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Unformatted text preview: CS/ECE 181B: Winter 2010 Introduction to Computer Vision B. S. Manjunath manj@ece.ucsb.edu (805) 893 7112 Tuesday, January 5, 2010 1 Web Site http:/ /www.ece.ucsb.edu/~manj/ece181b http:/ /www.ece.ucsb.edu/~manj/cs181b Tuesday, January 5, 2010 2 Text Book NO required text book Check out the free online text book by Richard Szelski (http://research.microsoft.com/en-us/um/people/szeliski/Book/) Other useful reference books Hartley & Zisserman, Multiview Geometry Forsyth & Ponce, Computer Vision Tuesday, January 5, 2010 3 Seeing is Believing Vision is unlike any other course you may have taken in engineering/science. There are MANY questions, very few answers! Remember this as we go through this subject. We are so familiar with seeing, that it takes a leap of imagination to realize that there are problems to be solved ---- Richard Gregory Tuesday, January 5, 2010 4 Tuesday, January 5, 2010 5 (human) vision is not perfect Tuesday, January 5, 2010 6 human vision is not perfect Tuesday, January 5, 2010 7 What is CV? Nice sunset! “Extracting descriptions from pictures” Tuesday, January 5, 2010 8 Some example CV tasks Make useful decisions about real physical objects and scenes based on sensed images Produce symbolic (perhaps task-dependent) descriptions from images Produce from images of the external world a useful description that is not cluttered with irrelevant information Support tasks that require visual information Tuesday, January 5, 2010 9 Example CV applications Photogrammetry (estimating depth from one or more images, a common task in the GIS/mapping industry) Robotics (inspection, manufacturing, navigation) Medical imaging Entertainment (graphics, animation) (have you seen “Avatar” in 3D?) Surveillance, Security (e.g., face detection/recognition) Multimedia databases (analyzing large scale collection) Tuesday, January 5, 2010 10 Why is vision difficult? Tuesday, January 5, 2010 11 Why is vision difficult? consider this input image... Tuesday, January 5, 2010 11 Why is vision difficult? consider this input image... Tuesday, January 5, 2010 11 Why is vision difficult? consider this input image... Tuesday, January 5, 2010 11 Why is vision difficult? consider this input image... Tuesday, January 5, 2010 computer “sees” this 11 01 03 03 03 02 00 03 00 04 00 04 02 02 02 03 02 04 01 02 02 02 01 03 01 05 02 03 02 04 00 03 00 03 01 03 01 04 00 04 00 06 02 05 01 07 00 05 00 07 00 07 00 0A 04 0B 01 08 00 0C 00 0B 03 09 00 0A 00 0C 00 0D 01 0F 03 0B 01 10 04 0F 04 0F 02 0D 01 30 3A 38 39 2D 1D 15 10 0E 0C 0A 0A 0A 09 06 08 07 06 06 05 05 07 07 04 05 04 04 06 02 01 02 02 02 02 07 01 02 02 Why is vision difficult? consider this input image... Tuesday, January 5, 2010 22 1B 16 14 0A 