里论外几-
Contrast Stretch
Since the data file values in a continuous raster layer often represent raw data, the range of data file values is often small. Therefore, a contrast stretch is usually performed, which stretches the range of the values to fit the range of the display.
See figure 4-5.
statistics Files
To perform a contrast stretch, certain statistics are necessary, such as the mean and the standard deviation of the data file values in each layer.
（这些statistics变换下章将完整分析，这里只涉及有关图像显示的部分。这里仅要求观察图像的statistics，e.g. the mean, standard deviation, middle, maximum, minimum）
Figure 4-7 illustrates the general process of displaying continuous raster layers on a 24-bit TureColor display.
Thematic Raster Layers
A thematic raster layer generally contains pixels that have been classified. It is stored in an image (.img) file. Only one data file value (the class value) is stored for each pixel. The class system gives the thematic layer a discrete look, in which each class can have its own color.
即：为每种类别指定一种RGB的组合（颜色），指定表称为Color scheme.
Color Table
When a thematic raster layer is displayed, ERDAS IMAGINE automatically creates a color table. The red, green, and blue brightness values for each class are stored in this table.
RGB Colors
Using the Viewer (未纳入教材，但对实习有指导意义)
意义：基于内存快速显示和进行某些图像处理（不保存）。
The Viewer not only makes digital images visible quickly, but it can also be used as a tool for image processing and raster GIS modeling.
Chapter 5 Mosaic
Introduction
The Mosaic process offers you the capability to stitch images together so one large, cohesive image of an area can be created. Because of the different features of the Mosaic Tool, you can smooth these images before mosaicking them together as well as color balance them, or adjust the histograms of each image in order to present a better large picture.
某些Mosaic Tool涉及到下章Enhancement的知识。
Chapter 6 Enhancement
重点：Radiometric Enhancement, Spatial Enhancement and Spectral Enhancement
Introduction
Image enhancement is the process of making an image more interpretable for a particular application. The techniques are often used instead of classification techniques for feature extraction.
Display vs. File Enhancement
Image enhancement may be performed:
• temporarily, upon the image that is displayed in the Viewer
• permanently, upon the image data in the data file.
Enhancing a displayed image is much faster than enhancing an image on disk.
Spatial Modeling Enhancements（基于组件应用建模）
• Graphical models（实习中有练习）
Correcting Data
Each generation of sensors shows improved data acquisition and image quality over previous generations. However, some anomalies still exist that are inherent to certain sensors and can be corrected by applying mathematical formulas derived from the distortions. In addition, the natural distortion that results from the curvature and rotation of the Earth in relation to the sensor platform produces distortions in the image data, which can also be corrected.
Generally, there are two types of data correction: radiometric and geometric.
这些辐射改正和几何改正（非10章之精校正）通常由卫星接收站完成。
Radiometric Enhancement（辐射增强）
Radiometric enhancement deals with the individual values of the pixels. It differs from spatial enhancement (discussed in “Spatial Enhancement”), which takes into account the values of neighboring pixels.
Radiometric enhancements that are applied to one band may not be appropriate for other bands.
简明地说：辐射增强用不涉及像元邻域性的统一变换关系来改变某个波段的 pixel value，。
Histograms concept（我加的小标题）
横坐标：pixel value（0-255）；
纵坐标：某像元值出现的Frequency
直方图是图像最基本的统计数据，它体现图的“调子”（明暗色调反差）。
Contrast Stretching
When radiometric enhancements are performed on the display device, the transformation of data file values into brightness values is illustrated by the graph of a lookup table.
Histograms和graph of a lookup table是分析辐射增强最有用的两种图。
Linear and Nonlinear
The terms linear and nonlinear, when describing types of spectral enhancement, refer to the function that is applied to the data to perform the enhancement. A piecewise linear stretch uses a polyline function to increase contrast to varying degrees over different ranges of the data, as in Figure 6-3.
Linear Contrast Stretch
Figure 4-5 就是用一种线性变换来进行反差拉伸。
注意149页底部有ERDAS提示：A two standard deviation linear contrast stretch is automatically applied to images displayed in the Viewer.
Histogram Equalization（直方图均衡化）
A nonlinear stretch that redistributes pixel values so that there is approximately the same number of pixels with each value within a range. The result approximates a flat histogram.
See figure 6-7
（补充：方法来源：累积直方图曲线作为变换关系。）
The resulting histogram is not exactly flat. 因为实际制作Histogram Equalization软件，是用差分代替微分（——教材中的Bin）
又：评论：最增强反差的方法，但并不等于效果最好。
Histogram Matching（直方图匹配）
Histogram matching is the process of determining a lookup table that converts the histogram of one image to resemble the histogram of another. Histogram matching is useful for matching data of the same or adjacent scenes that were scanned on separate days, or are slightly different because of sun angle or atmospheric effects. This is especially useful for mosaicking or change detection（后者指比较同区域两图以发现变化）.
