A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. DCTs are important to numerous applications in science and engineering, from lossy compression of audio (e.g. MP3) and images (e.g. JPEG) (where small high-frequency components can be discarded), to spectral methods for the numerical solution of partial differential equations. The use of cosine rather than sine functions is critical for compression, since it turns out (as described below) that fewer cosine functions are needed to approximate a typical signal, whereas for differential equations the cosines express a particular choice of boundary conditions.
In particular, a DCT is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), where in some variants the input and/or output data are shifted by half a sample. There are eight standard DCT variants, of which four are common.
DCT is the secret to JPEG's compression. Image Analyst Mike Pound explains how the compression works.
Colourspaces: https://youtu.be/LFXN9PiOGtY
JPEG 'files' & Colour: https://youtu.be/n_uNPbdenRs
Computer That Changed Everything (Altair 8800): https://youtu.be/6LYRgrqJgDc
Problems with JPEG: COMING SOON
Upside Down Trees (Huffman Encoding): https://youtu.be/umTbivyJoiI
Colourspaces: https://youtu.be/LFXN9PiOGtY
JPEG isn't a file format - JPEG pt1: https://youtu.be/n_uNPbdenRs
Upside Down Trees (Huffman Encoding): https://youtu.be/umTbivyJoiI
Problems with JPEG: COMING SOON!
Computer That Changed Everything (Altair 8800): https://youtu.be/6LYRgrqJgDc
http://www.facebook.com/computerphile
https://twitter.com/computer_phile
This video was filmed and edited by Sean Riley.
Comput...
published: 22 May 2015
The Two-Dimensional Discrete Cosine Transform
The two-dimensional discrete cosine transform (DCT) is used to represent images as weighted sums of cosines having different horizontal and vertical frequencies. The two-dimensional DCT plays a key role in JPEG image compression.
published: 23 Mar 2020
Discrete Cosine Transform and Haar transform with Examples
In this video we talk about Discrete Cosine Transform (DCT) and Haar transform with examples. These are important topics and are usually asked in exams.
Kindly like, subscribe and share the videos if you like them!
Check out our previous videos here!
Introduction to digital image processing - https://youtu.be/J-KxVvDRl18
Key stages in digital image processing - https://www.youtube.com/watch?v=8ekTeZD_lNY
Sampling and Quantization in digital image processing - https://youtu.be/KWc9SOOLfLw
Relationship between pixels Neighbourhood and Adjacency of Pixels-
https://youtu.be/JDsuds2oIF8
Distance Measures Between Pixels with examples- https://youtu.be/NOIlN9Bexpk
Arithmetic Operations and Logical Operations between Images in digital image processing-
https://youtu.be/kTxKca5i5tQ
Poin...
published: 01 Apr 2021
(DCT) Discrete Cosine Transform in image processing
My first Manim animation. How does the discrete cosine transform capture phase information if it has no sine components? I explain this here.
published: 05 Dec 2023
The Discrete Cosine Transform
The discrete cosine transform (DCT) is similar to the discrete Fourier transform, but describes signals as weighted sums of cosines rather than weighted sums of complex sinusoids. An advantage of the DCT is that the weights are real valued. The DCT is most often used in compression applications such as MP3 and JPEG.
Subject - Image Processing and Machine Vision
Video Name - Discrete Cosine Transform
Chapter - Image Transforms
Faculty - Prof. Vaibhav Pandit
Upskill and get Placements with Ekeeda Career Tracks
Data Science - https://ekeeda.com/career-track/data-scientist
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Get Free Notes of All Engineering Subjects & Technology
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Access the Complete Playlist of Subject I...
published: 08 Jan 2019
50 - Discrete Cosine Transform
published: 20 Feb 2022
Discrete Cosine Transformation
Brief introduction to Discrete Cosine Transform
published: 05 Oct 2022
The Unreasonable Effectiveness of JPEG: A Signal Processing Approach
Visit https://brilliant.org/Reducible/ to get started learning STEM for free, and the first 200 people will get 20% off their annual premium subscription.
Chapters:
00:00 Introducing JPEG and RGB Representation
2:15 Lossy Compression
3:41 What information can we get rid of?
