Image Compression. Review

Transcription

1 Image Compression Lossy Compression and JPEG Review Image Compression Image data too big in RAW pixel format Many redundancies We would like to reduce redundancies Three basic types Coding Redundancy Interpixel Redundancy Psycho-visual Redundancy 1

2 Review Compression for Coding Redundancies Variable Length Coding Huffman Encoding Use smaller codes for more probable symbols Compression for Interpixel Redundancies Introduce Mapping Often from pixel space to non-visual format Run-Length-Encoding (RLE) Predictive Coding Use previous pixel to predict the next pixel Encode the error Review f(x,y) Source encoder Channel encoder Channel Channel decoder Source decoder f (x,y) f(x,y) Mapper Quantizer Symbol Encoder Symbol decoder Inverse Mapper f (x,y) Loss 2

3 Objective Fidelity Criteria Commonly used fidelity criteria is mean-square signal-to-noise ratio defined as: SNR f(x,y) is original f (x,y) is compress/decompress x= 0 y= 0 ms = M 1 N 1 x= 0 y= 0 M 1 N 1 f '( x, y) [ f '( x, y) f ( x, y)] 2 2 Subjective Fidelity Criteria Human-based criteria Usually side-by-side comparison 3

4 Lossy Compression Compromise accuracy That is, we will allow for error Distortion In exchange for increased compression If the resulting distortion can be tolerated Then we can gain substantial compression Exploit psycho-visual Redundancy Our eye is less sensitive to high-frequencies Can can throw away a lot of detail and image looks the same Lossy Compression 4

5 Lossy vs. Lossless Lossy Schemes Image is still recognizable 30:1 Compression ratio Image is virtually indistinguishable from original 20:1-10:1 Compression ratio Lossless Rarely get more than 3:1 reduction Lossy Schemes Principal difference from lossless Introduction of quantization Quantization maps values to limited range The more the quantization More Compression (and more distortion) 5

6 Lossy Predictive Coding Predictive Coding Scheme Based on lossless predictive coding scheme Recall that lossless prediction was: e = f -f f is pixel f is the predicted pixel e is the error Loss Predictive Coding Error will be quantized to a limited range e = f f e = e/q f = f +e (resulting pixel with error) f is a pixel f is the predicted pixel e is the quantized error f is the resulting pixel note, we will need to use f in the predictor 6

7 Lossy Predictive Coding Model f(x,y) + e n Quantizer e n e (x,y) f Predictor f = f +e + Error is quantized Predictor uses lossy pixels (f ) to predict f Lossy/Lossless Comparison f(x,y) + e(x,y) Lossless Predictor Nearest integer f (x,y) f(x,y) + e n Quantizer e n e (x,y) f Predictor f = f +e + Lossy 7

8 Delta Modulation Simple and well-known technique Predictor = f n = αf n-1 e n = +γ e n > 0 -γ otherwise where γ is a constant γ = 6.5 Example 1-D scanline 8

9 2D Image Original γ = 6.5 α = 1.0 roughly 8:1 Compression Transform Coding Predictive coding operates directly on pixels Transform coding Uses a reversible, linear transform (FT for example) to map the pixels to new coefficients quantizes the coefficients Compresses these further (RLE, Variable Length Coding ) 9

10 Typical Transform Coding Scheme f(x,y) Construct nxn subimages Forward Transform Quantizer Other Compressed Image f (x,y) Merge nxn subimages Inverse Transform De-quantizer Other Compressed Image Idea behind Transform Coding Transform pixels to a new space The new space provides a more compact representation of the information Signal Power Packing Less susceptible to quantization 10

11 Remember this example 8 Original 2D Fourier Coefficients F(u,v) Transform Coding Transform Selection Discrete Fourier Transform One solution Discrete Cosine Transform Like FT But, no imaginary component DCT shown to have better power packing abilities over DFT 11

12 12 Discrete Cosine Transform (Forward DCT) 2 1 ) ( 2 1) (2 cos 2 1) (2 )cos, ( ) ( ) ( ), ( N N i N v y N u x y x f v u v u C N x N y = + + = = = α π π α α for i=0 for i=1,..., N-1 Discrete Cosine Transform (Inverse DCT) 2 1 ) ( 2 1) (2 cos 2 1) (2 )cos, ( ) ( ) ( ), ( N N i N v y N u x v u C v u y x f N u N v = + + = = = α π π α α for i=0 for i=1,..., N-1

13 DCT vs. FFT Why DCT and not FFT? Original Pixel Values +I(u) DCT behaves better under quantization... (and no complex math) Subimage size (n=?) f(x,y) Construct nxn subimages Forward Transform Quantizer Other Power of 2 Allows fast nlogn algorithms to be applied As the size n increases Level of compression achievable increases Level of computational complexity increases Empirical testing showed n=8 or n=16 gives the best overall performance (testing performed back in the early 90s) (today s compute power can probably handle larger n) 13

14 2D 8x8 DCT Basis 2D DCT using 8x8 block Transformed block is a linear combination of these basis 1 coefficient for each basis Example of compression with DCT matlab dctdemo divide image into 8x8 blocks quantization in this example means dropping coefficients coefficients are dropped based on their magnitude C(u,v) 14

