Computer Science 121. Scientific Computing Chapter 12 Images

Transcription

1 Computer Science 121 Scientific Computing Chapter 12 Images

2 Background: Images Signal (sound, last chapter) is a single (onedimensional) quantity that varies over time. Image (picture) can be thought of as a twodimensional quantity that varies over space. Otherwise, the computations for sounds and images are about the same!

3 12.1 Black-and-White Images

4 12.1 Black-and-White Images

5 12.1 Black-and-White Images

6 12.1 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 (2 N, N typically = 8) representing shades of gray.

7 12.1 Black-and-White Images To turn a b/w image into a grid of pixels, imagine running a stylus horizontally along the image:

8 12.1 Black-and-White Images This gives us a one-dimensional function showing gray level at each point along the horizontal axis:

9 12.1 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

10 12.2 Color Light comes in difference wavelengths (frequencies):

11 12.2 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. Energy (db) Intensity Frequency (Hz) Spectrum of vowel ee Wavelength (nm) Spectrum of star ICR3287

12 12.2 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:

13 12.2 Color =,, R G B

14 12.3 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

15 Quantization Problems With too few bits, we lose gradual shading: 1 bit 2 bits 3 bits 5 bits 6 bits 7 bits 4 bits 8 bits

16 12.4 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 Python and mathematical issues...

17 12.4 Sampling and Storing Images in Files imread command lets us read in files in several formats (automagically figures out which): >>> from matplotlib.pyplot import * >>> a = imread('sakidehli.png') # b/w >>> a.shape (269, 176) >>> b = imread('monalisalouvre.jpg') # color >>> b.shape (864, 560, 3)

18 12.4 Sampling and Storing Images in Files Then use imshow to display image: >>> imshow(b) >>> show()

19 12.4 Sampling and Storing Images in Files Then use imshow to display image: >>> imshow(b) >>> show() imsave saves it back out: >>> imsave('monalisacopy.jpg', b)

20 12.4 Sampling and Storing Images in Files Recall formula for sound signal: P(t) = Σ i A i sin(2πf i t + φ i ) For images, we just add another dimension: P(x, y) = Σ j,k A j,k sin(2π( f j x + f k y + φ j,k )) This means that the issues/algorithms for images are similar to those for sounds Aliasing Quantization ( clipping )

21 Spatial Frequency: Intuitive Version y : ~6 color changes x : ~18 color changes

22 12.4 Sampling and Storing Images in Files As with sound, aliasing becomes an issue if sampling frequency is inadequate:

23 12.4 Sampling and Storing Images in Files As with sound, aliasing becomes an issue if sampling frequency is inadequate:

24 12.4 Sampling and Storing Images in Files As with sound, aliasing becomes an issue if sampling frequency is inadequate:

25 12.4 Sampling and Storing Images in Files As with sound, aliasing becomes an issue if sampling frequency is inadequate: 0.04 mm samples

26 12.4 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

27 12.5 Manipulating and Synthesizing Images Probably want to use PhotoShop / GIMP or another image-processing package instead of Python. But writing code allows us to see the math behind the processing. Let's do some digital restoration of the faded colors in the Mona Lisa.

28 12.6 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

29 monalisalouvre.png

30 Split the image into its three component color planes (a.k.a. channels) >>> b = imread('monalisalouvre.png') >>> red = b[:,:,0] >>> green = b[:,:,1] >>> blue = b[:,:,2]

31 >>> from matplotlib import cm >>> imshow(red, cmap=cm.gray) >>> show() cmap stands for color map. It allows us to display the image using grayscale (or whatever color scheme we like) Plenty of red!

32 >>> imshow(green, cmap=cm.gray) >>> show() Nice green!

33 >>> imshow(blue, cmap=cm.gray) >>> show() We can see a clear pattern, but how can we quantify / modify it? Too little blue!

34 Quantifying the Data in an Image To get an overall sense of the data, we need to flatten the 3D (height x width x depth) array: >>> mona = imread('monalisalouvre.png') >>> flat = mona.flatten() >>> min(flat), max(flat) (0, ) So it appears that NumPy represents images as pixel intensities between zero and one (good!) Let's try a histogram on each color.

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37 Manipulating the Image Data Let's try to balance the pixels so that the histogram shows a more uniform distribution: >>> bf = blue.flatten() >>> index = argsort(bf) # indices of sorted >>> n = len(bf) >>> bf[index] = arange(n)/n >>> mix = 0.5 # half original, half new >>> bf = mix*bf + (1-mix)*blue.flatten() >>> hist(bf, 50) >>> ylim(0, ) >>> show()

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41 Then use NumPy's dstack function to put image planes back together: >>> imshow(dstack((equalize(red,.5), equalize(green,.5), equalize(blue,.5))))

42 Maybe tone down the red and green a little: >>> imshow(dstack((equalize(red,.25), equalize(green,.25), equalize(blue,.5))))

43 Original vs. Restored :