4:36 Introducing YCbCr
6:10 Chroma subsampling/downsampling
8:10 Images represented as signals
9:52 Introducing the Discrete Cosine Transform (DCT)
11:32 Sampling cosine waves
12:43 Playing around with the DCT
17:38 Mathematically defining the DCT
21:02 The Inverse DCT
22:45 The 2D DCT
23:49 Visualizing the 2D DCT
24:35 Introducing Energy Compaction
26:05 Brilliant Sponsorship
27:23 Building an image from the 2D DCT
28:20 Quantization
30:23 Run-length/Huffman Encoding within JPEG
32:56 How JPEG fits into the big pictu...
DCT is the secret to JPEG's compression. Image Analyst Mike Pound explains how the compression works.
Colourspaces: https://youtu.be/LFXN9PiOGtY
JPEG 'files' ...
DCT is the secret to JPEG's compression. Image Analyst Mike Pound explains how the compression works.
Colourspaces: https://youtu.be/LFXN9PiOGtY
JPEG 'files' & Colour: https://youtu.be/n_uNPbdenRs
Computer That Changed Everything (Altair 8800): https://youtu.be/6LYRgrqJgDc
Problems with JPEG: COMING SOON
Upside Down Trees (Huffman Encoding): https://youtu.be/umTbivyJoiI
Colourspaces: https://youtu.be/LFXN9PiOGtY
JPEG isn't a file format - JPEG pt1: https://youtu.be/n_uNPbdenRs
Upside Down Trees (Huffman Encoding): https://youtu.be/umTbivyJoiI
Problems with JPEG: COMING SOON!
Computer That Changed Everything (Altair 8800): https://youtu.be/6LYRgrqJgDc
http://www.facebook.com/computerphile
https://twitter.com/computer_phile
This video was filmed and edited by Sean Riley.
Computer Science at the University of Nottingham: http://bit.ly/nottscomputer
Computerphile is a sister project to Brady Haran's Numberphile. More at http://www.bradyharan.com
DCT is the secret to JPEG's compression. Image Analyst Mike Pound explains how the compression works.
Colourspaces: https://youtu.be/LFXN9PiOGtY
JPEG 'files' & Colour: https://youtu.be/n_uNPbdenRs
Computer That Changed Everything (Altair 8800): https://youtu.be/6LYRgrqJgDc
Problems with JPEG: COMING SOON
Upside Down Trees (Huffman Encoding): https://youtu.be/umTbivyJoiI
Colourspaces: https://youtu.be/LFXN9PiOGtY
JPEG isn't a file format - JPEG pt1: https://youtu.be/n_uNPbdenRs
Upside Down Trees (Huffman Encoding): https://youtu.be/umTbivyJoiI
Problems with JPEG: COMING SOON!
Computer That Changed Everything (Altair 8800): https://youtu.be/6LYRgrqJgDc
http://www.facebook.com/computerphile
https://twitter.com/computer_phile
This video was filmed and edited by Sean Riley.
Computer Science at the University of Nottingham: http://bit.ly/nottscomputer
Computerphile is a sister project to Brady Haran's Numberphile. More at http://www.bradyharan.com
The two-dimensional discrete cosine transform (DCT) is used to represent images as weighted sums of cosines having different horizontal and vertical frequencies...
The two-dimensional discrete cosine transform (DCT) is used to represent images as weighted sums of cosines having different horizontal and vertical frequencies. The two-dimensional DCT plays a key role in JPEG image compression.
The two-dimensional discrete cosine transform (DCT) is used to represent images as weighted sums of cosines having different horizontal and vertical frequencies. The two-dimensional DCT plays a key role in JPEG image compression.
In this video we talk about Discrete Cosine Transform (DCT) and Haar transform with examples. These are important topics and are usually asked in exams.
Kindl...
In this video we talk about Discrete Cosine Transform (DCT) and Haar transform with examples. These are important topics and are usually asked in exams.
Kindly like, subscribe and share the videos if you like them!
Check out our previous videos here!
