Discrete cosine transform
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The document discusses DCT/IDCT concepts and applications. It provides an introduction to DCT and IDCT, explaining that they are used widely in video and audio compression. It describes the DCT and IDCT functions and how they work to transform signals between spatial and frequency domains. Examples of one-dimensional and two-dimensional DCT/IDCT equations are also given. Finally, common applications of DCT/IDCT compression techniques are listed, such as in DVD players, cable TV, graphics cards, and medical imaging systems.
![DCT/IDCT Concept
One-Dimensional IDCT Equation
N-1
xc(k) = Σ c[u] Xn cos(k2πn/N),
n=0
where
k = 0, 1, 2, …, N-1,
Xn is the DCT result, and
c[u] = 1 for u=0, and c[u] = 2 for u=1,2,3,…N-1
Xilinx at Work in High Volume Applications ®
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![DCT/IDCT Concept
Two-Dimensional DCT Equation
N-1 N-1
F[u, v] = 1/N2 Σ Σ f[m, n] cos[ (2m + 1)uπ/ 2N] cos[ (2n + 1)vπ/2N ]
m=0 n=0
where:
u, v = discrete frequency variables (0, 1, 2, …, N - 1),
f[m, n] = N by N image pixels(0, 1, 2, …, N - 1), and
F[u, v] = the DCT result
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![DCT/IDCT Concept
Two-Dimensional IDCT Equation
N-1 N-1
f[m, n] = Σ Σ c[u] c[v] F[u, v] cos[ (2m + 1)uπ/ 2N] cos[ (2n + 1)vπ/2N ]
m=0 n=0
where:
m, n = image result pixel indices( 0, 1, 2, …, N – 1 ),
F[u, v] = N by N DCT result,
c[λ] = 1 for λ=0 and c[λ]=2 for λ=1,2,3,…N-1
f[m, n] = N by N IDCT result
Xilinx at Work in High Volume Applications ®
www.xilinx.com](https://image.slidesharecdn.com/13643133-120715032246-phpapp02/85/Discrete-cosine-transform-14-320.jpg)
Discrete cosine transform
- 1. ® The DCT/IDCT Solution Customer Tutorial February 2000 File Number Here
- 2. Agenda Introduction DCT/IDCT Concepts DCT/IDCT Applications Spartan-II DCT/IDCT IP Solutions Summary Xilinx at Work in High Volume Applications ® www.xilinx.com
- 3. Introduction Spartan-II FPGAs — 100,000 System Gates at under $10 — Extensive features: Block RAM, DLL, Select I/O — Vast IP Portfolio — Provide Density, Features, Performance at ASIC prices Xilinx at Work in High Volume Applications ® www.xilinx.com
- 4. DCT/IDCT Compression Compression allows increased throughput through transmission medium — Video and audio compression makes multimedia systems very efficient – Increases CPU bandwidth – Higher video frame rates – Better audio quality – Enables multimedia interactivity DCT and IDCT are widely used in video and audio compression Xilinx at Work in High Volume Applications ® www.xilinx.com
- 5. DCT/IDCT Overview DCT - Discrete Cosine Transform IDCT -Inverse Discrete Cosine Transform ®
- 6. DCT/IDCT Concept What is DCT? — X= DCT(video/audio input) – Returns the discrete cosine transform of ‘video/audio input’ – Can be referred to as the even part of the Fourier series – Converts an image or audio block into it’s equivalent frequency coefficients What is IDCT? — The IDCT function is the inverse of the DCT function — The IDCT reconstructs a sequence from its discrete cosine transform (DCT) coefficients Xilinx at Work in High Volume Applications ® www.xilinx.com