To achieve good results with histogram matching, the two input images should have similar characteristics:
• The general shape of the histogram curves should be similar.
• Relative dark and light features in the image should be the same.
• For some applications, the spatial resolution should be the same.
• The relative distributions of land covers should be about the same.
（补充：方法来源：用二者的累积直方图曲线两次变换）
Brightness Inversion
The brightness inversion functions produce images that have the opposite contrast of the original image.
Spatial Enhancement（空间增强）
While radiometric enhancements operate on each pixel individually, spatial enhancement modifies pixel values based on the values of surrounding pixels.
Spatial enhancement deals largely with spatial frequency, which is the difference between the highest and lowest values of a contiguous set of pixels.
See Figure 6-11
Convolution Filtering (卷积过滤)
The process of averaging small sets of pixels across an image. It is used to change the spatial frequency characteristics of an image.
A convolution kernel(卷积模板) is a matrix that is used to average the value of each pixel with the values of surrounding pixels in a particular way. The numbers (often called coefficients) in the matrix serve to weight this average toward particular pixels.
Convolution filtering is one method of spatial filtering.
See Figure 6-12
convolution相当于上学期GIS课中讲过的Moving windows 活动窗口法之移动平均法。
下面讲三种类型的Kernel:
Low-Frequency or low-pass Kernels
(讲义上放在后的第三种Kernels，这里提到前面先讲 )
This kernel simply averages the values of the pixels, causing them to be more homogeneous. The resulting image looks either more smooth or more blurred, decreases spatial frequency.
（补充：两种常用的 Low-Frequency Kernels——均值滤波和中值滤波，前者即滑动平均，后者取模板覆盖的像元值大小排列的中间像元的值。利用中值滤波可滤掉某些像元值异常的点。）
Zero-Sum Kernels
Zero-sum kernels are kernels in which the sum of all coefficients in the kernel equals zero.
This generally causes the output values to be:
• zero in areas where all input values are equal (no edges)
• low in areas of low spatial frequency
• extreme in areas of high spatial frequency
Therefore, a zero-sum kernel is an edge detector.
The resulting image often consists of only edges and zeros.
几种常用的 Zero-Sum Kernels例：
罗伯特算子： 索伯尔算子 拉普拉斯算子
Zero-sum kernels can be biased to detect edges in a particular direction.
其它方向的探测，如：
High-Frequency or high-pass Kernels
It has the effect of increasing spatial frequency and serves as edge enhancers. Unlike edge detectors, they highlight edges and do not necessarily eliminate other features.
When this kernel is used on a set of pixels, the relative low value gets lower, the relative high value becomes higher.
See example of text (page 160-161) .
补充: How to derove high-pass Kernels? ——某种算子与原图叠加
1、原图“减”拉普拉斯算子
拉普拉斯算子：(左-自)-（自-右）+(上-自)-（自-下）
即：
原图减拉普拉斯算子：
2、原图 + 原图减滑动平均
两原图减滑动平均：
Crisp
The Crisp filter sharpens the overall scene luminance without distorting the interband variance content of the image. This is a useful enhancement if the image is blurred.
在后面讲解主成分变换（Principal Component or PC）方法后回头来理解此段。
Wavelet Resolution Merge
low spatial resolution to be sharpened using a co-registered panchromatic image of relatively higher resolution. A primary intended target dataset is Landsat 7 ETM+. Increasing the spatial resolution of multispectral imagery in this fashion is, in fact, the rationale behind the Landsat 7 sensor design.
Aside from traditional Pan-Multispectral image sharpening, this algorithm can be used to merge any two images, for example, radar with SPOT Pan. Fusing information from several sensors into one composite image can take place on four levels; signal, pixel, feature, and symbolic. This algorithm works at the pixel level.
Wavelet Theory
Wavelet-based image reduction is similar to Fourier transform analysis. In the Fourier transform, long continuous (sine and cosine) waves are used as the basis. The wavelet transform uses short, discrete “wavelets” instead of a long wave. Thus the new transform is much more local. The wavelet can be parameterized as a finite size moving window.
（165和168页只要知道可通过小波变换可进行不同分辨率图像的融合就行）
Algorithm Theory
再也不做稀饭了-
Spectral Enhancement（多光谱变换）
The techniques require more than one band of data.
