Image Tiling Using Columnar Transposition

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Image Tiling Using Columnar Transposition
Rihartanto Rihartanto
Department of Information Technology
State Polytechnic of Samarinda
Samarinda, Indonesia
rihartanto@polnes.ac.id
Supriadi Supriadi
Department of Information Technology
State Polytechnic of Samarinda
Samarinda, Indonesia
supriadi.polnes@gmail.com
Didi Susilo Budi Utomo
Department of Information Technology
State Polytechnic of Samarinda
Samarinda, Indonesia
dsbudiutomo10@gmail.com
Abstract— Columnar transposition is a technique
commonly used in text encryption to produce cipher text which
is a permutation of its original text. In this study, the columnar
transposition and pixels reposition is applied to the image to
produce a tile structure where each tile segment is a miniature
of the original image. The number of tiles produced is n^2, n is
the key value which is the number of columns of the
transposition matrix. The size of each tile segment is n^(-2) of
its original image size. The similarity of each tile was measure
using SSIM. From the test result shows that SSIM values vary
depending on the given key value.
Keywords—Columnar transposition, pixel reposition, tile
structure, image scale down, SSIM
I. INTRODUCTION
Tiling is a segmentation effort of a large image. Tiling
an image segments it into a number of smaller rectangular
areas called tiles. The implementations that are popular
today include google maps and other map-based navigation
applications. This method is generally used to facilitate
computation[1][2], fast and efficient access[3], data
transmission[4] and information hiding [5]. For the purpose
of hiding data, it often implements steganography and
encryption.
An image is a digital object in the form of pictures,
photographs or other that can be seen visually. An image is a
collection of pixels that have different intensity values[6].
Each image consists of n×m pixels, where n denotes the
number of rows and m denotes the number of columns.
Pixels are small blocks that show gray intensity at a
particular position of an image. Generally pixels are
represented as integer values in the range 0 (black) to 255
(white). For example, an enlarged 15×15 pixels image in Fig.
1 shows gray intensity along with its intensity values.
Unlike the grayscale image, the colored or often called
RGB image is an image where each pixel has three color
components[7]. The three components are red (R), green (G)
and blue (B), so the RGB image is an array of n×m×3, each
of which has its own intensity value. The combination of the
intensity of each of these color components produces the
color of each pixel of the RGB image[8]. Since basically, the
image is a collection of numbers, then the image can be
manipulated for various purposes using various methods.
Differ from the research conducted by [2] and [3] where
tiling is carried out to produce uniform sized segments which
are part of the original image. In this research, it is conducted
to produce a number of tiles which are miniatures of the
original image. The tile generation process is carried out by
implementing columnar transposition on the image. Each tile
segment will show similar images in smaller dimension.
Fig. 1. An enlarged 15×15 image and its pixel intensity values
Transposition is one of the techniques used to perform
encryption. Generally, transpositions are used in text
encryption. The most widely used transposition techniques
include rail fence[9], route, columnar, double[10], [11] and
Myszkowsky transposition. In contrast to the substitution
technique, the transposition simply changes the position of
each letter in the text to produce a new arrangement that is
different from the original[12]. So it can be said that the
cipher of the transposed text is a permutation of the plain
text[9].
II. METHODS AND MATERIAL
The columnar transposition begins by specifying the key
which is the number of columns of the matrix to be
transposed. The letters on the plaintext are inserted into each
cell horizontally starting from the first row of the first
0 3 10 17 24 32 42 48 53 68 74 87 98 104 116
3 8 15 23 30 38 48 55 62 77 84 96 108 114 125
10 15 23 31 38 48 57 64 75 90 96 109 120 127 138
17 23 31 39 47 57 67 74 86 101 107 119 131 137 149
24 30 38 47 56 67 77 85 94 109 115 128 139 146 157
32 38 48 57 67 78 89 96 105 120 126 138 150 156 167
42 48 57 67 77 89 100 108 117 132 138 151 163 169 180
48 55 64 74 85 96 108 116 127 142 148 161 172 178 190
53 62 75 86 94 105 117 127 140 148 161 167 178 190 194
68 78 90 101 109 120 132 142 148 157 169 175 186 198 202
74 84 96 107 115 126 138 148 161 169 182 188 199 211 215
87 96 109 119 128 138 151 161 167 175 188 194 204 216 221
98 108 120 131 139 150 163 172 178 186 199 204 215 227 232
104 114 127 137 146 156 169 178 190 198 211 216 227 239 244
116 125 138 149 157 168 180 190 194 202 215 221 232 244 248
Research grant of State Polytechnic of Samarinda, contract number
2014/PL7/LT/2018
118
2018 International Conference on Applied Information Technology and Innovation (ICAITI)

column. When it reaches the last column, the process is
continued in the first column of the next row. And so on until
all the letters successfully inserted into the matrix. The
cipher text is created by transposing the matrix and then
reading it horizontally from the first row [13]. If there is an
empty cell, then the cell is ignored.
