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PR-395: Variational Image Compression with a Scale Hyperprior

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Hyeongmin Lee

제가 이번에 소개드릴 논문은 Variational Image Compression with a Scale Hyperprior라는 논문입니다. 지난 328번째 발표에 이어서 두번째 Deep Learning-based Image Compression이고, 지난번 발표때 다루지 못했던 Variational Autoencoder와의 관계와 이번 논문에서의 새 Contribution까지, Deep Learning을 이용한 Image Compression연구는 어떤 고민을 주로 하고 있는지 등을 전달해드리고자 노력하였습니다. 논문 링크: https://arxiv.org/abs/1802.01436 영상 링크: https://youtu.be/ne9ieHRsfCc

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Variational Image Compression with a Scale Hyperprior Hyeongmin Lee PR-395 2022.7.17
Entropy Coding
Entropy Coding  Image Compression 0011010100111...
Entropy Coding  Entropy Coding A,B,C,D: 4 Letters (need at least 2 bits per letter) Sample Bits A 00 B 10 C 01 D 11 100100001111110101000100010001 (30 bits) https://www.programiz.com/dsa/huffman-coding
Entropy Coding  Entropy Coding A,B,C,D: 4 Letters (need at least 2 bits per letter) Sample Bits A 11 B 100 C 0 D 101 1000111110110110100110110110 (28 bits) Lower Bound?? Huffman Coding
Entropy Coding  Entropy 𝑯𝑯 = − � 𝒑𝒑𝒊𝒊𝒍𝒍𝒍𝒍𝒈𝒈𝟐𝟐𝒑𝒑𝒊𝒊 Sample 𝒑𝒑𝒊𝒊 A 5/15 B 1/15 C 6/15 D 3/15
Image Compression
Image Compression  Image Compression 0011010100111... Lossless Coding: bmp  Coding the Integers Itself Lossy Coding: JPEG  Reducing the ‘Entropy’ of samples
Image Compression  JPEG 8x8 cutting Discrete Cosine Transform Quantization Entropy Coding... https://www.whydomath.org/node/wavlets/basicjpg.html
Image Compression  JPEG High Quality High Entropy High Bitrate Low Quality Low Entropy Low Bitrate
Image Compression  Bitrate-Distortion Tradeoff
Deep Learning-based Image Compression
Deep Learning-based Image Compression  Image Compression Pipeline Transform Quantization Entropy Coding Decoding Inverse Transform
Deep Learning-based Image Compression  Image Compression Pipeline 𝐿𝐿 = 𝑅𝑅 + 𝜆𝜆𝜆𝜆
Deep Learning-based Image Compression  Auto Encoder � 𝑦𝑦 = 𝑄𝑄(𝑦𝑦) 𝑦𝑦 = 𝑔𝑔𝑎𝑎 𝑥𝑥; 𝜙𝜙𝑔𝑔 � 𝑥𝑥 = 𝑔𝑔𝑠𝑠(� 𝑦𝑦; 𝜃𝜃𝑔𝑔) 𝑅𝑅 = 𝐸𝐸𝑥𝑥~𝑝𝑝𝑥𝑥 [− log2 𝑝𝑝� 𝑦𝑦(� 𝑦𝑦)] 𝐷𝐷 = 𝑥𝑥 − � 𝑥𝑥 2 2 ???
