(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 873035, 15528] NotebookOptionsPosition[ 864235, 15274] NotebookOutlinePosition[ 864670, 15293] CellTagsIndexPosition[ 864627, 15290] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Discrete Cosine Transform (and the JPEG Standard)\ \>", "Title"], Cell["\<\ M. E. Mauel Columbia University\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "The ", StyleBox["discrete Cosine transfrom", FontSlant->"Italic"], " (DCT) is commonly used in digital signal processing. It converts a \ sequence of ", StyleBox["real numbers", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " to a sequence of ", StyleBox["real numbers", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " of the same length. (Recall, the DFT converts a sequence of ", StyleBox["real numbers (or ", FontColor->RGBColor[0, 0, 1]], StyleBox["complex", FontColor->RGBColor[1, 0, 0]], StyleBox[")", FontColor->RGBColor[0, 0, 1]], " to a sequence of ", StyleBox["complex numbers", FontColor->RGBColor[1, 0, 0]], ".) The use of the DCT for image compression was introduced in 1974, and it \ plays an important role in JPEG and MPEG compression schemes." }], "Text"], Cell["\<\ In this notebook, the DCT and the processing of two dimensional graphics is \ examined. In particular, we explore \ \>", "Text"], Cell[TextData[{ "I've used two references. First is \"Image Compression Using the Discrete \ Cosine Transform\" by Andrew Watson, ", StyleBox["Mathematica Journal", FontSlant->"Italic"], " ", StyleBox["4", FontWeight->"Bold"], " (1994) p. 81. This is a wonderful collection of ", StyleBox["Mathematica", FontSlant->"Italic"], " routines from which I have based this lecture. Second is \"The JPEG Still \ Picture Compression Standard\" by Gregory Wallace, ", StyleBox["Communications of the ACM", FontSlant->"Italic"], " (1991) p. 35. Both articles are available online on our class website; \ however, you do not need to read these articles in order to understand this \ notebook." }], "Text"], Cell[CellGroupData[{ Cell["DSP Hardware", "Subsubsection"], Cell["\<\ Today, the DCT and other digital signal processing algorithms are encoded \ into special hardware that provides high-speed signal processing at low cost.\ \ \>", "Text"], Cell[GraphicsData["CompressedBitmap", "\<\ eJxsvHdfG8uaNYoSOQflRBBIAqGMBAgQOeeMicaAbRxwwBkDzjlnb8ftnU+e k+bMb957P8P9QuuuDsI+M+8fjUSru7r6qadWqK7q/qXTB1eOLp0+tLxk7zy5 tHHw0PIpe8fxk9ylUiQlKf4/bv+yJwnfkZS090fFTzWUimR+KsX/kpKuCN/F /UlJGigUGfIx2VAk5UKZpOWnsOXxezZUyhRkpOYgN80EQ4EFxiIdCjN1yNIU QpengzlfhwqrGy5nI5qre3BuaBFXhkZxtnccPXWdCBQH4bdEESgLwmb0w1ba CAP/t5oq0B5oxuZAMy6OtWJnpBVPT0/j5akp3D26H682xvD86BBuHO/H2e1R HDrZgsm+evQ11qHV1YqYKYgGaxC1PgdWOkOY8paj19OJkC2CBksVwjo9HEV2 lJgrYSuqhF/nQ50ngpbyMBoMYcSLgoi7gmiO1iNSFUKopJTlBlFnrUFdaT1s BZUoyLUiJbmAsRC2IjEuyiQ9Nx23IsasEEkK7lNqkZGsg4afav7vNDoQNttR 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