08 0B 0A 0D 0B 0B 0C 06 07 05 05 06 06 06 03 07 04 06 05 09 05 04 05 01 04 04 02 03 03 04 02 04 03 0F 0B 04 10 07 09 07 08 09 09 08 05 08 08 05 09 03 08 05 02 08 08 06 06 04 02 05 03 02 05 05 00 02 02 04 04 00 00 07 09 0E 0C 07 08 0A 0A 0B 0F 0A 0C 07 06 0B 07 0B 05 0B 08 09 07 03 08 04 04 02 00 04 02 04 00 04 03 08 00 06 09 0E 0C 09 09 08 08 07 08 09 09 0A 05 08 07 07 07 09 08 0A 08 09 06 0A 03 09 07 06 06 03 05 03 01 06 02 03 07 01 04 0C 0B 0A 05 08 09 0A 0C 0A 0A 08 0A 0A 06 08 06 06 04 06 02 06 07 04 04 04 06 09 05 05 08 06 04 05 04 06 01 0A 03 0B 14 0F 0F 0D 0A 0E 0A 0C 0C 0E 0A 0C 0B 09 0A 09 0A 0A 09 0B 0B 05 0C 0C 0A 04 07 06 03 05 07 04 05 03 02 01 06 10 12 0B 10 0A 0D 0D 0B 0D 0C 0B 0B 0C 0D 0B 0B 0A 0A 0A 0B 0C 17 15 1C 15 0D 08 09 08 05 05 05 04 02 05 04 04 00 computer “sees” this 15 0E 10 12 0C 0D 0C 0C 0A 0B 0B 09 0C 0F 09 09 0D 07 0B 08 15 60 5D 61 59 33 0D 0A 07 08 08 05 03 06 07 01 03 05 12 10 0F 0E 10 10 0B 0C 0F 0F 0E 0C 10 0D 15 10 09 12 11 12 50 68 66 89 71 5E 3F 08 09 0A 09 0A 03 03 02 05 05 04 11 12 0C 11 13 10 10 0B 10 0F 0C 11 11 13 0D 0F 0D 0D 0B 25 7A 7F 79 6D 80 6E 54 0C 0D 09 0A 06 04 02 05 00 05 04 10 0F 0D 12 0E 10 0E 0F 13 13 11 13 17 11 0F 14 11 11 14 39 84 88 7E 8C 73 7A 5C 1E 05 0A 0F 0E 0C 05 02 04 03 06 0F 15 0D 18 11 0D 11 14 10 12 12 14 19 13 17 13 16 16 20 73 68 87 89 93 8B 83 69 43 07 0A 12 0A 0B 06 06 03 04 05 13 14 14 16 11 13 13 17 12 17 17 28 1E 1A 17 19 14 12 4F 7D 74 85 91 93 8C 7F 6F 5F 0B 09 12 0D 0C 02 04 07 04 05 0F 16 0F 13 12 10 1D 12 21 15 1E 21 1F 1C 1D 2D 1A 2D 7C 7A 95 6B 30 48 62 87 71 5C 0A 08 11 0C 09 04 04 02 06 04 10 1C 10 11 1A 0D 1A 1A 25 28 33 30 26 2B 3E 29 35 6C 83 5E 7B 94 8A 5A 3D 42 76 5C 13 08 13 0F 0C 04 04 01 05 05 12 17 1A 19 18 15 20 29 20 3F 1F 37 29 39 49 24 33 8F 93 B4 AE 79 42 39 73 7D 89 46 12 06 12 12 0F 08 03 03 03 04 13 20 0F 14 26 1B 18 20 2F 3D 3E 42 3B 45 2E 48 70 96 9F 96 6B 24 0F 22 4B C3 A4 3F 4F 0C 18 16 0F 05 05 08 05 05 19 1C 13 13 21 1D 12 18 47 3D 47 45 3A 27 3B 33 A8 A6 91 81 4B A1 75 4B AC A1 B5 79 0C 0B 13 0F 0B 02 03 06 07 07 1B 1D 1C 1C 1C 1B 1B 1E 55 49 49 36 28 2A 24 9F AD AC AA B1 9C 8D 5F 3E 98 B7 B7 A3 31 11 14 0A 0D 04 08 07 07 07 21 18 15 16 1D 15 18 1E 36 5B 29 2C 19 29 4F AF BC AF AB 9E A1 97 82 70 9F AE AD A5 92 16 10 07 0E 0A 0C 08 05 0B 17 1B 1A 1A 2B 1B 2A 32 34 46 2C 1B 26 4C 40 BA BB B5 AE 95 94 84 7A 8A 9A B9 BB AD 9C 8A 15 09 09 05 0B 0D 0F 0B 1A 18 1C 1E 27 21 1D 3F 4E 32 25 1B 1B 93 46 AF AB B1 AC A4 93 89 91 86 90 AA 9F 91 97 AD 7F 0C 0B 0E 0B 0C 0C 09 15 1A 21 1E 2E 1B 23 47 4E 23 21 19 49 99 5B AA AC B7 AF A6 9A 93 8F 85 7F A0 A4 C2 9F 99 4E 09 08 0A 0D 0C 0A 0C 13 18 21 26 31 28 25 34 4C 1F 2B 1C 8B 9B 42 9B A7 A1 B4 B0 AA A0 9D 92 72 8E 97 71 A7 32 04 0A 0A 0D 0D 09 0D 0C 1A 1C 21 28 3A 30 26 40 4C 26 18 2C 90 A1 39 A0 97 B8 