Introduction to digital image processing - https://youtu.be/J-KxVvDRl18
Key stages in digital image processing - https://www.youtube.com/watch?v=8ekTeZD_lNY
Sampling and Quantization in digital image processing - https://youtu.be/KWc9SOOLfLw
Relationship between pixels Neighbourhood and Adjacency of Pixels-
https://youtu.be/JDsuds2oIF8
Distance Measures Between Pixels with examples- https://youtu.be/NOIlN9Bexpk
Arithmetic Operations and Logical Operations between Images in digital image processing-
https://youtu.be/kTxKca5i5tQ
Point operations in digital image processing with examples -
https://youtu.be/FMDmXz6ynvk
Contrast Stretching and intensity level Slicing in digital image processing with examples -
https://youtu.be/YJIgFMoC_yg
Logarithmic Transformation and power-law Transformation in digital image processing with examples -
https://youtu.be/jfoXoFoDb48
Image Enhancement in digital image processing with Histogram Equalization -
https://youtu.be/Yd6QISby8kk
Histogram matching in digital image processing - https://youtu.be/r565euxWZBs
Fundamentals of Spatial Filtering in digital image processing - https://youtu.be/eDyJsFHYFFI
Smoothing Spatial Filters in digital image processing - https://youtu.be/x6zoQ-a7A9U
few important theory questions- https://youtu.be/vRxgHn2KTtI
Sharpening Spatial filters - https://youtu.be/ElE_XcYxvCY
Unsharp Masking and High boost Filtering - https://youtu.be/Hutnurzr59o
First-order Derivative Filters- https://youtu.be/fhDBy-wV3ic
Image transform in digital image processing with examples: - https://youtu.be/2oEt5lbsyhM
In this video we talk about Discrete Cosine Transform (DCT) and Haar transform with examples. These are important topics and are usually asked in exams.
Kindly like, subscribe and share the videos if you like them!
Check out our previous videos here!
Introduction to digital image processing - https://youtu.be/J-KxVvDRl18
Key stages in digital image processing - https://www.youtube.com/watch?v=8ekTeZD_lNY
Sampling and Quantization in digital image processing - https://youtu.be/KWc9SOOLfLw
Relationship between pixels Neighbourhood and Adjacency of Pixels-
https://youtu.be/JDsuds2oIF8
Distance Measures Between Pixels with examples- https://youtu.be/NOIlN9Bexpk
Arithmetic Operations and Logical Operations between Images in digital image processing-
https://youtu.be/kTxKca5i5tQ
Point operations in digital image processing with examples -
https://youtu.be/FMDmXz6ynvk
Contrast Stretching and intensity level Slicing in digital image processing with examples -
https://youtu.be/YJIgFMoC_yg
Logarithmic Transformation and power-law Transformation in digital image processing with examples -
https://youtu.be/jfoXoFoDb48
Image Enhancement in digital image processing with Histogram Equalization -
https://youtu.be/Yd6QISby8kk
Histogram matching in digital image processing - https://youtu.be/r565euxWZBs
Fundamentals of Spatial Filtering in digital image processing - https://youtu.be/eDyJsFHYFFI
Smoothing Spatial Filters in digital image processing - https://youtu.be/x6zoQ-a7A9U
few important theory questions- https://youtu.be/vRxgHn2KTtI
Sharpening Spatial filters - https://youtu.be/ElE_XcYxvCY
Unsharp Masking and High boost Filtering - https://youtu.be/Hutnurzr59o
First-order Derivative Filters- https://youtu.be/fhDBy-wV3ic
Image transform in digital image processing with examples: - https://youtu.be/2oEt5lbsyhM
The discrete cosine transform (DCT) is similar to the discrete Fourier transform, but describes signals as weighted sums of cosines rather than weighted sums of...
The discrete cosine transform (DCT) is similar to the discrete Fourier transform, but describes signals as weighted sums of cosines rather than weighted sums of complex sinusoids. An advantage of the DCT is that the weights are real valued. The DCT is most often used in compression applications such as MP3 and JPEG.
The discrete cosine transform (DCT) is similar to the discrete Fourier transform, but describes signals as weighted sums of cosines rather than weighted sums of complex sinusoids. An advantage of the DCT is that the weights are real valued. The DCT is most often used in compression applications such as MP3 and JPEG.