Problem of Quantization
Problem of Quantization  Adding Uniform Noise [PR328] � 𝑦𝑦 = 𝑄𝑄 𝑦𝑦 � 𝑦𝑦 = 𝑦𝑦 + 𝑛𝑛, 𝑛𝑛~𝑈𝑈(− 1 2 , 1 2 ) Ballé, Johannes, Valero Laparra, and Eero P. Simoncelli. "End-to-end optimized image compression." 5th International Conference on Learning Representations, ICLR 2017. 𝑅𝑅 = 𝐸𝐸𝑥𝑥~𝑝𝑝𝑥𝑥 − log2 𝑝𝑝� 𝑦𝑦 � 𝑦𝑦 = 𝐸𝐸𝑥𝑥~𝑝𝑝𝑥𝑥 [− log2 𝑝𝑝� 𝑦𝑦(𝑄𝑄(𝑔𝑔𝑎𝑎(𝑥𝑥; 𝜙𝜙𝑔𝑔)))] 𝑅𝑅 = 𝐸𝐸𝑥𝑥~𝑝𝑝𝑥𝑥 − log2 𝑝𝑝� 𝑦𝑦 � 𝑦𝑦 = 𝐸𝐸𝑥𝑥~𝑝𝑝𝑥𝑥 [− log2 𝑝𝑝� 𝑦𝑦(𝑔𝑔𝑎𝑎 𝑥𝑥; 𝜙𝜙𝑔𝑔 + 𝑛𝑛)]
Rate Loss
Rate Loss  Applying Uniform Noise Approximation 𝑅𝑅 = 𝐸𝐸𝑥𝑥~𝑝𝑝𝑥𝑥 [− log2 𝑝𝑝� 𝑦𝑦(𝑔𝑔𝑎𝑎 𝑥𝑥; 𝜙𝜙𝑔𝑔 + 𝑛𝑛)] 𝑝𝑝� 𝑦𝑦 𝑅𝑅 = 𝐸𝐸𝑥𝑥~𝑝𝑝𝑥𝑥 [− log2 𝑝𝑝� 𝑦𝑦(� 𝑦𝑦)] Entropy Model
Non-parametric Density Model
Non-parametric Density Model  Defining the density as a function (Neural Network)
Non-parametric Density Model  Using Sigmoid at the last layer All Jacobian elements are positive
Non-parametric Density Model  Intermediate Activation Functions All Jacobian elements are positive 𝑎𝑎 > 0 −1 < 𝑎𝑎 < 0 𝑎𝑎 < −1
Non-parametric Density Model  Setting parameter constraints All Jacobian elements are positive
Non-parametric Density Model  Experiment on toy example
Non-parametric Density Model  Revisiting Loss Function 𝑅𝑅 = 𝐸𝐸𝑥𝑥~𝑝𝑝𝑥𝑥 [− log2 𝑝𝑝� 𝑦𝑦(� 𝑦𝑦)] 𝐷𝐷 = 𝑥𝑥 − � 𝑥𝑥 2 2 𝐿𝐿 = 𝑅𝑅 + 𝜆𝜆𝜆𝜆
Variational Auto Encoder
Variational Auto Encoder  Auto Encoder � 𝑦𝑦 = 𝑦𝑦 + 𝑛𝑛 𝑦𝑦 = 𝑔𝑔𝑎𝑎 𝑥𝑥; 𝜙𝜙𝑔𝑔 � 𝑥𝑥 = 𝑔𝑔𝑠𝑠(� 𝑦𝑦; 𝜃𝜃𝑔𝑔)
Variational Auto Encoder  Variational Auto Encoder � 𝑦𝑦~𝑝𝑝� 𝑦𝑦|𝑥𝑥(� 𝑦𝑦|𝑥𝑥) 𝑥𝑥~𝑝𝑝𝑥𝑥|𝑦𝑦(𝑥𝑥|� 𝑦𝑦)
Variational Auto Encoder  Defining Generative Model � 𝑦𝑦~𝑝𝑝� 𝑦𝑦|𝑥𝑥(� 𝑦𝑦|𝑥𝑥)
Variational Auto Encoder  Setting the parametric version of posterior
Variational Auto Encoder  Setting KL Divergence
Scale Hyperprior
Scale Hyperprior  Factorized Prior � 𝑦𝑦 = 𝑦𝑦 + 𝑛𝑛 𝑦𝑦 = 𝑔𝑔𝑎𝑎 𝑥𝑥; 𝜙𝜙𝑔𝑔 � 𝑥𝑥 = 𝑔𝑔𝑠𝑠(� 𝑦𝑦; 𝜃𝜃𝑔𝑔) 𝑅𝑅 = − log2 𝑝𝑝� 𝑦𝑦(� 𝑦𝑦) 𝐷𝐷 = 𝑥𝑥 − � 𝑥𝑥 2 2 𝐿𝐿 = 𝑅𝑅 + 𝜆𝜆𝜆𝜆 Inference Model Generative Model Entropy Model 𝐸𝐸𝐸𝐸𝐸𝐸[� 𝑦𝑦] 𝐷𝐷𝐷𝐷𝐷𝐷[� 𝑦𝑦]
Scale Hyperprior  Scale of Y Encoding Scale as ‘Additional Information’
Scale Hyperprior  Scale Hyperprior