AA B2 A5 A6 A3 98 76 92 96 98 6D 08 0D 07 08 0C 0B 0E 0D 0D 1E 29 1F 27 32 26 2E 41 4A 2C 34 46 8A A5 89 9E A3 B0 B7 AF AB AB 99 97 90 A4 94 85 7C 08 07 07 08 09 09 08 0C 0D 1F 29 27 27 2A 2C 36 4D 50 34 42 45 95 9B AA 7E AD B3 AA B2 A8 B2 92 98 8E 9E 8E 44 34 18 05 06 0A 0D 0D 0D 0F 0C 21 2E 23 29 2C 2A 34 44 5A 39 4F 29 90 9B A5 86 AA B2 B3 AE A0 A3 9C 94 79 43 2B 25 2D 07 0E 05 06 0C 0A 0F 0D 09 21 27 20 28 29 2F 2A 44 57 42 31 28 8C 93 A3 AC 60 BA BD B4 AE A8 A2 62 91 5F 52 4F 3F 09 0D 0D 09 0E 0E 0B 12 0B 30 2E 2C 29 2A 3B 30 4E 3C 40 40 49 5E AE 9F A4 B1 4E AA AA A0 A4 9C 94 A2 AB A8 93 52 0E 0E 09 0B 0D 10 0C 0C 10 30 32 2E 36 39 36 24 2D 5A 46 46 68 30 8B 8C A3 AC A5 3E A1 AF A8 82 A4 AC A2 96 71 73 08 10 0B 0B 0B 0E 0F 10 11 54 34 1E 3C 3F 3E 29 27 56 38 4C 5C 44 26 94 9A A2 A2 A6 8E 4E 70 99 AC A6 A2 89 7E 5B 11 0E 10 10 17 12 0D 0C 0D 4B 30 23 36 44 48 3C 2E 2D 34 35 29 58 5B 0D 36 50 34 52 9C A8 B5 AA B3 AE A0 9C 8C 62 0A 12 14 0D 16 14 11 10 0E 38 2C 24 2E 51 59 4B 30 27 39 2B 2B 24 29 69 37 25 29 82 97 A1 AB AC B2 A6 A6 A0 89 69 0F 10 1C 18 14 10 10 0F 0C 21 2A 27 22 5C 44 31 3F 33 1F 37 24 23 36 27 24 2B 4D 50 85 90 96 86 A3 A5 99 8D 7A 4E 0E 1B 15 20 0F 0F 16 12 13 1D 1F 2B 20 21 48 2F 40 2F 2D 2A 25 2B 2C 20 25 25 26 3E 55 5E 62 6D 6D 6E 68 5E 43 0D 10 21 18 32 1A 13 10 13 15 27 2F 2A 28 21 3B 45 2E 3A 40 33 2D 2F 1F 1E 1B 20 37 3C 3F 3C 34 30 24 17 0D 0B 0E 11 1E 23 1B 25 14 0D 10 0F 12 22 27 37 33 1A 1B 35 4A 1D 20 2C 2F 1F 1F 3B 34 1A 2A 38 44 1E 0C 0C 06 0C 10 12 1B 21 21 34 32 20 0B 0E 10 0D 0D 32 22 33 29 20 22 19 30 35 1D 1E 16 19 18 1C 16 18 23 39 10 13 0E 0E 1A 15 15 13 1A 18 2C 2E 19 0F 0D 10 0E 0E 14 33 36 23 31 29 20 19 1B 1E 17 1C 1F 1F 1F 1C 31 23 1C 2F 13 11 16 10 12 16 13 19 1B 17 19 1D 13 14 10 10 12 11 12 11 01 03 03 03 02 00 03 00 04 00 04 02 02 02 03 02 04 01 02 02 02 01 03 01 05 02 03 02 04 00 03 00 03 01 03 01 04 00 04 00 06 02 05 01 07 00 05 00 07 00 07 00 0A 04 0B 01 08 00 0C 00 0B 03 09 00 0A 00 0C 00 0D 01 0F 03 0B 01 10 04 0F 04 0F 02 0D 01 30 3A 38 39 2D 1D 15 10 0E 0C 0A 0A 0A 09 06 08 07 06 06 05 05 07 07 04 05 04 04 06 02 01 02 02 02 02 07 01 02 02 Why is vision difficult? consider this input image... Tuesday, January 5, 2010 22 1B 16 14 0A 08 0B 0A 0D 0B 0B 0C 06 07 05 05 06 06 06 03 07 04 06 05 09 05 04 05 01 04 04 02 03 03 04 02 04 03 0F 0B 04 10 07 09 07 08 09 09 08 05 08 08 05 09 03 08 05 02 08 08 06 06 04 02 05 03 02 05 05 00 02 02 04 04 00 00 07 09 0E 0C 07 08 0A 0A 0B 0F 0A 0C 07 06 0B 07 0B 05 0B 08 09 07 03 08 04 04 02 00 04 02 04 00 04 03 08 00 06 09 0E 0C 09 09 08 08 07 08 09 09 0A 05 08 07 07 07 09 08 0A 08 09 06 0A 03 09 07 06 06 03 05 03 01 06 02 03 07 01 04 0C 0B 0A 05 08 09 0A 0C 0A 0A 08 0A 0A 06 08 06 06 04 06 02 06 07 04 04 04 06 09 05 05 08 06 04 05 04 06 01 0A 03 0B 14 0F 0F 0D 0A 