Subject - Image Processing and Machine Vision
Video Name - Discrete Cosine Transform
Chapter - Image Transforms
Faculty - Prof. Vaibhav Pandit
Upskill and g...
Subject - Image Processing and Machine Vision
Video Name - Discrete Cosine Transform
Chapter - Image Transforms
Faculty - Prof. Vaibhav Pandit
Upskill and get Placements with Ekeeda Career Tracks
Data Science - https://ekeeda.com/career-track/data-scientist
Software Development Engineer - https://ekeeda.com/career-track/software-development-engineer
Embedded & IoT Engineer - https://ekeeda.com/career-track/embedded-and-iot-engineer
Get FREE Trial for GATE 2023 Exam with Ekeeda GATE - 20000+ Lectures & Notes, strategy, updates, and notifications which will help you to crack your GATE exam.
https://ekeeda.com/catalog/competitive-exam
Coupon Code - EKGATE
Get Free Notes of All Engineering Subjects & Technology
https://ekeeda.com/digital-library
Access the Complete Playlist of Subject Image Processing and Machine Vision -
https://www.youtube.com/playlist?list=PLm_MSClsnwm8vk9HCc8WOQyrZz3VzEHsW
Happy Learning
Social Links:
https://www.instagram.com/ekeeda_official/
https://in.linkedin.com/company/ekeeda.com
#DiscreteCosineTransform #ImageTransforms #ImageProcessing
Subject - Image Processing and Machine Vision
Video Name - Discrete Cosine Transform
Chapter - Image Transforms
Faculty - Prof. Vaibhav Pandit
Upskill and get Placements with Ekeeda Career Tracks
Data Science - https://ekeeda.com/career-track/data-scientist
Software Development Engineer - https://ekeeda.com/career-track/software-development-engineer
Embedded & IoT Engineer - https://ekeeda.com/career-track/embedded-and-iot-engineer
Get FREE Trial for GATE 2023 Exam with Ekeeda GATE - 20000+ Lectures & Notes, strategy, updates, and notifications which will help you to crack your GATE exam.
https://ekeeda.com/catalog/competitive-exam
Coupon Code - EKGATE
Get Free Notes of All Engineering Subjects & Technology
https://ekeeda.com/digital-library
Access the Complete Playlist of Subject Image Processing and Machine Vision -
https://www.youtube.com/playlist?list=PLm_MSClsnwm8vk9HCc8WOQyrZz3VzEHsW
Happy Learning
Social Links:
https://www.instagram.com/ekeeda_official/
https://in.linkedin.com/company/ekeeda.com
#DiscreteCosineTransform #ImageTransforms #ImageProcessing
Visit https://brilliant.org/Reducible/ to get started learning STEM for free, and the first 200 people will get 20% off their annual premium subscription.
Cha...
Visit https://brilliant.org/Reducible/ to get started learning STEM for free, and the first 200 people will get 20% off their annual premium subscription.
Chapters:
00:00 Introducing JPEG and RGB Representation
2:15 Lossy Compression
3:41 What information can we get rid of?
4:36 Introducing YCbCr
6:10 Chroma subsampling/downsampling
8:10 Images represented as signals
9:52 Introducing the Discrete Cosine Transform (DCT)
11:32 Sampling cosine waves
12:43 Playing around with the DCT
17:38 Mathematically defining the DCT
21:02 The Inverse DCT
22:45 The 2D DCT
23:49 Visualizing the 2D DCT
24:35 Introducing Energy Compaction
26:05 Brilliant Sponsorship
27:23 Building an image from the 2D DCT
28:20 Quantization
30:23 Run-length/Huffman Encoding within JPEG
32:56 How JPEG fits into the big picture of data compression
The JPEG algorithm is rather complex and in this video, we break down the core parts of the algorithm, specifically color spaces, YCbCr, chroma subsampling, the discrete cosine transform, quantization, and lossless encoding. The majority of the focus is on the mathematical and signal processing insights that lead to advancements in image compression and the big themes in compression as a whole that we can take away from it.
Animations created jointly by Nipun Ramakrishnan and Jesús Rascón.
