### 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Digital Halftoning

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## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Ordered Dithering

Considering the fact that the HVS is a low-pass system that only perceives the result of local average, one can design a distributed multilevel quantizer in a small region. For example, using a $2 \times 2$ block, we may design a five-level quantizer with thresholds $(1 / 8,3 / 8,5 / 8,7 / 8)$ and reconstruction points $(0,1 / 4,2 / 4,3 / 4,1)$. Then the thresholds are placed in a $2 \times 2$ block as:
$$\mathbf{T}=\left[\begin{array}{lll} 3 / 8 & 5 / 8 \ 7 / 8 & 1 / 8 \end{array}\right]$$
Suppose that the input value is a constant $2 \times 2$ block $\mathbf{x}$ :
$$\mathbf{x}=\left[\begin{array}{ll} c & c \ c & c \end{array}\right]$$
then we perform sample-wise quantization as in (3.2) as:
$$y[i, j]= \begin{cases}1, & \text { if } c>T[i, j], \ 0, & \text { otherwise }\end{cases}$$
So, the number of black pixels in $y$ will be approximately proportional to the value of $c$ in $\mathbf{x}$. Or, one can think of the proportion of black pixels in $\mathbf{y}$ as representing the reconstruction levels $(0,1 / 4,2 / 4,3 / 4,1)$. Thus, by arranging the quantization levels in a small block of thresholds, we get a distributed multi-level quantizer. The reconstruction levels are the proportion of black pixels in the output block. This is usually referred to as Ordered Dithering since we can consider the thresholds in $\mathbf{T}$ as dithering of the constant threshold $1 / 2$.

In general, for ordered dithering, the thresholds are specified by an index matrix. For a $L \times L$ block, the index matrix contains a list of all the $L^{2}$ integer numbers $\left{0, \ldots, L^{2}-1\right}$. For example, for $L=2$, the index matrix can be
$$\mathbf{I}=\left[\begin{array}{ll} 3 & 0 \ 2 & 1 \end{array}\right]$$
which specifies the sequence of turning on the corresponding pixels to black. From it, the threshold matrix can be determined:

$$T[i, j]=\frac{I[i, j]+1 / 2}{L^{2}}$$
which gives the threshold matrix:
$$\mathbf{T}=\left[\begin{array}{lll} 7 / 8 & 1 / 8 \ 5 / 8 & 3 / 8 \end{array}\right]$$
In general, the size of the image is larger than the size of the threshold matrix. So, the image is segmented into blocks having the same size of the threshold matrix, and then we use the threshold matrix to quantize each block.
$$y[i, j]= \begin{cases}1, & \text { if } x[i, j]>T[i \bmod L, j \bmod L], \ 0, & \text { otherwise. }\end{cases}$$
The size of the threshold matrix is usually much larger than $2 \times 2$, such as $8 \times$ $8,16 \times 16$ or even the size of the image to be processed. For such a large size, the arrangement of thresholds is not a trivial issue. Different arrangements of the quantization levels lead to different halftoning effects. If nearby quantization levels are spatially close to each other, then we have clustered dot dithering. If nearby quantization levels are located far away from each other, then we have dispersed dot dithering.

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Clustered Dot Dithering

The term ‘clustered dot’ comes from the fact that when using this dithering, the black dots are clustered together when the image pixels are smoothly varying. For this reason, changing the input gray level is equivalent to changing the size of the whole clustered dot, and is usually called Amplitude Modulation (AM) [9].

To construct the index matrix, we start from the center of the matrix, and put the integers from 0 to $L^{2}-1$ into it, by following a spiral curve with increasing radius. This is shown in Fig.3.2.
Similarly, we can get index matrix for $L=8$ :

which is shown as an image in Fig.3.3
Using this index matrix, we halftone a test image having continuous varying gray scale, as shown in Fig.3.4. Visually, the size of the marco-dot is changing with incrcasing blackncss.

Onc drawback of clustcred dot dithcring, duc to the dot-sizc modulation, is that the halftone result has reduced resolution. The larger the size of the block, the lower the resolution. This drawback can be remedied by dispersed dot dithering.

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Dispersed Dot Dithering

For dispersed dot dithering, such as Bayer’s dithering, adjacent thresholds are separated as far as possible from each other. Bayer’s index matrix is defined recursively [2], starting from trivial $2 \times 2$ index matrix
$$\mathbf{I}{2}=\left[\begin{array}{ll} 1 & 2 \ 3 & 0 \end{array}\right]$$ and $$\mathbf{I}{2 n}=\left[\begin{array}{lc} 4 \mathbf{I}{n}+1 & 4 \mathbf{I}{n}+2 \ 4 \mathbf{I}{n}+3 & 4 \mathbf{I}{n} \end{array}\right]$$
for $n=2,4, \ldots$ The $8 \times 8$ threshold matrix for dispersed dot dithering is shown in Fig. 3.5a, and the halftone result for the grayscale image in Fig.3.4a is shown in Fig. 3.5b. Since the thresholds are separated from each other, different blackness corresponds to different frequency of black dots. The resolution is significantly improved.

X=[CC CC]

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