0E 0A 0C 0C 0E 0A 0C 0B 09 0A 09 0A 0A 09 0B 0B 05 0C 0C 0A 04 07 06 03 05 07 04 05 03 02 01 06 10 12 0B 10 0A 0D 0D 0B 0D 0C 0B 0B 0C 0D 0B 0B 0A 0A 0A 0B 0C 17 15 1C 15 0D 08 09 08 05 05 05 04 02 05 04 04 00 computer “sees” this 15 0E 10 12 0C 0D 0C 0C 0A 0B 0B 09 0C 0F 09 09 0D 07 0B 08 15 60 5D 61 59 33 0D 0A 07 08 08 05 03 06 07 01 03 05 12 10 0F 0E 10 10 0B 0C 0F 0F 0E 0C 10 0D 15 10 09 12 11 12 50 68 66 89 71 5E 3F 08 09 0A 09 0A 03 03 02 05 05 04 11 12 0C 11 13 10 10 0B 10 0F 0C 11 11 13 0D 0F 0D 0D 0B 25 7A 7F 79 6D 80 6E 54 0C 0D 09 0A 06 04 02 05 00 05 04 10 0F 0D 12 0E 10 0E 0F 13 13 11 13 17 11 0F 14 11 11 14 39 84 88 7E 8C 73 7A 5C 1E 05 0A 0F 0E 0C 05 02 04 03 06 0F 15 0D 18 11 0D 11 14 10 12 12 14 19 13 17 13 16 16 20 73 68 87 89 93 8B 83 69 43 07 0A 12 0A 0B 06 06 03 04 05 13 14 14 16 11 13 13 17 12 17 17 28 1E 1A 17 19 14 12 4F 7D 74 85 91 93 8C 7F 6F 5F 0B 09 12 0D 0C 02 04 07 04 05 0F 16 0F 13 12 10 1D 12 21 15 1E 21 1F 1C 1D 2D 1A 2D 7C 7A 95 6B 30 48 62 87 71 5C 0A 08 11 0C 09 04 04 02 06 04 10 1C 10 11 1A 0D 1A 1A 25 28 33 30 26 2B 3E 29 35 6C 83 5E 7B 94 8A 5A 3D 42 76 5C 13 08 13 0F 0C 04 04 01 05 05 12 17 1A 19 18 15 20 29 20 3F 1F 37 29 39 49 24 33 8F 93 B4 AE 79 42 39 73 7D 89 46 12 06 12 12 0F 08 03 03 03 04 13 20 0F 14 26 1B 18 20 2F 3D 3E 42 3B 45 2E 48 70 96 9F 96 6B 24 0F 22 4B C3 A4 3F 4F 0C 18 16 0F 05 05 08 05 05 19 1C 13 13 21 1D 12 18 47 3D 47 45 3A 27 3B 33 A8 A6 91 81 4B A1 75 4B AC A1 B5 79 0C 0B 13 0F 0B 02 03 06 07 07 1B 1D 1C 1C 1C 1B 1B 1E 55 49 49 36 28 2A 24 9F AD AC AA B1 9C 8D 5F 3E 98 B7 B7 A3 31 11 14 0A 0D 04 08 07 07 07 21 18 15 16 1D 15 18 1E 36 5B 29 2C 19 29 4F AF BC AF AB 9E A1 97 82 70 9F AE AD A5 92 16 10 07 0E 0A 0C 08 05 0B 17 1B 1A 1A 2B 1B 2A 32 34 46 2C 1B 26 4C 40 BA BB B5 AE 95 94 84 7A 8A 9A B9 BB AD 9C 8A 15 09 09 05 0B 0D 0F 0B 1A 18 1C 1E 27 21 1D 3F 4E 32 25 1B 1B 93 46 AF AB B1 AC A4 93 89 91 86 90 AA 9F 91 97 AD 7F 0C 0B 0E 0B 0C 0C 09 15 1A 21 1E 2E 1B 23 47 4E 23 21 19 49 99 5B AA AC B7 AF A6 9A 93 8F 85 7F A0 A4 C2 9F 99 4E 09 08 0A 0D 0C 0A 0C 13 18 21 26 31 28 25 34 4C 1F 2B 1C 8B 9B 42 9B A7 A1 B4 B0 AA A0 9D 92 72 8E 97 71 A7 32 04 0A 0A 0D 0D 09 0D 0C 1A 1C 21 28 3A 30 26 40 4C 26 18 2C 90 A1 39 A0 97 B8 AA B2 A5 A6 A3 98 76 92 96 98 6D 08 0D 07 08 0C 0B 0E 0D 0D 1E 29 1F 27 32 26 2E 41 4A 2C 34 46 8A A5 89 9E A3 B0 B7 AF AB AB 99 97 90 A4 94 85 7C 08 07 07 08 09 09 08 0C 0D 1F 29 27 27 2A 2C 36 4D 50 34 42 45 95 9B AA 7E AD B3 AA B2 A8 B2 92 98 8E 9E 8E 44 34 18 05 06 0A 0D 0D 0D 0F 0C 21 2E 23 29 2C 2A 34 44 5A 39 4F 29 90 9B A5 86 AA B2 B3 AE A0 A3 9C 94 79 43 2B 25 2D 07 0E 05 06 0C 0A 0F 0D 09 21 27 20 28 29 2F 2A 44 57 42 31 28 8C 93 A3 AC 60 BA BD B4 AE A8 A2 62 91 5F 52 4F 3F 09 0D 0D 09 0E 0E 0B 12 0B 30 2E 2C 29 2A 3B 30 4E 3C 40 40 49 5E AE 9F A4 B1 4E AA AA