To further facilitate the understanding, suppose the
columnar transposition is done on plaintext "POLITEKNIK
NEGERI SAMARINDA" with 6 as the key which means to
form a matrix consisting of 6 columns. Removing all spaces
in the original text and taking only the letters, the
transposition process can be illustrated as in Fig. 2. It should
be noted that in the process of forming the cipher text, if
there are empty cells in the matrix then these cells will be
ignored.
Implementation of columnar transpositions on images
requires a number of modifications to ensure that the data
remains intact. In the image, this transposition process also
involves the process of image reshaping before and after
transposition. In addition, there is also a process of pixel
reposition because the value of 0 in a pixel cannot be ignored
since 0 represents black. The implementation of the
columnar transposition on the image follows the flowchart
shown in Fig. 3.
Fig. 2. Illustration of the columnar transposition
Start
Read Image
(img)
Input number
of column
(clm)
Compute number of
row (lns)
Reshape img to
lns × clm
Tranpose img
(lns × clm)
> (m ×n)
Pixels reposition
Reshape img to its
original size
End
false
Get image size (n,m)
Fig. 3. Implementation of columnar transposition on image
Fig. 4. Reshaped (a) and transposed (b) from the 15x15 image
Fig. 5. Pixels reposition
Suppose that the transposition will be performed on the
15×15 image in Fig. 1 with the key used as the column is 4,
stored as clm. From the image, it is known that the number of
columns is 15 stored as n and the number of rows is stored as
m. The number of rows in the new matrix is the roundup of
(n×m)/clm, so that 57 is obtained as lns. There are 3 blank
cells in the new matrix. The number of empty cells is
obtained from the reduction (lns×clm) with (n×m).
Furthermore, the image (img) was reshape from its
original size of n×m into lns×clm. The transposition is
conducted after the shape size changes have been done, i.e.
on the lns×clm image. The transposition result is a new
image of clm×lns. This process is shown in Fig. 4 where the
last three cells that are zero (red) are the empty cells.
Before being returned to its original shape, the pixel
value in the transpose matrix needs to be repositioned to
ensure that the empty cell is always at the end of the matrix.
This repositioning process is illustrated in Fig. 5. The value
of 10 in the third row of the first column is moved to the
second row of the last column. The value in the second
column and so on in the third row is moved forward to
occupy the column in front of it. The same way is done on
the next row and column.
After the pixel reposition has been completed, the matrix
is converted back to its original shape to obtain the tile
structure of the image. The similarity of each tile segment in
the resulting image is measured using Structure Similarity
Index Measurement (SSIM). The SSIM value will between -
1 and 1. The closer the value to 1 (one) the higher the
similarity between the two tested images.
0 3 10 17
24 32 42 48
53 68 74 87
98 104 116 3
815 23 30
… … … …
… … … …
239 244 116 125
138 149 157 168 024 53 98 8…239 138 180 215 248
180 190 194 202 332 68 104 15 …244 149 190 221 0
215 221 232 244 10 42 74 116 23 …116 157 194 232 0
248 000 17 48 87 330 …125 168 202 244 0
(b) 4 x 57
(a) 57 x 4
024 53 98 8…239 138 180 215 248
332 68 104 15 …244 149 190 221 0
10 42 74 116 23 …116 157 194 232 0
17 48 87 330 …125 168 202 244 0
024 53 98 8…239 138 180 215 248
332 68 104 15 …244 149 190 221 10
42 74 116 23 55 …157 194 232 17 48
87 330 62 108 …202 244 0 0 0
(a) Before
(b) After
119

a) b) c)
d) e) f)
Fig. 6. Sample Images (scaled down for visibility: a). cactus.jpg (297 x 297); b). rainbow.jpg (766 x 1080); c). temple.jpg (1440 x 900)); d;
lena.jpg (225 x 225); e). baloon.jpg (339 x 458); f). sunshine.jpg (1280 x 960)
Fig. 7. The original and the tile image
The SSIM index is a measurement or prediction of image
quality. It is based on an initial uncompressed or distortion-
free image as reference. SSIM is designed to improve on
traditional methods such as peak signal-to-noise
ratio (PSNR) and mean squared error (MSE)[14]. SSIM is a
perception-based model that considers image degradation
as perceived change in structural information, while also
incorporating important perceptual phenomena, including
both luminance masking and contrast masking terms.