References/Additional Resources:
https://www.red.com/red-101/video-chroma-subsampling - great resource on different types of chroma subsampling
http://weitz.de/dct/ - play around with the DCT
https://www.cse.iitd.ac.in/~pkalra/col783-2017/DCT-History.pdf - paper referenced in video
http://www.ee.ic.ac.uk/hp/staff/dmb/courses/DSPDF/00300_Transforms.pdf - a more rigorous signal processing approach to the DCT
https://www.impulseadventure.com/photo/jpeg-huffman-coding.html - great landing point for learning more about how huffman codes work in the context of JPEG
https://www.youtube.com/watch?v=CPT4FSkFUgs&list=PLpsTn9TA_Q8VMDyOPrDKmSJYt1DLgDZU4&ab_channel=DanielHarding - a great playlist I recommend that dives deep into actually implementing a JPEG decoder
This video wouldn't be possible without the open source library manim created by 3blue1brown and maintained by Manim Community.
The Manim Community Developers. (2021). Manim – Mathematical Animation Framework (Version v0.11.0) [Computer software]. https://www.manim.community/
Here is link to the repository that contains the code used to generate the animations in this video: https://github.com/nipunramk/Reducible
All music in the video is from Aakash Gandhi
Visit https://brilliant.org/Reducible/ to get started learning STEM for free, and the first 200 people will get 20% off their annual premium subscription.
Chapters:
00:00 Introducing JPEG and RGB Representation
2:15 Lossy Compression
3:41 What information can we get rid of?
4:36 Introducing YCbCr
6:10 Chroma subsampling/downsampling
8:10 Images represented as signals
9:52 Introducing the Discrete Cosine Transform (DCT)
11:32 Sampling cosine waves
12:43 Playing around with the DCT
17:38 Mathematically defining the DCT
21:02 The Inverse DCT
22:45 The 2D DCT
23:49 Visualizing the 2D DCT
24:35 Introducing Energy Compaction
26:05 Brilliant Sponsorship
27:23 Building an image from the 2D DCT
28:20 Quantization
30:23 Run-length/Huffman Encoding within JPEG
32:56 How JPEG fits into the big picture of data compression
The JPEG algorithm is rather complex and in this video, we break down the core parts of the algorithm, specifically color spaces, YCbCr, chroma subsampling, the discrete cosine transform, quantization, and lossless encoding. The majority of the focus is on the mathematical and signal processing insights that lead to advancements in image compression and the big themes in compression as a whole that we can take away from it.
Animations created jointly by Nipun Ramakrishnan and Jesús Rascón.
References/Additional Resources:
https://www.red.com/red-101/video-chroma-subsampling - great resource on different types of chroma subsampling
http://weitz.de/dct/ - play around with the DCT
https://www.cse.iitd.ac.in/~pkalra/col783-2017/DCT-History.pdf - paper referenced in video
http://www.ee.ic.ac.uk/hp/staff/dmb/courses/DSPDF/00300_Transforms.pdf - a more rigorous signal processing approach to the DCT
https://www.impulseadventure.com/photo/jpeg-huffman-coding.html - great landing point for learning more about how huffman codes work in the context of JPEG
https://www.youtube.com/watch?v=CPT4FSkFUgs&list=PLpsTn9TA_Q8VMDyOPrDKmSJYt1DLgDZU4&ab_channel=DanielHarding - a great playlist I recommend that dives deep into actually implementing a JPEG decoder
This video wouldn't be possible without the open source library manim created by 3blue1brown and maintained by Manim Community.
The Manim Community Developers. (2021). Manim – Mathematical Animation Framework (Version v0.11.0) [Computer software]. https://www.manim.community/
Here is link to the repository that contains the code used to generate the animations in this video: https://github.com/nipunramk/Reducible
All music in the video is from Aakash Gandhi
DCT is the secret to JPEG's compression. Image Analyst Mike Pound explains how the compression works.
Colourspaces: https://youtu.be/LFXN9PiOGtY
JPEG 'files' & Colour: https://youtu.be/n_uNPbdenRs
Computer That Changed Everything (Altair 8800): https://youtu.be/6LYRgrqJgDc
Problems with JPEG: COMING SOON
Upside Down Trees (Huffman Encoding): https://youtu.be/umTbivyJoiI
Colourspaces: https://youtu.be/LFXN9PiOGtY
JPEG isn't a file format - JPEG pt1: https://youtu.be/n_uNPbdenRs
Upside Down Trees (Huffman Encoding): https://youtu.be/umTbivyJoiI
Problems with JPEG: COMING SOON!
