image compression ppt
TRANSCRIPT
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IMAGE COMPRESSION
Under Guidance of: Presented by:-
Jayash Sharma Sir Shivangi Saxena M.tech,Semester-II
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• The problem of reducing the amount of data required to represent a digital image.
• From a mathematical viewpoint: transforming a 2-D pixel array into a statistically uncorrelated data set.
Image Compression?
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• For data STORAGE and data TRANSMISSION• DVD• Remote Sensing• Video conference• FAX• Control of remotely piloted vehicle
• The bit rate of uncompressed digital cinema data exceeds one Gbps.
Why do We Need Compression?
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REDUNDANTDATA
INFORMATION
DATA = INFORMATION + REDUNDANT DATA
Information vs Data
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• Spatial redundancy
• Neighboring pixels are not independent but correlated
Temporal redundancy
Why Can We Compress?
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• Basic data redundancies:1. Coding redundancy2. Inter-pixel redundancy3. Psycho-visual redundancy
Fundamentals
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Let us assume, that a discrete random variable rk in the interval [0,1] represent the gray level of an image:
If the number of bits used to represent each value of rk is l(rk), then the average number of bits required to represent each pixel:
The total number bits required to code an MxN image:
1,,2,1,0)( −== Lkn
nrp kkr
avgLNM ..
Coding Redundancy
∑−
=
=1
0
)()(L
kkrkavg rprlL
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Coding Redundancy
bits
rprlLk
krkavg
7.2
)02.0(6)03.0(6)06.0(5
)08.0(4)16.0(3)21.0(2)25.0(2)19.0(2
)()(7
02
=+++
++++=
= ∑=
2
1
n
nCR =
Rd CR
11−=
Compression ratio:
Relative data redundancy:
11.17.2
3 ==RC 099.011.1
11 =−=dR
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Inter-pixel Redundancy
Spatial Redundancy
Geometric Redundancy
Inter-frame Redundancy
Here the two pictures have
Approximately the same
Histogram.
We must exploit Pixel
Dependencies.
Each pixel can be estimated
From its neighbors
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Psycho-visual RedundancyElimination of psych-visual redundant data results in a loss of quantitative information ,it is commonly referred as quantization.
Improved Gray-Scale
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Psycho-visual Redundancy
IGS Quantization
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Fideli ty Criteria
The general classes of criteria :1. Objective fidelity criteria
2. Subjective fidelity criteria
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Fideli ty CriteriaObjective f ideli ty:
Level of information loss can be expressed as a function of the original and the compressed and subsequently decompressed image.
2/1
21
0
1
0
]),(),(ˆ[1
−= ∑∑
−
=
−
=
M
x
N
yrms yxfyxf
MNe
∑∑
∑∑−
=
−
=
−
=
−
=
−= 1
0
1
0
2
1
0
1
0
2
)],(),(ˆ[
),(ˆ
M
x
N
y
M
x
N
yms
yxfyxf
yxf
SNR
Root-mean-square error
Mean-square signal-to-noise ratio
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Fideli ty Criteria
25.10
93.6
==
rm
rms
SNR
e
39.10
78.6
==
rm
rms
SNR
e
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Fideli ty CriteriaSubjective f idelity (Viewed by Human):
• By absolute rating
• By means of side-by-side comparison of and),( yxf ),(ˆ yxf
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Image Compression Model
Remove Input Redundancies
Increase the Noise Immunity
The source encoder is responsible for removing redundancy
(coding, inter-pixel, psycho-visual)
The channel encoder ensures robustness against channel noise.
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Classif ication Lossless compression
lossless compression for legal and medical documents, computer programs
exploit only code and inter-pixel redundancy Lossy compression
digital image and video where some errors or loss can be tolerated
exploit both code and inter-pixel redundancy and sycho-visual perception properties
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Error-Free Compression
Applications:
• Archive of medical or business documents
• Satellite imaging
• Digital radiography
They provide: Compression ratio of 2 to 10.
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Huffman coding
Error-Free CompressionVariable-length Coding
The most popular technique for removing coding redundancy is due
to Huffman (1952)
Huffman Coding yields the smallest number of code symbols per
source symbol
The result ing code is optimal
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Error-Free CompressionVariable-length Coding
Huffman coding (optimal code)
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Huffman coding
Error-Free CompressionVariable-length Coding
symbolbitsentropy
symbolbits
Lavg
/14.2
/2.2
)5)(04.0()5)(06.0()4)(1.0()3)(1.0()2)(3.0()1)(4.0(
==
+++++=
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Error Free Compression Technique.
Remove Inter-pixel redundancy.
Requires no priori knowledge of probability distribution of pixels.