A0 A4 9C 94 A2 AB A8 93 52 0E 0E 09 0B 0D 10 0C 0C 10 30 32 2E 36 39 36 24 2D 5A 46 46 68 30 8B 8C A3 AC A5 3E A1 AF A8 82 A4 AC A2 96 71 73 08 10 0B 0B 0B 0E 0F 10 11 54 34 1E 3C 3F 3E 29 27 56 38 4C 5C 44 26 94 9A A2 A2 A6 8E 4E 70 99 AC A6 A2 89 7E 5B 11 0E 10 10 17 12 0D 0C 0D 4B 30 23 36 44 48 3C 2E 2D 34 35 29 58 5B 0D 36 50 34 52 9C A8 B5 AA B3 AE A0 9C 8C 62 0A 12 14 0D 16 14 11 10 0E 38 2C 24 2E 51 59 4B 30 27 39 2B 2B 24 29 69 37 25 29 82 97 A1 AB AC B2 A6 A6 A0 89 69 0F 10 1C 18 14 10 10 0F 0C 21 2A 27 22 5C 44 31 3F 33 1F 37 24 23 36 27 24 2B 4D 50 85 90 96 86 A3 A5 99 8D 7A 4E 0E 1B 15 20 0F 0F 16 12 13 1D 1F 2B 20 21 48 2F 40 2F 2D 2A 25 2B 2C 20 25 25 26 3E 55 5E 62 6D 6D 6E 68 5E 43 0D 10 21 18 32 1A 13 10 13 15 27 2F 2A 28 21 3B 45 2E 3A 40 33 2D 2F 1F 1E 1B 20 37 3C 3F 3C 34 30 24 17 0D 0B 0E 11 1E 23 1B 25 14 0D 10 0F 12 22 27 37 33 1A 1B 35 4A 1D 20 2C 2F 1F 1F 3B 34 1A 2A 38 44 1E 0C 0C 06 0C 10 12 1B 21 21 34 32 20 0B 0E 10 0D 0D 32 22 33 29 20 22 19 30 35 1D 1E 16 19 18 1C 16 18 23 39 10 13 0E 0E 1A 15 15 13 1A 18 2C 2E 19 0F 0D 10 0E 0E 14 33 36 23 31 29 20 19 1B 1E 17 1C 1F 1F 1F 1C 31 23 1C 2F 13 11 16 10 12 16 13 19 1B 17 19 1D 13 14 10 10 12 11 12 12 What your brain “does” Tuesday, January 5, 2010 13 Appearance variations Tuesday, January 5, 2010 14 Vision is not easy ★ We know little about human vision ★ human vision appears effortless ★ just open the eyes to “see” ★ much of the human brain is devoted to vision ★ Computer vision is about 50 years old ★ child of AI Tuesday, January 5, 2010 15 what makes vision difficult? Underspecified problem - 3D world projected onto 2D “images” Environment - Lighting, background, motion, camera Varying appearance of objects Computational complexity others.... Tuesday, January 5, 2010 16 Areas of study Calibration and image formation Low level vision - edge detection, segmentation, texture, shape from shading, stereo, optic flow Mid-level vision - tracking, depth computation, detection, shape analysis Recognition and image understanding Tuesday, January 5, 2010 17 Edge detection Tuesday, January 5, 2010 18 Stereopsis Depth maps Tuesday, January 5, 2010 19 multi-camera stereo 3D reconstruction from multiple views Tuesday, January 5, 2010 20 Object detection and recognition faces vehicles Model, detect and recognize Issues: feature extraction (what features?), representation (how?), matching (how to compare?), learning from data (how to learn new features/metrics) Tuesday, January 5, 2010 21 This quarter as we go through some of the computational methods underlying vision, keep in mind that there are quite a few issues that we know very little about in vision more questions than answers! will have some fun solving/implementing a few basic techniques Tuesday, January 5, 2010 22 your tasks