sampling and Storing Images in Files
• Scanners usually output images in one of several formats– JP(E)G (Joint Photographic Experts Group)– GIF (Graphics Interchange Format)– PNG (Portable Network Graphics)– TI(F)F (Tagged Image File Format)
• As with sound formats, main issues in image formats are compression scheme (how images are stored and transmitted to save space/time) and copyright
• We'll focus on Matlab and mathematical issues....
14.4 Sampling and Storing Images in Files
• Matlab imread commands lets us read in files in several formats (automagically figures out which):
>> a = imread('sakidehli.png'); % b/w>> size(a)ans = 269 176>> b = imread('monaLisaLouvre.jpg'); % color>> size(b)ans = 864 560 3
14.4 Sampling and Storing Images in Files
• Then use image to display image:
>> image(b)
14.4 Sampling and Storing Images in Files
14.4 Sampling and Storing Images in Files
• Then use image to display image:
>> image(b)
• imwrite writes it back out:
>> imwrite(b, 'monaLisaCopy.jpg')
14.4 Sampling and Storing Images in Files
• Recall formula for sound signal:
P(t) = Σi Ai sin(2πfi t + φi)
• For images, we just add another dimension:
P(x, y) = Σj,k Aj,k sin(2π( fj x + fk y + φj,k))
• This means that the issues/algorithms for images are similar to those for sounds• Aliasing• Frequency transforms, filters (next lecture)
Spatial Frequency: Intuitive Version
x : ~18 color changes
y : ~
6 co
lor
chan
ges
14.4 Sampling and Storing Images in Files
• As with sound, aliasing becomes an issue if sampling frequency is inadequate:
14.4 Sampling and Storing Images in Files
• As with sound, aliasing becomes an issue if sampling frequency is inadequate:
14.4 Sampling and Storing Images in Files
• As with sound, aliasing becomes an issue if sampling frequency is inadequate:
14.4 Sampling and Storing Images in Files
• As with sound, aliasing becomes an issue if sampling frequency is inadequate:
0.04 mm samples
14.4 Sampling and Storing Images in Files
• As with sound, aliasing becomes an issue if sampling frequency is inadequate:
0.04 mm samples
0.25 mm samples
14.5 Manipulating and Synthesizing Images
• Probably want to use PhotoShop or another image-processing package instead of Matlab.
• But Matlab allows us to see the math behind the processing:
>> b = imread('monaLisaLouvre.jpg');>> red = b(:,:,1);>> green = b(:,:,2);>> blue = b(:,:,3);>> image(red)>> colormap(gray) % just show intensity
14.5 Manipulating and Synthesizing Images
>> image(red)
14.5 Manipulating and Synthesizing Images
>> image(green)
14.5 Manipulating and Synthesizing Images
>> image(blue)
14.5 Manipulating and Synthesizing Images
• Put them back together with cat:>> image(cat(3, red, green, blue))
14.5 Manipulating and Synthesizing Images
• Put them back together with cat:>> image(cat(3, red, green, blue))
14.5 Manipulating and Synthesizing Images
>> max(max(a/3))ans = 68>> image(68-a/3), colormap(gray)
• Make a negative:
14.5 Manipulating and Synthesizing Images
>> max(max(a/3))ans = 68>> image(68-a/3), colormap(gray)
• Make a negative:
14.6 Example: Image Restoration
• Mona Lisa has faded and yellowed over 500 years.
• Can we get an idea of what it looked like originally?
• Basic idea: Balance out R, G, B
>> a = imread('monaLisaLouvre.jpg');>> mona = image2double(a); % kaplan>> min(min(min(mona)))ans = 0>> max(max(max(mona)))ans = 1>> r = mona(:,:,1);>> hist(r(:), 50) % histogram
>> a = imread('monaLisaLouvre.jpg');>> mona = image2double(a); % kaplan>> min(min(min(mona)))ans = 0>> max(max(max(mona)))ans = 1>> r = mona(:,:,1);>> hist(r(:), 50) % histogram
>> newr = equalize(r); % kaplan>> hist(newr(:), 50)
>> newr = equalize(r); % kaplan>> hist(newr(:), 50)
>> newr = equalize(r); >> newg = equalize(g);>> newb = equalize(b);>> newmona = cat(3, newr, newg, newb);>> image(newmona)
>> newr = equalize(r); >> newg = equalize(g);>> newb = equalize(b);>> newmona = cat(3, newr, newg, newb);>> image(newmona)
>> image(newmona*.5 + mona*.5);