Assigns fixed length code words to variable length sequences.
Patented Algorithm US 4,558,302.
Included in GIF and PDF file formats.
LZW Coding
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Coding Technique
A codebook or a dictionary has to be constructed
For an 8-bit monochrome image, the first 256 entries are
assigned to the gray levels 0,1,2,..,255.
As the encoder examines image pixels, gray level sequences
that are not in the dictionary are assigned to a new entry.
For instance sequence 255-255 can be assigned to entry 256,
the address following the locations reserved for gray levels 0 to
255.
LZW Coding
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LZW Coding
Example
Consider the following 4 x 4 8 bit image
39 39 126 12639 39 126 12639 39 126 12639 39 126 126
Dict ionary Location Entry
0 0
1 1
. .
255 255
256 -
511 -
Initial Dictionary
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LZW Coding
39 39 126 12639 39 126 12639 39 126 12639 39 126 126
•Is 39 in the dict ionary……..Yes
•What about 39-39………….No
•Then add 39-39 in entry 256
•And output the last recognized symbol…39
Dict ionary Location Entry
0 0
1 1
. .
255 255
256 -
511 -
39-39
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H.R. Pourreza
Error-Free CompressionBit-plane coding
Bit-plane coding is based on decomposing a multilevel image into a series of binary images and compressing each binary image .
a b c d e f
f
ed
cb
a
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Error-Free CompressionBit-plane coding
Binary Bit-planes Gray Bit-planes
11
1
0121
00
22
11
20
:
222:
−−
+
−−
−−
−−
=−≤≤⊕=
+++−
mm
lll
mm
mm
mm
ag
mlaag
ggggcodeGray
aaascalegraybitm
Bit-plane decomposit ion
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Error-Free CompressionBit-plane coding
Binary Bit-planes Gray Bit-planes
Bit-plane decomposit ion
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Error-Free CompressionBit-plane coding
• Constant area Coding
• One-dimensional run-length coding10
10
LL
HHHRL +
+=
Average values of black and white run
lengths
Relative Address Coding (RAC) is based on tracking the binary transitions.
• Two-dimensional RLC
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Error-Free CompressionLoss-less Predictive Coding
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Error-Free CompressionLoss-less Predictive Coding
= ∑
=−
m
llnln froundf
1
ˆ α
In most cases, the predict ion is formed by a l inear combination of m previous pixels. That is:
1-D Linear Predict ive coding:
−= ∑
=
m
iin iyxfroundyxf
1
),(),(ˆ α
m is the order of l inear predictor
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H.R. Pourreza
Error-Free CompressionLoss-less Predictive Coding
[ ])1,(),(ˆ −= yxfroundyxf α First-order l inear predictor
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Lossy Compression
Lossy encoding is based on the concept of compromising the accuracy of the reconstructed image in exchange for increased compression.
Lossy encoding techniques are capable of reproducing recognizable mono-chrome images from data that have been compressed by more than 100:1 and images that are virtually indistinguishable from the original at 10:1 to 50:1 .
Lossy Compression:
1. Spatial domain methods
2. Transform coding
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Lossy CompressionLossy Predictive Coding
• Predictive Coding : transmit the difference between estimate of future sample & the sample itself.
• Delta modulation• DPCM• Adaptive predictive coding• Differential frame coding
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Lossy CompressionLossy Predictive Coding
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Lossy CompressionLossy Predictive Coding – Delta Modulation (DM)
−>+
=
= −
otherwise
efore
ff
nn
nn
ζξ
α0
ˆ1
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Lossy CompressionOptimal Prediction
∑
∑
∑
=
=−
=−
≤
−=
=
=+≈+=
−=
m
ii
m
iininn
m
iinin
nnnnnn
nnn
ffEeE
ff
ffefef
ffEeE
1
2
1
2
1
22
1
}{
ˆ
ˆˆ
}]ˆ{[}{
α
α
α
Differential Pulse Code Modulation (DPCM)
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Lossy CompressionOptimal Prediction
)1,1()1,()1,1(),1(
),1(97.0
)1,(97.0),(ˆ
)1,1(5.0),1(75.0)1,(75.0),(ˆ
),1(5.0)1,(5.0),(ˆ
)1,(97.0),(ˆ
−−−−=∆−−−−=∆
−∆≤∆−
=
−−−−+−=
−+−=
−=
yxfyxfvandyxfyxfh
otherwiseyxf
vhifyxfyxf
yxfyxfyxfyxf
yxfyxfyxf
yxfyxf
Predict ion Error
Pred. #1
Pred. #2
Pred. #3
Pred. #4
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Lossy CompressionOptimal Prediction
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H.R. Pourreza
Lossy CompressionPrediction Error for dif ferent predictors
Pred. #1
Pred. #3
Pred. #2
Pred. #4
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Lossy CompressionOptimal Quantization
)()( functionoddanisqsqt =
Decision Levels
Reconstruction Levels
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Lossy CompressionOptimal Quantization
)()( functionoddanisqsqt =
Minimization of the mean-square quantization error
iiii
iii
s
s
i
ttss
Li
Li
tti
s
Lidssptsi
i
−=−=
=∞
−=+=
=
=−
−−
+
∫−
2
12,,2,1
2
00
2/,,2,1)()(
1
1
Addit ional constraint for optimum uniform quantizer:
θ=−=− −− 11 iiii sstt
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Lossy CompressionOptimal Quantization
Unit variance Laplacian probabil i ty density function
As this table constructed for a unit variance distr ibution, the reconstruction and decision levels
for the case of are obtained by mult iplying the tabulated values by the standard deviat ion 0of
the probabil i ty density function
1≠σ
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Lossy CompressionDPCM RMSE
The best of four possible quantizers is selected for each block of 16 pixels.