Scale Hyperprior  Factorized Prior � 𝑦𝑦 = 𝑦𝑦 + 𝑛𝑛 𝑦𝑦 = 𝑔𝑔𝑎𝑎 𝑥𝑥; 𝜙𝜙𝑔𝑔 � 𝑥𝑥 = 𝑔𝑔𝑠𝑠(� 𝑦𝑦; 𝜃𝜃𝑔𝑔) 𝑅𝑅 = − log2 𝑝𝑝� 𝑦𝑦 � 𝑦𝑦 − log2 𝑝𝑝� 𝑦𝑦|� 𝑧𝑧(� 𝑦𝑦| ̃ 𝑧𝑧) 𝐷𝐷 = 𝑥𝑥 − � 𝑥𝑥 2 2 𝐿𝐿 = 𝑅𝑅 + 𝜆𝜆𝜆𝜆 Inference Model Generative Model Entropy Model 𝑧𝑧 = ℎ𝑎𝑎 𝑦𝑦; 𝜙𝜙ℎ ̃ 𝑧𝑧 = 𝑧𝑧 + 𝑛𝑛 � 𝜎𝜎 = ℎ𝑠𝑠 ̃ 𝑧𝑧; 𝜃𝜃ℎ 𝐸𝐸𝐸𝐸𝐸𝐸[� 𝑦𝑦; � 𝜎𝜎] 𝐷𝐷𝐷𝐷𝐷𝐷[ ̃ 𝑧𝑧] � 𝜎𝜎 = ℎ𝑠𝑠 ̃ 𝑧𝑧; 𝜃𝜃ℎ 𝐸𝐸𝐸𝐸𝐸𝐸[ ̃ 𝑧𝑧] 𝐷𝐷𝐷𝐷𝐷𝐷[� 𝑦𝑦; � 𝜎𝜎]
Scale Hyperprior  Network Architecture
Scale Hyperprior  Experiments
Scale Hyperprior  Experiments
Thank You!

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PR-395: Variational Image Compression with a Scale Hyperprior

  • 1. Variational Image Compression with a Scale Hyperprior Hyeongmin Lee PR-395 2022.7.17
  • 3. Entropy Coding  Image Compression 0011010100111...
  • 4. Entropy Coding  Entropy Coding A,B,C,D: 4 Letters (need at least 2 bits per letter) Sample Bits A 00 B 10 C 01 D 11 100100001111110101000100010001 (30 bits) https://www.programiz.com/dsa/huffman-coding
  • 5. Entropy Coding  Entropy Coding A,B,C,D: 4 Letters (need at least 2 bits per letter) Sample Bits A 11 B 100 C 0 D 101 1000111110110110100110110110 (28 bits) Lower Bound?? Huffman Coding
  • 6. Entropy Coding  Entropy 𝑯𝑯 = − � 𝒑𝒑𝒊𝒊𝒍𝒍𝒍𝒍𝒈𝒈𝟐𝟐𝒑𝒑𝒊𝒊 Sample 𝒑𝒑𝒊𝒊 A 5/15 B 1/15 C 6/15 D 3/15
  • 8. Image Compression  Image Compression 0011010100111... Lossless Coding: bmp  Coding the Integers Itself Lossy Coding: JPEG  Reducing the ‘Entropy’ of samples
  • 9. Image Compression  JPEG 8x8 cutting Discrete Cosine Transform Quantization Entropy Coding... https://www.whydomath.org/node/wavlets/basicjpg.html
  • 10. Image Compression  JPEG High Quality High Entropy High Bitrate Low Quality Low Entropy Low Bitrate
  • 13. Deep Learning-based Image Compression  Image Compression Pipeline Transform Quantization Entropy Coding Decoding Inverse Transform
  • 14. Deep Learning-based Image Compression  Image Compression Pipeline 𝐿𝐿 = 𝑅𝑅 + 𝜆𝜆𝜆𝜆
  • 15. Deep Learning-based Image Compression  Auto Encoder � 𝑦𝑦 = 𝑄𝑄(𝑦𝑦) 𝑦𝑦 = 𝑔𝑔𝑎𝑎 𝑥𝑥; 𝜙𝜙𝑔𝑔 � 𝑥𝑥 = 𝑔𝑔𝑠𝑠(� 𝑦𝑦; 𝜃𝜃𝑔𝑔) 𝑅𝑅 = 𝐸𝐸𝑥𝑥~𝑝𝑝𝑥𝑥 [− log2 𝑝𝑝� 𝑦𝑦(� 𝑦𝑦)] 𝐷𝐷 = 𝑥𝑥 − � 𝑥𝑥 2 2 ???