15 JPEG Compression Story of JPEG Joint Photographic Expert Group International Organization for Standards (ISO) 1988: ISO got together a group of experts to develop a good image compression scheme Geared towards photographs of natural imagery Color and monochrome Easy to use (spin-dial quality control) Through empirical testing, the following scheme proved to be the best Standardized in August 1990 JPEG Encoding Scheme f(x,y) (normalize between 128 to 127) FDCT (Forward DCT) on each block Quantize DCT coefficients via Quantization Table C (u,v) = round(c(u,v)/t(u,v)) f(x,y) Divided into 8x8 blocks Differential coding DC component 0 T(u,v) JPEG bitstream Huffman Encode RLE AC Vector Zig-zag Order Coefficients 15

16 JPEG Decoding Scheme f(x,y) IDCT (Inverse DCT) on each block De-quantize DCT coefficients via Quantization Table C(u,v) = C (u,v)*t(u,v) f(x,y) Divided into 8x8 blocks Differential decoding DC component 0 T(u,v) JPEG bitstream Huffman decode RLE (decode) AC Vector Zig-zag order Coefficients Example Quantization Table T(u,v) = DC (low) AC (higher) We are more sensitive to low-frequencies Quantize the high-frequencies more Usage: C(u,v) is DCT of f(x,y) 8x8 block C (u,v) = C(u,v)./ T(u,v) Element-wise division Note: non-uniform quantization Example via matlab 16

17 Example x8 block of pixels DCT C (u,v) = C(u,v)./ T(u,v) Example Difference Reconstructed 8x8 pixels +128 C (u,v) = C(u,v).* T(u,v) IDCT 17

18 JPEG Quality JPEG uses the term quality for its compression Different quality factors correspond to different quantization tables Lower Quality corresponds to larger values in the quantization table Quality 100% is quantization table of T(u,v) s 1 a little info lost from rounding, but usually less than 1 pixel Example Using XV JPEG Quantization is analogous to blurring You also get a blocking effect from the 8x8 blocks JPEG isn t well-suited for graphics and text images 18

19 JPEG Quantization Tables Code for generating the Quantization tables used by JPEG Color Image Compression via JPEG Luminance C o l o r Note: Quantization of color info 2:1:1 y y y y q i Y = I Q R G B 19

20 JPEG Several Implementations of JPEG JFIF (JPEG File Interchange Format) is what we typically think of as JPEG Some viewers/decoders dither the reconstructed blocks to try to remove blocking effects JPEG is also the bases for MPEG and M- JPEG M-JPEG (Motion-JPEG) is just a sequence of JPEGS MPEG a little more complicated Compressor/Decompressor CODEC Short for Compressor + Decompressor Engines for compression Different CODECs might give slightly different results Some approximate FDCT/IDCT with integer math and/or lookup tables 20

21 Graphics Interchange Format (GIF) short slide about GIF another popular compression scheme GIF is lossless, uses Lempel-Ziv-Welch scheme, a variation on Huffman encoding However, some implementations of GIF only allow 255 colors It quantizes the color In this sense it is lossy Image Compression Summary Idea Reduce Redundancy Maintain information Lossless Compression Huffman Encoding RLE (Mapping) Predictive Encoding Lossy Compression Quantization Step Transform Coding 21

22 Image Compression Summary Loss is due to quantization Gives much higher compression Especially useful with transform coding schemes DCT is one of the most commonly used transforms JPEG puts it all together (DCT, RLE, Huffman.. ) There are criterions to compare CODECs Objective Signal to Noise Ratio Subjective Human-based rating system Active Research Areas Mainly focused on video Compression constant bit-rate compression for networks Layered Compression Quality of Service (QoS) Very low-bandwidth compression teleconferencing over networking 22

23 Active Research Areas Content-based compression Watermarking Add some information into the image so you can tell who is the author Make it invariant or resistant to filtering Encode messages in imagery Steganography \Steg`a*nog"ra*phy\ Active Research Areas Compressed-Domain Processing Imagine a database of 100,000 images All are compressed with JPEG to save space You want to compare each image with a single image (perhaps diff the two images) 23

24 Pixel-Domain Processing You would have to decompress 100,000 image Compare the images using the pixels Imagine if you made a change (added the value of 10 to each image) Decompress -> process -> compress Compressed-Domain Processing Process the compressed-data For example DCT is a linear transform A B ~ IDCT( DCT_A DCT_B ) You can perform this operation directly on the DCT coefficients -(matlab example) 24

25 Semi-Compressed Processing IDCT (Inverse DCT) on each block De-quantize DCT coefficients via Quantization Table C(u,v) = C (u,v)*t(u,v) T(u,v) Differential decoding DC component 0 Huffman decode RLE (decode) AC Vector Decompress to RLE Zig-zag order Coefficients Evolving Compression Schemes Something to consider Compression reduces the image size However there is overhead Compression + decompression processing time Bandwidth vs. Computer Power It also makes the data harder to process Future: Semantic preserving compression Key features of the original image are easy to obtain in the compressed format Allow for compressed-domain processing 25