Scaling factors: 0.5, 1.0, 1.75 and 2.5
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Lossy CompressionDPCM result images
2-level Lloyd-Max quantizer
1.0 bits/pixel
4-level Lloyd-Max quantizer
2.0 bits/pixel
8-level Lloyd-Max quantizer
3.0 bits/pixel
2-level adaptive quantizer
1.125 bits/pixel
4-level adaptive quantizer
2.125 bits/pixel
8-leveladaptive quantizer
3.125 bits/pixel
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Lossy CompressionDPCM Predict ion Error
2-level Lloyd-Max quantizer
1.0 bits/pixel
4-level Lloyd-Max quantizer
2.0 bits/pixel
8-level Lloyd-Max quantizer
3.0 bits/pixel
2-level adaptive quantizer
1.125 bits/pixel
4-level adaptive quantizer
2.125 bits/pixel
8-leveladaptive quantizer
3.125 bits/pixel
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Lossy CompressionTransform Coding
The goal of the transformation process is to decorrelate the pixels of each sub-image, or to pack as much information as possible into the smallest number of transform coeff icients
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1,,1,0,),,,(),(),(
1,,1,0,),,,(),(),(
1
0
1
0
1
0
1
0
−==
−==
∑∑
∑∑−
=
−
=
−
=
−
=
NyxvuyxhvuTyxf
NvuvuyxgyxfvuT
N
u
N
v
N
u
N
v
Forward kernel is Separable if :
Lossy CompressionTransform Coding
),().,(),,,( 21 vyguxgvuyxg =
Forward kernel is Symmetric i f:
),().,(),,,( 1121 vyguxgvuyxggg =⇒=
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Discrete Fourier Transform (DFT):
Lossy CompressionTransform Coding
Nvyuxj
Nvyuxj
evuyxh
eN
vuyxg
/)(2
/)(2
),,,(
1),,,(
+
+−
=
=
π
π
Walsh-Hadamard Transform (WHT):
)2()1(1
),,,(),,,(
1
0
)]()()()([m
vpybupxb
NN
vuyxhvuyxg
m
iiiii
=∑
−==
−
=
+
bk(z) is the kth bit (from right to left) in the binary representation of z.
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)()()(
)()()(
)()()(
)()(
011
322
211
10
ububup
ububup
ububup
ubup
m
mm
mm
m
+=
+=+=
=
−
−−
−−
−
Lossy CompressionTransform Coding
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Discrete Cosin Transform (DCT):
Lossy CompressionTransform Coding
−=
==
+
+=
=
1,,2,12
01
)(
2
)12(cos
2
)12(cos)()(
),,,(),,,(
NuforN
uforNuwhere
N
vy
N
uxvu
vuyxhvuyxg
α
ππαα
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Lossy CompressionTransform Coding
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Lossy CompressionTransform Coding
1. Dividing the image into sub-images of size 8x8
2. Representing each sub-image using one of the transforms
3. Truncating 50% of the result ing coeff icients
4. Taking the inverse Transform of the truncated coeff icients
DFT
WHT
DCT
rmse=1.28
rmse=0.86
rmse=0.68
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H.R. Pourreza
Lossy CompressionTransform Coding• DCT Advantages:
1. Implemented in a single integrated circuit (IC)
2. Packing the most information into the fewest coeff icients
3. Minimizing the block-like appearance (blocking art ifact)
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Lossy CompressionTransform Coding
Sub-image size selection
Truncating 75% of the result ing coefficients
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Lossy CompressionTransform Coding
Truncating 75% of the result ing coefficients.