  • 17. Problem of Quantization  Adding Uniform Noise [PR328] � 𝑦𝑦 = 𝑄𝑄 𝑦𝑦 � 𝑦𝑦 = 𝑦𝑦 + 𝑛𝑛, 𝑛𝑛~𝑈𝑈(− 1 2 , 1 2 ) Ballé, Johannes, Valero Laparra, and Eero P. Simoncelli. "End-to-end optimized image compression." 5th International Conference on Learning Representations, ICLR 2017. 𝑅𝑅 = 𝐸𝐸𝑥𝑥~𝑝𝑝𝑥𝑥 − log2 𝑝𝑝� 𝑦𝑦 � 𝑦𝑦 = 𝐸𝐸𝑥𝑥~𝑝𝑝𝑥𝑥 [− log2 𝑝𝑝� 𝑦𝑦(𝑄𝑄(𝑔𝑔𝑎𝑎(𝑥𝑥; 𝜙𝜙𝑔𝑔)))] 𝑅𝑅 = 𝐸𝐸𝑥𝑥~𝑝𝑝𝑥𝑥 − log2 𝑝𝑝� 𝑦𝑦 � 𝑦𝑦 = 𝐸𝐸𝑥𝑥~𝑝𝑝𝑥𝑥 [− log2 𝑝𝑝� 𝑦𝑦(𝑔𝑔𝑎𝑎 𝑥𝑥; 𝜙𝜙𝑔𝑔 + 𝑛𝑛)]
  • 19. Rate Loss  Applying Uniform Noise Approximation 𝑅𝑅 = 𝐸𝐸𝑥𝑥~𝑝𝑝𝑥𝑥 [− log2 𝑝𝑝� 𝑦𝑦(𝑔𝑔𝑎𝑎 𝑥𝑥; 𝜙𝜙𝑔𝑔 + 𝑛𝑛)] 𝑝𝑝� 𝑦𝑦 𝑅𝑅 = 𝐸𝐸𝑥𝑥~𝑝𝑝𝑥𝑥 [− log2 𝑝𝑝� 𝑦𝑦(� 𝑦𝑦)] Entropy Model
  • 21. Non-parametric Density Model  Defining the density as a function (Neural Network)
  • 22. Non-parametric Density Model  Using Sigmoid at the last layer All Jacobian elements are positive
  • 23. Non-parametric Density Model  Intermediate Activation Functions All Jacobian elements are positive 𝑎𝑎 > 0 −1 < 𝑎𝑎 < 0 𝑎𝑎 < −1
  • 24. Non-parametric Density Model  Setting parameter constraints All Jacobian elements are positive
  • 25. Non-parametric Density Model  Experiment on toy example
  • 26. Non-parametric Density Model  Revisiting Loss Function 𝑅𝑅 = 𝐸𝐸𝑥𝑥~𝑝𝑝𝑥𝑥 [− log2 𝑝𝑝� 𝑦𝑦(� 𝑦𝑦)] 𝐷𝐷 = 𝑥𝑥 − � 𝑥𝑥 2 2 𝐿𝐿 = 𝑅𝑅 + 𝜆𝜆𝜆𝜆