Structural information is the idea that the pixels have strong
inter-dependencies especially when they are spatially close.
These dependencies carry important information about the
structure of the objects in the visual scene. Luminance
masking is a phenomenon whereby image distortions (in this
context) tend to be less visible in bright regions, while
contrast masking is a phenomenon whereby distortions
255 255 255 255 255 255 255 255 255 250 255 255 255 252 253 255 254 255 255 0255 255 255 255 255 255 255 253 254 255 255 255 255 255 250 255 255 255 251 0
255 255 255 255 255 255 255 255 250 255 255 241 255 255 255 236 251 253 0 1 255 255 255 255 255 254 255 254 255 6255 255 255 255 255 255 254 245 0 5
255 255 255 255 255 255 255 255 255 255 254 255 255 246 254 255 255 0 6 6 255 255 255 255 255 255 249 255 0 0 255 255 255 255 252 246 245 1 0 1
255 255 255 255 255 255 255 255 255 241 255 247 254 255 245 237 0 0 5 0 255 255 255 255 253 254 255 0 4 2 255 255 255 255 255 255 15 0 6 2
255 255 255 255 255 255 255 255 255 255 255 254 249 246 255 15 0 1 0 1 255 255 255 253 247 233 0 0 1 1 250 255 255 255 255 5 7 4 1 1
255 255 255 255 255 255 255 255 252 255 246 255 245 252 1 0 0 0 1 0 255 254 255 254 233 7 0 0 1 1 255 255 254 245 22 019 4 1 1
255 255 255 255 255 255 255 255 253 255 254 245 255 1 0 0 4 0 2 4 255 255 249 255 0 0 0 6 1 1 252 246 245 1 3 7 2 8 1 1
255 255 255 255 255 255 255 255 255 236 255 237 15 0 0 0 6 0 2 1 253 254 255 0 0 0 6 0 1 1 255 255 15 0 0 0 0 10 1 1
255 250 255 255 255 252 253 255 247 255 233 22 0 3 0 0 1 1 1 1 254 255 0 4 1 1 1 1 1 1 255 0 1 0 1 1 1 1 1 1
250 255 255 241 255 255 255 236 255 245 5 3 7 0 4 1 1 1 1 1 255 6 0 2 1 1 1 1 1 1 0 6 1 4 1 1 1 1 1 1
255 255 254 255 255 246 254 255 233 5 7 0 0 7 0 0 1 1 1 1 255 255 255 255 250 255 252 255 255 0255 255 255 255 255 241 255 236 253 1
255 241 255 247 254 255 245 237 22 3 0 5 19 0 4 10 1 1 1 1 255 255 255 255 255 255 246 255 0 6 255 255 255 255 241 247 255 237 0 0
255 255 255 254 249 246 255 15 0 7 0 19 0 2 6 0 1 1 1 1 255 255 255 255 255 254 246 15 1 1 255 255 255 255 255 255 252 0 0 0
252 255 246 255 245 252 1 0 3 0 7 0 2 0 8 1 1 1 1 1 255 255 255 255 255 245 1 0 0 4 255 255 255 255 236 237 0 0 0 1
253 255 254 245 255 1 0 0 0 4 0 4 6 8 0 10 1 1 1 1 250 255 252 255 255 22 3 0 1 1 255 241 255 236 245 3 0 1 1 1
255 236 255 237 15 0 0 0 0 1 0 10 0 1 10 0 1 1 1 1 255 255 246 255 5 0 7 0 1 1 241 247 255 237 3 5 0 10 1 1
254 251 255 0 0 0 4 6 1 1 1 1 1 1 1 1 1 1 1 1 255 254 246 15 719 2 0 1 1 255 255 252 0 0 0 0 1 1 1
255 253 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 255 245 1 0 4 4 8 10 1 1 236 237 0 0 1 10 1 0 1 1
255 0 6 5 0 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 251 0 0 6 1 1 1 1 1 1 253 0 0 0 1 1 1 1 1 1
0 1 6 0 1 0 4 1 1 1 1 1 1 1 1 1 1 1 1 1 0 5 1 2 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1
(a) Original image
(b) The image after transposed and reshaped
120

become less visible where there is significant activity or
"texture" in the image. The SSIM is calculated using the
formula (1).