Sub-images size:
8x8
4x4
2x2
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Lossy CompressionTransform Coding
Bit allocation87.5% of the DCT coeff. Of each 8x8 subimage.
Threshold coding (8 coef)
Zonal coding
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H.R. Pourreza
Lossy CompressionTransform Coding- Bit allocation
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Lossy CompressionTransform Coding- Bit allocation• Zonal coding
1. Fixed number of bits / coeff icient
- Coeff icients are normalized by their standard deviat ions and uniformly quantized
2. Fixed number of bits is distributed among the coeff icients unequally.
- A quantizer such as an optimal Lloyed-Max is designed for each coeff. :
- DC coeff. Is modeled by Rayleigh density func.
- The remaining coeff. Are modeled by Laplcian or
Gaussian
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Lossy CompressionTransform Coding- Bit allocation• Threshold coding
1. Single global threshold
2. Different threshold for each subimage (N-Largest coding)
3. Threshold can be varied as a function of the location of each coeff.
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H
Lossy CompressionTransform Coding
Bit allocation 67:1
rmse: 6.33
34:1
rmse: 3.42
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Lossy CompressionTransform Coding
Wavelet Coding
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Lossy CompressionTransform Coding
Wavelet Coding34:1
rmse: 2.29
67:1
rmse: 2.96
No blocking
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Lossy CompressionTransform Coding
108:1
rmse: 3.72
167:1
rmse: 4.73
Wavelet Coding
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Lossy CompressionTransform Coding
Quantizer selectionEffectiveness of the quantization can be improved by:
• introducing an enlarge quantization interval around zero
• Adapting the size of the quantization interval from scale to scale
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Why Do We Need International Standards?
International standardization is conducted to achieve inter-operability . Only syntax and decoder are specified. Encoder is not standardized and its optimization is left to the
manufacturer. Standards provide state-of-the-art technology that is
developed by a group of experts in the field. Not only solve current problems, but also anticipate the future
application requirements. Most of the standards are sanction by the International
Standardization Organization (ISO) and the Consultative Committee of the International Telephone and Telegraph (CCITT)
Image Compression Standards
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CCITT Group 3 and 4
They are designed as FAX coding methods. The Group 3 applies a non-adaptive 1-D run length
coding and optionally 2-D manner. Both standards use the same non-adaptive 2-D coding
approach, similar to RAC technique. They sometime result in data expansion. Therefore, the
Joint Bilevel Imaging Group (JBIG), has adopted several other binary compression standards, JBIG1 and JBIG2.
Image Compression StandardsBinary Image Compression Standards
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What Is JPEG?
"Joint Photographic Expert Group". Voted as
international standard in 1992.
Works with color and grayscale images, e.g., satellite,
medical, ...
Lossy and lossless
Image Compression StandardsContinues Tone Sti l l Image Comp.
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First generation JPEG uses DCT+Run length Huffman entropy coding.
Second generation JPEG (JPEG2000) uses wavelet transform + bit plane coding + Arithmetic entropy coding.
Image Compression StandardsContinues Tone Sti l l Image Comp. - JPEG
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Still-image compression standard Has 3 lossless modes and 1 lossy mode
sequential baseline encoding encode in one scan input & output data precision is limited to 8 bits, while quantized
DCT values are restricted to 11 bits progressive encoding hierarchical encoding lossless encoding
Can achieve compression ratios of up-to 20 to 1 without noticeable reduction in image quality
Image Compression StandardsContinues Tone Sti l l Image Comp. - JPEG
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Work well for continuous tone images, but not good for cartoons or computer generated images.
Tend to filter out high frequency data. Can specify a quality level (Q)
with too low Q, resulting images may contain blocky, contouring and ringing structures.
5 steps of sequential baseline encoding transform image to luminance/chrominance space (YCbCr) reduce the color components (optional) partition image into 8x8 pixel blocks and perform DCT on each
block quantize resulting DCT coefficients variable length code the quantized coefficients
Image Compression StandardsContinues Tone Sti l l Image Comp. - JPEG
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Original JPEG 27:1
Image Compression StandardsJPEG Encoding
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Video compression standards:1. Video teleconferencing standards
H.261 (Px64) H.262 H.263 (10 to 30 kbit/s) H.320 (ISDN bandwidth)
2. Multimedia standards MPEG-1 (1.5 Mbit/s) MPEG-2 (2-10 Mbit/s) MPEG-4 (5 to 64 kbit/s for mobile and PSTN and uo to 4
Mbit/s for TV and film application)
Image Compression StandardsVideo Compression Standards
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THANK YOU