  • 28. Variational Auto Encoder  Auto Encoder � 𝑦𝑦 = 𝑦𝑦 + 𝑛𝑛 𝑦𝑦 = 𝑔𝑔𝑎𝑎 𝑥𝑥; 𝜙𝜙𝑔𝑔 � 𝑥𝑥 = 𝑔𝑔𝑠𝑠(� 𝑦𝑦; 𝜃𝜃𝑔𝑔)
  • 29. Variational Auto Encoder  Variational Auto Encoder � 𝑦𝑦~𝑝𝑝� 𝑦𝑦|𝑥𝑥(� 𝑦𝑦|𝑥𝑥) 𝑥𝑥~𝑝𝑝𝑥𝑥|𝑦𝑦(𝑥𝑥|� 𝑦𝑦)
  • 30. Variational Auto Encoder  Defining Generative Model � 𝑦𝑦~𝑝𝑝� 𝑦𝑦|𝑥𝑥(� 𝑦𝑦|𝑥𝑥)
  • 31. Variational Auto Encoder  Setting the parametric version of posterior
  • 32. Variational Auto Encoder  Setting KL Divergence
  • 34. Scale Hyperprior  Factorized Prior � 𝑦𝑦 = 𝑦𝑦 + 𝑛𝑛 𝑦𝑦 = 𝑔𝑔𝑎𝑎 𝑥𝑥; 𝜙𝜙𝑔𝑔 � 𝑥𝑥 = 𝑔𝑔𝑠𝑠(� 𝑦𝑦; 𝜃𝜃𝑔𝑔) 𝑅𝑅 = − log2 𝑝𝑝� 𝑦𝑦(� 𝑦𝑦) 𝐷𝐷 = 𝑥𝑥 − � 𝑥𝑥 2 2 𝐿𝐿 = 𝑅𝑅 + 𝜆𝜆𝜆𝜆 Inference Model Generative Model Entropy Model 𝐸𝐸𝐸𝐸𝐸𝐸[� 𝑦𝑦] 𝐷𝐷𝐷𝐷𝐷𝐷[� 𝑦𝑦]
  • 35. Scale Hyperprior  Scale of Y Encoding Scale as ‘Additional Information’
  • 37. Scale Hyperprior  Factorized Prior � 𝑦𝑦 = 𝑦𝑦 + 𝑛𝑛 𝑦𝑦 = 𝑔𝑔𝑎𝑎 𝑥𝑥; 𝜙𝜙𝑔𝑔 � 𝑥𝑥 = 𝑔𝑔𝑠𝑠(� 𝑦𝑦; 𝜃𝜃𝑔𝑔) 𝑅𝑅 = − log2 𝑝𝑝� 𝑦𝑦 � 𝑦𝑦 − log2 𝑝𝑝� 𝑦𝑦|� 𝑧𝑧(� 𝑦𝑦| ̃ 𝑧𝑧) 𝐷𝐷 = 𝑥𝑥 − � 𝑥𝑥 2 2 𝐿𝐿 = 𝑅𝑅 + 𝜆𝜆𝜆𝜆 Inference Model Generative Model Entropy Model 𝑧𝑧 = ℎ𝑎𝑎 𝑦𝑦; 𝜙𝜙ℎ ̃ 𝑧𝑧 = 𝑧𝑧 + 𝑛𝑛 � 𝜎𝜎 = ℎ𝑠𝑠 ̃ 𝑧𝑧; 𝜃𝜃ℎ 𝐸𝐸𝐸𝐸𝐸𝐸[� 𝑦𝑦; � 𝜎𝜎] 𝐷𝐷𝐷𝐷𝐷𝐷[ ̃ 𝑧𝑧] � 𝜎𝜎 = ℎ𝑠𝑠 ̃ 𝑧𝑧; 𝜃𝜃ℎ 𝐸𝐸𝐸𝐸𝐸𝐸[ ̃ 𝑧𝑧] 𝐷𝐷𝐷𝐷𝐷𝐷[� 𝑦𝑦; � 𝜎𝜎]