Understanding JPEG

1. Understanding JPEG MIT-CETI Xi’an ‘99 Lecture 10 Ben Walter, Lan Chen, Wei Hu

2. What is JPEG? • JPEG is a method for compressing image data so it takes less space to store or transmit across a network. • JPEG is very efficient. A file that was 1Mb in size could be compressed to as little 25Kb (1:40)! • JPEG achieves such good compression ratios because it is lossy - but the loss is not visually perceptible.

3. Overview • Images contain different frequencies; low frequencies correspond the slowly varying colors, high frequencies correspond to fine detail. • The low frequencies are much more important than the high frequencies; we can throw away some high frequencies to compress our data!

4. Overview • Note that we aren’t talking about the frequencies of light, but of the light and dark areas in the image! • We need a way to go from the color of pixels, which is essentially a number, to frequencies… • This way is called the Discrete Cosine Transform (DCT).

5. A JPEG Encoder DCT Quantizer Entropy Encoder

6. The Discrete Cosine Transform =

7. The Discrete Cosine Transform

8. The Discrete Cosine Transform = x1 + x2 + … + x15 + x16

9. The Discrete Cosine Transform

10. The 2D DCT • So far we’ve been talking about one-dimensional images, just one line of the picture… but an image has two dimensions. • We can talk about frequencies in two dimensions, although it’s much harder to visualize.

11. Basis • Remember we saw that every 16-pixel line can be written as the sum of 16 different waves? • Those 16 waves formed a basis for the set of 16-pixel lines.

12. Basis • When we are compressing a JPEG, we work in blocks of 8x8 pixels. That’s 64 numbers, so there are 64 different basis images. • This means we can describe any 8x8 image as a combination (a sum) of those 64 images.

13. Basis

14. The 2D DCT

15. The 2D DCT

16. Summary • The Discrete Cosine Transform (DCT) allows us to determine what frequencies make up an image. • Into this stage we have 8x8 numbers that are the values of each pixel. • Out of this stage we have 8x8 numbers that represent how much of each frequency (or how much of each basis) is in the image.

17. A JPEG Encoder DCT Quantizer Entropy Encoder

18. Quantization • So we still have 64 numbers to work with - we haven’t reduced the size at all! • The reason we wanted the numbers as frequencies was because some frequencies are more important than others. • The low frequencies are the most important, the high frequencies are not very important (think back to building up the image).

19. Quantization • Before quantization, each frequency can be between 0 and 255. • To quantize, we divide frequencies by a number so that the range is reduced. For example, it becomes 0 to 31. For high frequencies we divide by a higher number.

20. Quantization • Before we had, say: 134,113,145,117,32,11,17,5… 4. • After quantization, we might have: 116, 55, 55, 30, 1, 0, 0, … 0.

21. 124 15 5 0 71 25 113 7 9 56 34 21 1 19 3 27 17 49 7 110 Quantization

22. Quantization Original Medium Quality JPEG 300 kB 75 kB

23. Quantization Original Low Quality JPEG 300 kB 35 kB

24. Summary • The degree of quantization, dictates the amount of information “thrown away”. • If you throw away more information, you will get better compression, but the picture will start to look bad. • When you adjust the quality of a JPEG save from Photoshop, you are changing the quantization!

25. A JPEG Encoder DCT Quantizer Entropy Encoder

26. Entropy Encoding • Entropy encoding is another stage of compression, that relies on statistical properties of the data, e.g. most frequently occuring numbers, lots of the same number in a row. • So the take the 64 numbers, do Run Length Encoding, then follow that with Huffman Coding! (Remember yesterday?)

27. Entropy Encoding • These compression schemes now work very well, because quantization turns numbers like 132, 117, 78 into numbers more like 31, 31, 15. • After quantization, the range of numbers is smaller, and there are often large runs of numbers - so it can be highly compressed! • This is where all of the compression happens!

28. Summary DCT Quantizer Entropy Encoder

29. Summary • We break up the image into 8x8 blocks. • We calculate the frequencies in each block, this allows us to identify the important and less important data. • We throw away some less important data. • We compress the resulting data. • The result: ~ 1:40 compression!