 




 (1)
In this study, two images used as reference are the
resized-image and the first tile image. The resized-image is
generated from original image using openCV to have
dimension that match to the tile dimension. The sample
images used in this study is shown in Fig. 6. It consist of six
different images both in size and characteristic.
III. RESULT AND DISCUSSION
In this study the program for image tiling was built using
Python programming, utilizing openCV, numPy and scikit-
image modules. The type of image used can be either
grayscale image or RGB image. Transposition is conducted
to form the tile structure of the image. The tiled structure
referred to herein, is the resulting image that visually similar
to the original image in a smaller size and laid out like a tile
installation. Suppose we have a grayscale image of 20×20
with its intensity values shown in Fig. 7(a). Using a key
value of 2 which means the original image is converted into
2 columns, transposed, and if necessary reposition pixels,
then restored to its original shape to form a tile structure
(Fig. 7(b)).
The transpose results show four segments (marked with
different colors) that show a pattern similar to the original
image, but in smaller sizes. Furthermore, the similarity
between segments is measured using the SSIM, where the
first segment (blue) is used as the reference. The SSIM
between the first and the second segment (blue-red), the first
and the third segment (blue-yellow), the first and the fourth
segment (blue-green) respectively 0.74743, 0.74748, 0.72847
so that the average SSIM is 0.74112. The same way is used
to test the sample images, the results of the tests are shown in
Table I.
The key value used in the transposition affects the
number of tile structures produced. The number of tiles is the
square of the key. A key value of two will produce 2^2 tiles,
a key of three will produce 3^2 tiles and so on. The shape of
tile will follow the shape of its original but smaller.
TABLE I. THE SSIM VALUE BY KEY
key
#Tile
Compare to the resized image
Compare to the first tile
cactus
lena
balloon
temple
rainbow
sunshine
cactus
lena
Balloon
temple
rainbow
sunshine
1
1
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
2
4
0.6757
0.8531
0.9503
0.9539
0.9859
0.9733
0.4716
0.7400
0.8768
0.8977
0.9656
0.9355
3
9
0.6013
0.8298
0.9036
0.9111
0.9526
0.8941
0.4132
0.7136
0.8597
0.8517
0.9042
0.8370
4
16
0.4023
0.7322
0.7467
0.9022
0.9431
0.9163
0.2334
0.5826
0.8071
0.8302
0.8827
0.8179
5
25
0.3980
0.7681
0.6354
0.8682
0.9040
0.8667
0.2599
0.6282
0.7723
0.8136
0.8300
0.7692
6
36
0.2355
0.5362
0.8435
0.8715
0.8809
0.8161
0.2407
0.5814
0.7074
0.8009
0.8321
0.7074
7
49
0.2414
0.6421
0.7648
0.7304
0.7997
0.6331
0.1938
0.4656
0.7104
0.7803
0.6572
0.5869
8
64
0.2509
0.6134
0.7585
0.8316
0.7875
0.8326
0.1475
0.4485
0.6726
0.7296
0.7810
0.7251
9
81
0.3163
0.6982
0.3487
0.8366
0.8300
0.5967
0.2034
0.5687
0.5058
0.7749
0.7060
0.6896
10
100
0.0735
0.3048
0.2418
0.8428
0.8096
0.8042
0.1333
0.4303
0.3764
0.7665
0.7334
0.7056
11
121
0.3112
0.4064
0.2643
0.4728
0.6128
0.6055
0.1915
0.3825
0.4072
0.5553
0.5476
0.4581
12
144
0.0168
0.0142
0.5766
0.8331
0.6048
0.7077
0.0447
0.1258
0.5236
0.7535
0.6828
0.6813
13
169
0.0106
0.3455
0.5764
0.5431
0.5113
0.3436
0.0267
0.2364
0.4911
0.5288
0.5268
0.4476
14
196
0.0884
0.5122
0.4783
0.4733
0.4830
0.5909
0.0768
0.3291
0.5702
0.4779
0.4772
0.6441
15
225
-0.0056
0.6652
0.3631
0.8115
0.7552
0.7027
0.0155
0.4268
0.5151
0.7393
0.5806
0.6117
16
256
0.0897
0.4997
0.5858
0.6479
0.4809
0.7925
0.1293
0.2598
0.6145
0.5475
0.5324
0.6689
17
289
0.0804
0.2364
-0.0046
0.3028
0.5411
0.6281
0.0754
0.1540
0.0561
0.3595
0.5694
0.4921
18
324
0.0602
0.0763
0.0967
0.8139
0.7390
0.4922
0.0544
0.1532
0.1386
0.7329
0.6334
0.3706
19
361
0.0062
-0.0269
0.0808
0.3802
0.2774
