ΝΤUA Τεχνολογία Πολυμέσων

Transcript

1 ΝΤUA Τεχνολογία Πολυμέσων

2 Rate Distortion Theory D may be the MSE or some human perceived measure of distortion

3 Types of Lossy Compression VBR Variable Bit Rate CBR Constant Bit Rate we Discuss Later

4 Fig. 16-7, p. 356

5

6 Color components

7 Color image Three basic colors Primitives e.g., RGB, YCrCb,.. Video series of frames per second Europe PAL system 25fps USA NTSC system 30fps No improvement in video quality if we add more frames per second

8 Contents 2. Lesson 3: Transform Coding

9 Κωδικοποίηση μετασχηματισμού (transform coding) Στη κωδικοποίηση μετασχηματισμού, το σήμα υφίσταται ένα μαθηματικό μετασχηματισμό από το αρχικό πεδίο του χρόνου ή του χώρου σε ένα αφηρημένο πεδίο το οποίο είναι πιο κατάλληλο για συμπίεση. x(t) --> g(λ) Αυτή η διαδικασία είναι αντιστρεπτή, δηλαδή υπάρχει ο αντίστροφος μετασχηματισμός που θα επαναφέρει το σήμα στην αρχική του μορφή.

10 Πχ Μετασχηματισμός Fourier Έστω σήμα x(t), τότε ο μετασχηματισμός Fourier X(f) ορίζεται: j2 ft X ( f ) x( t) e dt Ο αντίστροφος μετασχηματισμός Fourier ορίζεται: 1 2 F X ( f ) x( t) X ( f ) e j ft df

11 Ασχετο: Συστηματα, Συνελιξη, κλπ

12 Μετασχηματισμός Fourier (2) - Εδω ο μετασχηματισμός μας νοιάζει για άλλον λόγο: θα μεταδώσουμε τους Fourier συντελεστες αντι για το σημα (αγνοώντας τους μικρούς). - Στην πραγματικότητα θα χρησιμοποιήσουμε έναν άλλον μετασχηματισμό, τον discrete cosine transform (DCT) και όχι τον Fourier, γιατί ο DCT δουλεύει καλύτερα (σε εικόνες που περιέχουν ευθείες γραμμές κλπ) Πεδίο χρόνου ή χώρου Πεδίο συχνοτήτων Τ Οι συντελεστές αυτοί αγνοούνται t Οι σημαντικότεροι συντελεστές για πολλούς τύπους Πληροφορίας συγκεντρώνονται στις χαμηλές συχνότητες f

13

14 ΠΑΡΑΔΕΙΓΜΑΤΑ (1) Θεωρούμε τη ΘΕΜΕΛΙΩΔΗ ΣΥΧΝΟΤΗΤΑ : α 1 =1 και α 3 =1/3, α 5 =1/5 και α 7 =1/7 Το παραπάνω προσεγγίζει το τετραγωνικό παλμό 16

15 (fundamental + third + fifth) + seventh harmonic + =

16 (fundamental + third + fifth + seventh) + ninth harmonic + =

17 (fundamental + third + fifth + seventh + ninth) + eleventh harmonic + =

18 (fundamental + third + fifth + seventh + ninth + eleventh) + thirteenth harmonic + =

19 Joint Photographic Experts Group (JPEG) compression algorithm The JPEG compression algorithm is at its best on photographs and paintings of realistic scenes with smooth variations of tone and color. JPEG may not be as well suited for line drawings and other textual or iconic graphics, where there are sharp contrasts between adjacent pixels (such images may be better saved in a lossless graphics format such as TIFF, GIF, PNG, or a raw image format)

20 The Discrete Cosine Transform (DCT) The discrete cosine transform (DCT) helps separate the image into parts (or spectral sub-bands) of differing importance (with respect to the image's visual quality). The DCT is similar to the discrete Fourier transform: it transforms a signal or image from the spatial domain to the frequency domain The general equation for a 1D (N data items) DCT is defined by the following equation: where The general equation for a 2D (N by M image) DCT is defined as: Φυσικά υπάρχει και ο αντίστοιχος μετασχηματισμός για να πάρουμε πίσω το σήμα

