MS works in computer science in the Open University.

Efficient realization for geometric transformation of digital images in

run length encoding.

By Shlomo Pongratz.

Instructor: Dr Jack Weinstein.

Seven by six pixels digital image of the letter A.

0 0 0 0 0 0 6(0)0 0 1 0 0 0 2(0)1(1)3(0)0 1 0 1 0 0 1(0)1(1)1(0)1(1)2(0)0 1 0 1 0 0 1(0)1(1)1(0)1(1)2(0)0 1 1 1 0 0 1(0)3(1)2(0)0 1 0 1 0 0 1(0)1(1)1(0)1(1)2(0)0 0 0 0 0 0 6(0)

Image serialization.

• height=6,width=7,6(0)2(0)1(1)3(0)1(0)1(1)1(0)2(0)1(0)1(1)1(0)2(0)1(0)3(1)2(0)1(0)1(1)1(0)2(0)6(0).

Create directory is linearoffset = 0;for (line = 0; line < hight; line++){

{length = 0;while (length < width){length += run.getLength();run.next();offset++;}output(offset);}

}

The Geometric Operations OP Procedure

No op Copy source to destination row by row run by run.

Column flip Read input columns in reverse order.

Row flip Read input rows in reverse order.

Rotate 180° Do row flip and column flip.

Diagonal flip Swap rows with columns.

Rotate 90° (CCW) Do column flip and then diagonal flip.

Rotate 270° (CW) Do row flip and then diagonal flip.

Opposite diagonal flip Do 180° rotation and diagonal flip.

Rotate by 0° < α ≤ 45°  

Rotate by 0° > α ≥ 45°  

Rotate by any angle  

The No Op operation.

The Columns flip operation. • for (line = 0; line < height; i++)• {• start = directory[line];• ends = directory[line + 1];• // go backwards• for (index = ends – 1; index >= start; index--)• {• output_run(image[index]);• }• }

The Rows flip operation. • for (line = 0; line < height; line++)• {• start = directory[height - line - 1];• ends = directory[height - line];• // go forwards• for (index = start; index < end; index++)• {• output_run(image[index]);• }• }

The Rotation by 180° operation.

• for (line = 0; line < height; line++)• {• start = directory[height - line - 1];• ends = directory[height - line];• // go backwards• for (index = ends – 1; index >= start; index--)• {• output_run(image[index]);• }• }

The Diagonal Flip operation.

• Merging process• For each line sequence runi={leni,colouri}

• Create ordered list {Σij=0 lenj,colouri}

• Use regular merging algorithm• Compare colours while merging.

Same colour runs

Different colour runs

States in merge

  1 0 3 2 1 1 1 0 0

0 1 0 0 0 0 0 0 0 0

1 0 0 0 0 1 1 1 0 0

2 0 0 1 1 1 1 1 0 0

3 0 0 0 1 1 1 0 0 0

4 0 0 0 0 1 1 0 0 0

5 0 0 0 0 0 0 0 0 0

6 0 0 0 0 0 0 0 0 0

  0 1 2 3 4 5 6 7 8

  1 0 3 4 1 1 1 0 0

0 1 0 0 0 0 0 0 0 0

1 0 0 0 0 1 1 1 0 0

2 0 0 1 1 1 1 1 0 0

3 0 0 0 1 1 1 0 0 0

4 0 0 0 0 1 1 0 0 0

5 0 0 0 0 0 0 0 0 0

6 0 0 0 0 0 0 0 0 0

  0 1 2 3 4 5 6 7 8

Source to destination problem• for (I= 0; I < 10; I++)• {

– for (J = 0; J< 10; J++)– {

• map pixel at [I,J] to• [round(I * cos(30) – J*sin(30)) , round(I*sin(30) + J * cos(30)) }

• }

(0:0) (0:1) (0:2) (0:3) (-1:3) (-1:4) (-2:5) (-2:6) (-3:7) (-3:8)

(1:1) (0:1) (0:2) (0:3) (0:4) (-1:5) (-1:6) (-2:7) (-2:7) (-3:8)

(2:1) (1:2) (1:3) (0:4) (0:4) (0:5) (0:6) (-1:7) (-1:8) (-2:9)

(3:1) (2:2) (2:3) (1:4) (1:5) (0:6) (0:7) (0:8) (0:8) (-1:9)

(3:2) (3:3) (2:4) (2:5) (1:5) (1:6) (0:7) (0:8) (0:9) (0:10)

(4:2) (4:3) (3:4) (3:5) (2:6) (2:7) (1:8) (1:9) (0:9) (0:10)

(5:3) (5:4) (4:5) (4:6) (3:6) (3:7) (2:8) (2:9) (1:10) (1:11)

(6:3) (6:4) (5:5) (5:6) (4:7) (4:8) (3:9) (3:10) (2:10) (2:11)

(7:4) (6:5) (6:6) (5:7) (5:7) (4:8) (4:9) (3:10) (3:11) (2:12)

(8:4) (7:5) (7:6) (6:7) (6:8) (5:9) (5:10) (4:11) (4:11) (3:12)

Tanaka et-al algorithm

102

tan1*1sin01

*10

2tan1

cossinsincos a

a

a

aaaa

Eliminate Vertical sheer.

0110

*10

sin1*

0110

1sin01 a

a

Combining it all together

What’s wrong

Not linear in the worst case

One pass algorithm                                        1 1 1 1 1

                              0 0 0 1 1 0 0 0 0 0

                    1 1 1 1 1 1 1 1 0 0          

          0 0 1 1 1 1 1 1 1 1                    

0 0 0 0 0 0 0 1 1 1                              

0 0 0 0 0                                        

                                                 

1st case

2nd case

3rd case

4th case

5th case

6th case

7th case

1st case

2nd case

3rd case

4th case

5th case

Cohen algorithm

What’s wrong

cossinsincos

tan1tantan1

10tan1

*1tan01

2

Fixing Cohen's algorithm

cossinsincos

10tan1

sec00sin

1tan01

Note – non reversible