- 7. DCT/IDCT Concept The DCT transform of an image brings out a set of numbers called coefficients. A coefficient’s usefulness is determined by its variance over a set of images as in video’s case. If a coefficient has a lot of variance over a set, then it cannot be removed without affecting the picture quality. Xilinx at Work in High Volume Applications ® www.xilinx.com
- 8. DCT/IDCT Concept DCT Frequency Coefficients Compared to Magnitude Original Image Thresholds Resulting in Compressed IDCT Data Streams The image is broken into 8x8 groups, each containing 64 pixels. Three of these 8x8 groups are enlarged in this Recovered Image figure, showing the values of (Notice Lesser Image Quality) the individual pixels, a single byte value between 0 and 255. Courtesy: The Scientist and Engineer's Guide to Digital Signal Processing by Steven W. Smith Xilinx at Work in High Volume Applications ® www.xilinx.com
- 9. DCT/IDCT Concept Divide picture into 16 by 16 blocks. (macroblocks) Each block is 8 pixels by 8 lines. Each macroblock is 16 pixels by 16 lines. (4 blocks) DCT 8 X 8 Block Frequency Coefficients Xilinx at Work in High Volume Applications ® www.xilinx.com
- 10. DCT/IDCT Usage Areas of Use: — One-Dimensional DCT/IDCT – Dolby AC2 & AC3 – Biomedical signals like EEG & ECG – Speech information compression — Two-Dimensional DCT/IDCT – JPEG Encoders – MPEG-1 & MPEG-2 – Image & Pattern Recognition Xilinx at Work in High Volume Applications ® www.xilinx.com
- 11. DCT/IDCT Concept One-Dimensional DCT Equation N-1 Xc(k) = (1/N) Σ xn cos(k2πn/N), n=0 where k = 0, 1, 2, …, N-1 Xilinx at Work in High Volume Applications ® www.xilinx.com
- 12. DCT/IDCT Concept One-Dimensional IDCT Equation N-1 xc(k) = Σ c[u] Xn cos(k2πn/N), n=0 where k = 0, 1, 2, …, N-1, Xn is the DCT result, and c[u] = 1 for u=0, and c[u] = 2 for u=1,2,3,…N-1 Xilinx at Work in High Volume Applications ® www.xilinx.com
- 13. DCT/IDCT Concept Two-Dimensional DCT Equation N-1 N-1 F[u, v] = 1/N2 Σ Σ f[m, n] cos[ (2m + 1)uπ/ 2N] cos[ (2n + 1)vπ/2N ] m=0 n=0 where: u, v = discrete frequency variables (0, 1, 2, …, N - 1), f[m, n] = N by N image pixels(0, 1, 2, …, N - 1), and F[u, v] = the DCT result Xilinx at Work in High Volume Applications ® www.xilinx.com
- 14. DCT/IDCT Concept Two-Dimensional IDCT Equation N-1 N-1 f[m, n] = Σ Σ c[u] c[v] F[u, v] cos[ (2m + 1)uπ/ 2N] cos[ (2n + 1)vπ/2N ] m=0 n=0 where: m, n = image result pixel indices( 0, 1, 2, …, N – 1 ), F[u, v] = N by N DCT result, c[λ] = 1 for λ=0 and c[λ]=2 for λ=1,2,3,…N-1 f[m, n] = N by N IDCT result Xilinx at Work in High Volume Applications ® www.xilinx.com
- 15. DCT/IDCT Concept Example of a Simplistic one-Dimensional DCT — Data is transformed first and the newly calculated values are threshold limited to a magnitude of 0.375 — Assuming a data sequence to be {1, 2, 0, 5} Applying the one-Dimensional DCT formula,the resultant DCT sequence is {2, 0.25, -6, 0.25} The values that above the threshold (|values| > 0.375) are 2 and –6 This results in a 50% reduction in data size with minimal loss in quality Xilinx at Work in High Volume Applications ® www.xilinx.com
- 16. DCT/IDCT Applications ®
- 17. DCT/IDCT Applications List of Some End Applications — DVD/Video CD Players — Cable TV — DBS Systems — HDTV — Graphics/Image Processing Cards — Ultrasound/MRI Systems — Digital VCRs — Set-Top Boxes — Digital Camera Xilinx at Work in High Volume Applications ® www.xilinx.com