0.6289
0.0069
-0.0457
0.1140
0.3766
0.3361
0.6079
20
400
0.0168
0.1574
-0.0250
0.8118
0.5957
0.7830
0.0077
0.0751
0.0094
0.7199
0.4879
0.6604
21
441
0.0406
-0.0381
0.0930
0.2755
0.4766
0.2092
0.0303
0.0011
0.1738
0.3353
0.4003
0.2666
22
484
0.0440
0.1288
0.0436
0.3076
0.3185
0.3746
0.0455
0.0565
0.1321
0.3722
0.3381
0.4774
23
529
0.0091
-0.0345
0.0269
0.3919
0.1665
0.2043
0.0093
-0.0172
0.0407
0.4092
0.2074
0.2892
24
576
0.0343
0.0629
0.4251
0.7807
0.3589
0.6507
0.0070
0.0469
0.3506
0.6508
0.4463
0.5691
25
625
0.0109
0.5890
0.1149
0.6317
0.3603
0.4529
0.0109
0.3073
0.1223
0.6712
0.3052
0.3298
26
676
0.0250
-0.0622
0.1919
0.4438
0.3973
0.1086
0.0326
0.0103
0.2716
0.5436
0.4810
0.1588
27
729
0.1629
0.0783
0.0209
0.4196
0.5555
0.5226
0.0822
0.0208
0.0860
0.3298
0.4579
0.6164
28
784
0.0049
0.3346
0.3365
0.3909
0.3878
0.2413
0.0187
0.1437
0.2133
0.3544
0.4612
0.2062
29
841
0.0358
0.0119
0.0130
0.2569
0.3436
0.3878
0.0187
0.0136
0.0567
0.2820
0.2465
0.3055
30
900
0.0159
0.0702
0.1496
0.7867
0.7014
0.5534
0.0116
0.0545
0.0551
0.6891
0.5756
0.5768
121

Average
0.1751
0.3668
0.3867
0.6508
0.6187
0.6105
0.1397
0.2964
0.4077
0.6225
0.5862
0.5738
a) b)
Fig. 8. Tile MSSIM, a). the resized image as reference, b) the first tile image as reference
a). key=3 b). key=16 c). key=17
Fig. 9. Tiles of balloon image using different keys (scaled down for visibility)
Fig. 10. MSSIM of ballon.jpg
From Fig. 8 can be seen that each image is unique. They
have different pattern in decreasing or increasing their SSIM.
SSIM conducted from resized image as reference is more
fluctuate than those from the first tile image. The three
images that relatively have higher SSIM value than other
three are temple, rainbow and sunshine.
The most decrease in SSIM value occurred in baloon.jpg
when the key changed from 16 to 17 as shown in Fig. 10.
Visually, the tile produce using key of 16 and 17 looks
similar as shown in Fig. 9. Even though simple, columnar
transposition shows the ability to scale down an image to
n^(-2) of its original size, where n is an integer number
greater than one. Each tile segment as the result of
transposition visually shows an image similar to the original
image in a smaller dimension. So it can be said that each
122

segment is a miniature of the original image. The more
columns for transposition, the smaller the tiles are generated.
It is important to note since this tiling process does not
change any intensity value, then the process can be reversed
to get its original image. The image that produces from the
reverse process will be exactly matched to the original.
The implementation of the columnar transposition in this
study which produces tile is completely different from the
common use of transposition, that is, in the data encryption.
Where in the encryption, transposition is used to improve
data security[10], [11], [15]. However, the ability of the
columnar transposition to produce similar tiles in the image
can further be used to improve data security and information
hiding.
IV. CONCLUSION
Implementation of columnar transposition on the image
succeeds in forming the tile structure. The key value used to
determine the number of columns of a transposition matrix
determines the number of tiles produced, which is the square
of the key value. The SSIM value varies depending on the
given key. But this increase and decrease did not occur
consistently
Determining the optimal size of the smallest tile to
improve data security and information hiding is a follow-up
study of this study. The goal is to increase confidentiality and
efficiency in data transmission.
ACKNOWLEDGMENT
The authors would like to express their heartfelt thanks to
P3M of State Polytechnic of Samarinda, for providing all
their support.
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