You are expected to attendall lectures, discussion sessions. you are responsible for everything that transpires in the class and discussion sessions (not just what is on the slides) ask questions and participate! do the assignments on time and with integrity give us feedback during the quarter Tuesday, January 5, 2010 23 grading (tentative) there will be HWs, in-class quizzes, quizzes during discussion sessions and projects (40%) one midterm exam on Feb 11 (15%) one final exam (45%) Tuesday, January 5, 2010 24 projects/groups work in groups of at most four students. finalize your groups by Thursday, Jan 7. send me an email, at manj@ece.ucsb.edu, w ith full names of the members of the group. Make sure that you have the subject as CS/ECE 181b project group so that I can filter it properly. Tuesday, January 5, 2010 25 why study vision? “Engineering” reasons images are ubiquitous many useful applications Good mix of applied mathematics, signal processing and CS Solving interesting and challenging real world problems “Science” reasons how does the brain do it? understanding the biology and neural circuitry deep computational issues (if the above is not enough) you can invent the next google for images (and become very rich) Tuesday, January 5, 2010 26 Homework #1 We explore a little bit about binocular vision here. Binocular (twoeyes) vision is critical to human vision, we all have two eyes and we use it to perceive depth. Have you thought about what needs to be done (by the computing machinery in our brains) to solve this depth perception problem? Tuesday, January 5, 2010 1.each eye takes an instantaneous picture of the 3D world. Let us assume that the visual brain is seeing a replica of this projected image coming from the left and the right eyes. 2.You will notice (by closing alternatively the left and the right eyes) that the pictures coming from the L & R are slightly displaced with respect to each other. 3.The brain needs to compute this relative displacement at each point to estimate the depth. Explain, in your own words, the problem that the human brain needs to solve (or compute) in order to analyze the two images coming from L & R eyes, and thus perceive the depth. (no more than two pages, including any graphics you may want to include in your explanation). Due: Thursday, Jan 7, in class. 27 topics (tentative) basic image processing/image formation overview (Ch. 2, 3) feature detection and matching (Ch. 4) projective geometry (Ch.6) stereo (Ch. 11) optic flow (Ch. 8) Recognition (Ch. 14) Tuesday, January 5, 2010 28 ...
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This note was uploaded on 12/29/2011 for the course ECE 181b taught by Professor Staff during the Fall '08 term at UCSB.

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