21 DCT on real image block

22 Properties of Transform Coding The basic operation of the DCT is as follows: The input image is N by M; f(i,j) is the intensity of the pixel in row i and column j; F(u,v) is the DCT coefficient in row u and column v of the DCT matrix. For most images, much of the signal energy lies at low frequencies; these appear in the upper left corner of the DCT. Compression is achieved since the lower right values represent higher frequencies, and are often small enough to be neglected with little visible distortion. The DCT input is an 8 by 8 array of integers. This array contains each pixel's gray scale level; 8 bit pixels have levels from 0 to 255. The output array of DCT contains a set of coefficients (between and 2048). It is computationally easier to regard the DCT as a set of basis functions which given a known input array size (8 x 8) can be precomputed and stored. This involves simply computing values for a convolution mask (8 x8 window) that gets applied. The values are simply calculated from the DCT formula.

23 DCT Discrete Cosine Transform The 64 (8 x 8) DCT basis functions

24 Transform coding

25 Computing the 2D DCT Problem can be reduced to a series of 1D DCTs apply 1D DCT (Vertically) to Columns apply 1D DCT (Horizontally) to resultant Vertical DCT above. or alternatively Horizontal to Vertical.

26 Fundamentals of JPEG Encoder DCT Quantizer Entropy coder Compressed image data IDCT Dequantizer Entropy decoder Decoder

27 Compression in JPEG="Joint Photographic Expert Group" JPEG works on 8 8 blocks Extract 8 8 block of pixels Convert to DCT domain Quantize each coefficient Different stepsize for each coefficient Based on sensitivity of human visual system Order coefficients in zig-zag order Entropy code the quantized values

28 Zig-Zag Scanning Purpose of the Zig-zag Scan: to group low frequency coefficients in top of vector. Maps 8 x 8 to a 1 x 64 vector

29 Zig-Zag Scanning and DCT coefficiant variation

30 Approximation by DCT basis

31 Transform Coding

32 Quantization of DCT coefficients

33 Default quantization table in JPEG A common quantization table (for luminance component) is

34 Example: quantized indices

35 Example: quantized coefficients

36 Example: reconstructed image

37 Zig-zag ordering Fundamentals of JPEG

38 Entropy coding Fundamentals of JPEG Run length encoding followed by Huffman Arithmetic DC term treated separately Differential Pulse Code Modulation (DPCM) 2-step process 1. Convert zig-zag sequence to a symbol sequence 2. Convert symbols to a data stream

39 Modes Sequential Progressive Fundamentals of JPEG Spectral selection Send lower frequency coefficients first Successive approximation Send lower precision first, and subsequently refine Lossless Hierarchical Send low resolution image first

40 Transform coding Encoder Image block Decoder T Transform Coefficients Q Zigzag Scan (2D->1D) Entropy coding Bitstream Reconstructed Image block T -1 Reconstructed Transform Coefficients Q -1 Entropy coding Inverse Zigzag Scan (1D->2D) Bitstream

41 Why divide to blocks? Image->Blocks Block-Based Coding

42 Example of JPEG Coding(Encoder) Transform coding(dct) Quantization Zigzag Scan Entropy Coding (bit stream) -415/16 = -26 2D->1D Number->binary EOB EOB

43 Example of JPEG Coding(decoder) Inverse Transform coding(dct) Inverse Quantization Inverse Zigzag Scan Inverse Entropy Coding (bit stream) 1D->2D Binary->number EOB EOB

44 Example of JPEG Coding(Encoder) DCT

45 Example of JPEG Coding(Encoder) -415/16 =

46 Example of JPEG Coding (Encoder)

47 General Transform coding Encoder Image block T Transform Coefficients Q Zigzag Scan (2D->1D) Entropy coding Bitstream Decoder Reconstructed Image block T -1 Reconstructed Transform Coefficients Q -1 Entropy coding Inverse Zigzag Scan (1D->2D) Bitstream

48 Example of Zig Zag Scanning Transform coding(dct) Quantization Zigzag Scan Entropy Coding (bit stream) 2D->1D EOB

49 Effect of QP Nearly all software implementations of JPEG permit user control over the compression-ratio (as well as other optional parameters), allowing the user to trade off picture-quality for smaller file size.

50 Summary