- 18. DCT/IDCT in JPEG Encoding Compressed Zig-Zag Data Pixel Data Coefficient Run-Length Huffman DCT Encoding Encoding Quantization Decoding Reconstructed Compressed Huffman Zig-Zag Pixel Data Run-Length Coefficient Data Decoding IDCT Expansion Denormalization JPEG Codec Block Diagram Xilinx at Work in High Volume Applications ® www.xilinx.com
- 19. DCT/IDCT in Bio-Medical 1-D DCT is commonly used on a sequence of digital information like voice or heartbeat information in an ECG Xilinx at Work in High Volume Applications ® www.xilinx.com
- 20. DCT/IDCT in DVD/VCD Reconstructed (Decoded) Frame Inverse IDCT Quantizer Video to Coded Video be Coded Bitstream Bitstream DCT Quantizer Coder + - Motion Motion Estimator Compensator Anchor Frame Storage Courtesy: C-Cube Corporation MPEG-2 Block Diagram in a Typical DVD System Xilinx at Work in High Volume Applications ® www.xilinx.com
- 21. DCT/IDCT in DVD/VCD Master Variable Bit Rate or Video Tape MPEG-2 Audio Encoder Quality Control Multiplexing Emulation Human and Formatting Assist Master Recoding Disc Master Dolby AC3 or Audio Tape MPEG-2 Audio Encoder Courtesy: C-Cube Corporation DVD/VCD Mastering Xilinx at Work in High Volume Applications ® www.xilinx.com
- 22. DCT/IDCT in DVD/VCD Digital Audio/Video Decoder MPEG-2 Video Decoder NTSC/ PAL To TV DVD Subpicture OSD Encoder DSP DEMUX Processing Graphics To Audio System Dolby AC3 or MPEG-2 Audio Decoder Front Panel Microcontroller 10:00:23 || Courtesy: C-Cube Corporation DVD Player Xilinx at Work in High Volume Applications ® www.xilinx.com
- 23. DCT/IDCT in DVD/VCD DVD Players Xilinx at Work in High Volume Applications ® www.xilinx.com
- 24. DCT/IDCT in Digital Cameras 2-D DCT/IDCT is applied generally on data sets that have a naturally two-dimensional characteristic, like a digital image Digital Cameras Xilinx at Work in High Volume Applications ® www.xilinx.com
- 25. Spartan-II DCT/IDCT Solution DCT/IDCT Cores — Available Separately or Combined AllianceCORE Xentec DCT/IDCT Core Xilinx at Work in High Volume Applications ® www.xilinx.com
- 26. Spartan-II DCT/IDCT Solution Features Features Spartan-II Device XC2S100-6 CLBs 1026 Clock IOBs 1 IOBs 28 Performance (MHz) 33.3 AllianceCORE Xentec DCT/IDCT Core Xilinx at Work in High Volume Applications ® www.xilinx.com
- 27. Spartan-II DCT/IDCT Solution Performance Low cost Spartan-II FPGA with soft IP from Xentec has High Performance — 180 times faster 32-bit mainstream processor operating at 266MHz Xilinx at Work in High Volume Applications ® www.xilinx.com
- 28. Spartan-II DCT/IDCT Solution - Features The Xilinx solution is efficient and cost-effective compared to DCT/IDCT software solution being run by a high performance 32-bit processor The Xilinx Xentec core solution is capable of operating either as DCT or IDCT by the use of a single mode pin Xilinx at Work in High Volume Applications ® www.xilinx.com
- 29. Summary DCT/IDCT Solutions are Widely Used in Multimedia, Video, Audio, and Imaging Applications The Spartan-II Family has Significant Strengths in its DCT/IDCT Solution: — Features — Performance — Scalability and Flexibility — Cost effectiveness Xilinx at Work in High Volume Applications ® www.xilinx.com