Computer That Changed Everything (Altair 8800): https://youtu.be/6LYRgrqJgDc
http://www.facebook.com/computerphile
https://twitter.com/computer_phile
This video was filmed and edited by Sean Riley.
Computer Science at the University of Nottingham: http://bit.ly/nottscomputer
Computerphile is a sister project to Brady Haran's Numberphile. More at http://www.bradyharan.com
The two-dimensional discrete cosine transform (DCT) is used to represent images as weighted sums of cosines having different horizontal and vertical frequencies. The two-dimensional DCT plays a key role in JPEG image compression.
In this video we talk about Discrete Cosine Transform (DCT) and Haar transform with examples. These are important topics and are usually asked in exams.
Kindly like, subscribe and share the videos if you like them!
Check out our previous videos here!
Introduction to digital image processing - https://youtu.be/J-KxVvDRl18
Key stages in digital image processing - https://www.youtube.com/watch?v=8ekTeZD_lNY
Sampling and Quantization in digital image processing - https://youtu.be/KWc9SOOLfLw
Relationship between pixels Neighbourhood and Adjacency of Pixels-
https://youtu.be/JDsuds2oIF8
Distance Measures Between Pixels with examples- https://youtu.be/NOIlN9Bexpk
Arithmetic Operations and Logical Operations between Images in digital image processing-
https://youtu.be/kTxKca5i5tQ
Point operations in digital image processing with examples -
https://youtu.be/FMDmXz6ynvk
Contrast Stretching and intensity level Slicing in digital image processing with examples -
https://youtu.be/YJIgFMoC_yg
Logarithmic Transformation and power-law Transformation in digital image processing with examples -
https://youtu.be/jfoXoFoDb48
Image Enhancement in digital image processing with Histogram Equalization -
https://youtu.be/Yd6QISby8kk
Histogram matching in digital image processing - https://youtu.be/r565euxWZBs
Fundamentals of Spatial Filtering in digital image processing - https://youtu.be/eDyJsFHYFFI
Smoothing Spatial Filters in digital image processing - https://youtu.be/x6zoQ-a7A9U
few important theory questions- https://youtu.be/vRxgHn2KTtI
Sharpening Spatial filters - https://youtu.be/ElE_XcYxvCY
Unsharp Masking and High boost Filtering - https://youtu.be/Hutnurzr59o
First-order Derivative Filters- https://youtu.be/fhDBy-wV3ic
Image transform in digital image processing with examples: - https://youtu.be/2oEt5lbsyhM
The discrete cosine transform (DCT) is similar to the discrete Fourier transform, but describes signals as weighted sums of cosines rather than weighted sums of complex sinusoids. An advantage of the DCT is that the weights are real valued. The DCT is most often used in compression applications such as MP3 and JPEG.
Subject - Image Processing and Machine Vision
Video Name - Discrete Cosine Transform
Chapter - Image Transforms
Faculty - Prof. Vaibhav Pandit
Upskill and get Placements with Ekeeda Career Tracks
Data Science - https://ekeeda.com/career-track/data-scientist
Software Development Engineer - https://ekeeda.com/career-track/software-development-engineer
Embedded & IoT Engineer - https://ekeeda.com/career-track/embedded-and-iot-engineer
Get FREE Trial for GATE 2023 Exam with Ekeeda GATE - 20000+ Lectures & Notes, strategy, updates, and notifications which will help you to crack your GATE exam.
https://ekeeda.com/catalog/competitive-exam
Coupon Code - EKGATE
Get Free Notes of All Engineering Subjects & Technology
https://ekeeda.com/digital-library
Access the Complete Playlist of Subject Image Processing and Machine Vision -
https://www.youtube.com/playlist?list=PLm_MSClsnwm8vk9HCc8WOQyrZz3VzEHsW
Happy Learning
Social Links:
https://www.instagram.com/ekeeda_official/
https://in.linkedin.com/company/ekeeda.com
#DiscreteCosineTransform #ImageTransforms #ImageProcessing
Visit https://brilliant.org/Reducible/ to get started learning STEM for free, and the first 200 people will get 20% off their annual premium subscription.
