Digital Image Processing

Homework II Fast Discrete Cosine TransformJain-Yi Lu (呂健益 )

Visual Communications Laboratory Department of Communication Engineering

National Central University ChungLi, Taiwan

2

Implement Fast Discrete Cosine Transform Implement Fast Discrete Cosine Inverse Transform

The Goal of This Homework

3

Discrete Cosine Transform (DCT)

1

0 2)12(cos)()()(

N

u NuxuCuaxf

00

2

1

)(

uforufor

N

Nua

N : 1-D Sequence Length : 0,1,2…N-1 (u=0 -> DC)f(x) : Spatial Domain IntensityC(u) : Frequency Domain intensity

1

0 2)12(cos)()()(

N

x NuxxfuauC

u

4

Discrete Cosine Transform

DCT

Spatial Domain

Frequency Domain

Spatial Domain

IDCT

5

Discrete Cosine Transform (8 ×8 Blocks)

W. H. Chen, C. H. Smith and S. C. Fralick, “A Fast Computational Algorithm for the Discrete Cosine Transform,” IEEE Trans. Communications, Vol. COM-25, No. 9, September 1977

Fast Discrete Cosine Transform(FDCT)

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Fast Discrete Cosine Transform 8×8 Block

0i1i2i3i4i5i6i7i

j k F

Stage1 Stage2 Stage3 Stage4

fInitial Signal

Result

8

Fast Discrete Cosine Transform 8×8 Block Stage1

f i707

616

525

434

433

522

611

700

ffiffiffiffiffiffiffiffi

Stage1 Stage2 Stage3 Stage4

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Fast Discrete Cosine Transform 8×8 Block Stage2

Stage1 Stage2 Stage3 Stage4

i j77

566

565

44

303

212

211

300

)4

cos()(

)4

cos()(

ij

iij

iij

ijiijiijiijiij

Fast Discrete Cosine Transform 8×8 Block Stage3

Stage1 Stage2 Stage3 Stage4

j k

677

676

545

544

233

322

101

100

)83sin()

8sin(

)83sin()

8sin(

)4

cos()(

)4

cos()(

jjkjjkjjkjjk

jjk

jjk

jjk

jjk

Fast Discrete Cosine Transform 8×8 Block Stage4

Stage1 Stage2 Stage3 Stage4

k F

2/))167sin()

167cos((

2/

2/))165sin()

165cos((

2/

2/))163sin()

163cos((

2/

2/))16

cos()16

sin((

2/

477

36

655

14

563

22

741

00

kkF

kF

kkF

kF

kkF

kF

kkF

kF

12

Result

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We will provide grayscale Lena 512×512 pixels image.

Your program (Image size, bugs, some questions) (2 points)

The accuracy (2 points)

Computation perplexity (Bonus) Other comment (Bonus)

Note and Report

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Appendix-A