Posted on Categories:Digital image processing, 图像处理, 数学代写

# 数学代写|图像处理代写Digital image processing代考|EE637 Quantization

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## 数学代写|图像处理代写Digital image processing代考|Quantization

For use with a computer, the measured irradiance at the image plane must be mapped onto a limited number $Q$ of discrete gray values. This process is called quantization. The number of required quantization levels in image processing can be discussed with respect to two criteria.
First, we may argue that no gray value steps should be recognized by our visual system, just as we do not see the individual pixels in digital images. Figure $2.7$ shows images quantized with 2 to 16 levels of gray values. It can be seen clearly that a low number of gray values leads to false edges and makes it very difficult to recognize objects that show slow spatial variation in gray values. In printed images, 16 levels of gray values seem to be sufficient, but on a monitor we would still be able to see the gray value steps.

Generally, image data are quantized into 256 gray values. Then each pixel occupies 8 bits or one byte. This bit size is well adapted to the architecture of standard computers that can address memory bytewise. Furthermore, the resolution is good enough to give us the illusion of a continuous change in the gray values, since the relative intensity resolution of our visual system is no better than about $2 \%$.

The other criterion is related to the imaging task. For a simple application in machine vision, where homogeneously illuminated objects must be detected and measured, only two quantization levels, i. e., a binary image, may be sufficient. Other applications such as imaging spectroscopy or medical diagnosis with $\mathrm{x}$-ray images require the resolution of faint changes in intensity. Then the standard 8-bit resolution would be too coarse.

## 数学代写|图像处理代写Digital image processing代考|Signed Representation of Images

Normally we think of “brightness” (irradiance or radiance) as a positive quantity. Consequently, it appears natural to represent it by unsigned numbers ranging in an 8-bit representation, for example, from 0 to 255 . This representation causes problems, however, as soon as we perform arithmetic operations with images. Subtracting two images is a simple example that can produce negative numbers. Since negative gray values cannot be represented, they wrap around and appear as large positive values. The number $-1$, for example, results in the positive value 255 given that $-1$ modulo $256=255$. Thus we are confronted with the problem of two different representations of gray values, as unsigned and signed 8-bit numbers. Correspondingly, we must have several versions of each algorithm, one for unsigned gray values, one for signed values, and others for mixed cases.
One solution to this problem is to handle gray values always as signed numbers. In an 8-bit representation, we can convert unsigned numbers into signed numbers by subtracting 128 :
$$q^{\prime}=(q-128) \bmod 256, \quad 0 \leq q<256 .$$
Then the mean gray value intensity of 128 becomes the gray value zero and gray values lower than this mean value become negative. Essentially, we regard gray values in this representation as a deviation from a mean value.
This operation converts unsigned gray values to signed gray values which can be stored and processed as such. Only for display must we convert the gray values again to unsigned values by the inverse point operation
$$q=\left(q^{\prime}+128\right) \bmod 256, \quad-128 \leq q^{\prime}<128,$$
which is the same operation as in Eq. (2.7) since all calculations are performed modulo $256 .$

## 数学代写|图像处理代写 数字图像处理代考|图像的有符号表示

$$q^{\prime}=(q-128) \bmod 256, \quad 0 \leq q<256 .$$

$$q=\left(q^{\prime}+128\right) \bmod 256, \quad-128 \leq q^{\prime}<128,$$

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