Chapters:
00:00 Introducing JPEG and RGB Representation
2:15 Lossy Compression
3:41 What information can we get rid of?
4:36 Introducing YCbCr
6:10 Chroma subsampling/downsampling
8:10 Images represented as signals
9:52 Introducing the Discrete Cosine Transform (DCT)
11:32 Sampling cosine waves
12:43 Playing around with the DCT
17:38 Mathematically defining the DCT
21:02 The Inverse DCT
22:45 The 2D DCT
23:49 Visualizing the 2D DCT
24:35 Introducing Energy Compaction
26:05 Brilliant Sponsorship
27:23 Building an image from the 2D DCT
28:20 Quantization
30:23 Run-length/Huffman Encoding within JPEG
32:56 How JPEG fits into the big picture of data compression
The JPEG algorithm is rather complex and in this video, we break down the core parts of the algorithm, specifically color spaces, YCbCr, chroma subsampling, the discrete cosine transform, quantization, and lossless encoding. The majority of the focus is on the mathematical and signal processing insights that lead to advancements in image compression and the big themes in compression as a whole that we can take away from it.
Animations created jointly by Nipun Ramakrishnan and Jesús Rascón.
References/Additional Resources:
https://www.red.com/red-101/video-chroma-subsampling - great resource on different types of chroma subsampling
http://weitz.de/dct/ - play around with the DCT
https://www.cse.iitd.ac.in/~pkalra/col783-2017/DCT-History.pdf - paper referenced in video
http://www.ee.ic.ac.uk/hp/staff/dmb/courses/DSPDF/00300_Transforms.pdf - a more rigorous signal processing approach to the DCT
https://www.impulseadventure.com/photo/jpeg-huffman-coding.html - great landing point for learning more about how huffman codes work in the context of JPEG
https://www.youtube.com/watch?v=CPT4FSkFUgs&list=PLpsTn9TA_Q8VMDyOPrDKmSJYt1DLgDZU4&ab_channel=DanielHarding - a great playlist I recommend that dives deep into actually implementing a JPEG decoder
This video wouldn't be possible without the open source library manim created by 3blue1brown and maintained by Manim Community.
The Manim Community Developers. (2021). Manim – Mathematical Animation Framework (Version v0.11.0) [Computer software]. https://www.manim.community/
Here is link to the repository that contains the code used to generate the animations in this video: https://github.com/nipunramk/Reducible
All music in the video is from Aakash Gandhi
A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. DCTs are important to numerous applications in science and engineering, from lossy compression of audio (e.g. MP3) and images (e.g. JPEG) (where small high-frequency components can be discarded), to spectral methods for the numerical solution of partial differential equations. The use of cosine rather than sine functions is critical for compression, since it turns out (as described below) that fewer cosine functions are needed to approximate a typical signal, whereas for differential equations the cosines express a particular choice of boundary conditions.
In particular, a DCT is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), where in some variants the input and/or output data are shifted by half a sample. There are eight standard DCT variants, of which four are common.
Last night on the 2, ride home. I swear man I saw an angel. But when I looked again man, it was just another stranger. Last night I saw liquor land, man I had a big mission. 13 horsemen riding on and off wind, so I turn my color to the gold. Got back in my ride, turn up that stereo. Man what do you have, that you don't need. We need it all down here! Temptation, redemption, the Bottle and the Cross. Life's illusion, from everything you've done. And somewhere out there, I hear a SCREAM! Last night in brigham circle the devil tried to mug me. And I said: "Man what do I got, you don't got allready?" You know he looked up to me and said: "Maybe you're right" Maybe I got something else for you tonight, He said: "Man what do you need, that you don't own?" "We got it all down here!" Temptation, redemption, the Bottle and the Cross. Life's illusion, from everything you've done. And somewhere out there over the airwaves, I hear a SCREAM! And when the music takes control, it's allright by me. Ooh another moon rise competing with the lights of the bright city. Caught between the drum and the baseline, you know it's allright. SAM System homing like a 737's come to take me back home.