处理思路（英文讲义上未解释）：
辐射增强和空间增强对一个波段或其局部进行变换，多光谱变换则对多波段图像的整体进行变换。具体方法与GIS课中所讲空间统计分析类似。
假定有n个变量、m个样点数据，则可组成一个n行、m列的数据矩阵：
具有各自地理位置的 样点1 样点2 …… 样点j …… 样点m
空间变量 X1
X2
…
Xi
…
Xn x11 x12 …… x1j …… x1m
x21 x22 …… x2j …… x2m
…
xi1 xi2 …… xij …… xim
…
xn1 xn2 …… xnj …… xnm
同样，我们这里将每一波段视为一维变量，将多波段图像视为多维空间，采用多维空间线性变换的方法，形成新的多维变量，即新的多波段图像。假定图像有m波段，每波段K列L行，共n个像元（n = KL）则m波段图像组成m维向量矩阵：
不少人对多光谱变换概念模糊就是因为将此矩阵与图像本身矩阵相混淆，多光谱空间概念对下章讲分类提取也很重要。
Spectral Enhancement 就是利用一个变换矩阵A，将原图像（矩阵X）变换为一个新的多维变量或多波段图象（矩阵Y）。即：Y = AX，或
下面讲述几种多光谱变换的方法
Principal Components Analysis (PCA，主成分分析)
This is often used as a method of data compression. It allows redundant data to be compacted into fewer bands—that is, the dimensionality of the data is reduced. The bands of PCA data are noncorrelated and independent, and are often more interpretable than the source data.
补充：有关统计概念
n
Variance = [ ∑(xip-mxi) 2 ] /n-1（第i波段第p像元）
P=1
方差是信息丰富程度的一种度量。
n
Covariance = [ ∑(xip-mxi) (xjp-mxj) ]/n-1
P=1
协方差能体现变量间相似性或重复性，是信息冗余程度的一种度量。
为此，协方差矩阵与直方图一样，成为图像处理中的一个基本分析数据和工具。
COV =
式中,V ij为第 I 波段与第 j 波段之间的协方差。
例：TM图像fj2的协方差矩阵为
1 2 3 4 5 6 7
1 87.431 50.610 84.522 18.195 96.021 35.029 80.202
2 50.610 32.399 53.773 23.158 74.913 22.151 54.773
3 84.522 53.773 93.785 30.101 127.842 36.951 95.805
4 18.195 23.158 30.101 183.716 191.943 20.116 65.046
5 96.021 74.913 127.842 191.943 389.152 62.380 198.104
6 35.029 22.151 36.951 20.116 62.380 27.646 43.380
7 80.202 54.773 95.805 65.046 198.104 43.380 127.426
问题：哪波段信息量最大（小）？哪两波段间信息冗余最大（小）？
主成分分析旨在寻求一个矩阵变换Y = AX，使所得新图像Y的协方差矩阵为：
COV =
式中，V11 > V 22 >…… > V mn，即新图像Y的各波段间的协方差为零，集中了图像大多数方差的前几个波段，分别称为PC1（第一主成分）、PC2、PC3…
线性代数表明，以m个特征向量组成的mm方阵，作为变换矩阵A，即可达到上述目的，且变换中新旧图像Y和X的方差总量不变。
例如：上面给出协方差矩阵的TM图像fj2，经主成分变换后形成新的图像fj2_pc，其协方差矩阵为
703.318 0 0 0 0 0 0
0 174.407 0 0 0 0 0
0 0 43.816 0 0 0 0
0 0 0 11.242 0 0 0
0 0 0 0 4.652 0 0
0 0 0 0 0 3.459 0
0 0 0 0 0 0 0.661
几何意义：
线性变换——坐标轴平移或旋转。
The process is easily explained graphically with an example of data in two bands. （讲解Page 153 - 155）
评论：数学的力量。反过来，图像处理加深对数学的理解。
PC方法常用于与其它方法相结合，如前面的Crisp和Resolution Merge。
英语总结：The different bands in a multispectral image can be visualized as defining an N-dimensional space where N is the number of bands. Each pixel, positioned according to its DN value in each band, lies within the N-dimensional space. This clustering of the pixels is termed the data structure.
The data structure can be considered a multidimensional hyperellipsoid. The principal axes of this data structure are not necessarily aligned with the axes of the data space. They are more directly related to the absorption spectra. You could view the axes that are largest for the data structure produced by the absorption peaks of special interest for a application.
For example, a geologist and a botanist are interested in different absorption features. They would want to view different data structures and therefore, different data structure axes. Both would benefit from viewing the data in a way that would maximize visibility of the data structure of interest.
Tasseled Cap（缨帽变换）
The Tasseled Cap transformation offers a way to optimize data viewing for vegetation studies. Research has produced three data structure axes that define the vegetation information content.
数学统计上最优的Principal Components方法不一定对各种应用都最优。这里的Tasseled Cap也是一种多维线性变换，但它的坐标轴平移或旋转朝最有利于观测植被地物的方向进行平移或旋转。
• Brightness—a weighted sum of all bands, defined in the direction of the principal variation in soil reflectance.
• Greenness—orthogonal to brightness, a contrast between the near-infrared and visible bands. Strongly related to the amount of green vegetation in the scene.
• Wetness—relates to canopy and soil moisture.
These rotations are sensor-dependent。For TM4 (第页).
缨帽变换系数的分析
RGB to IHS 及其逆计算
It is possible to define an alternate color space that uses intensity (I), hue (H), and saturation (S). This system is advantageous in that it presents colors more nearly as perceived by the human eye.
雷达图像的应用。
Indices
Indices are used to create output images by mathematically combining the DN values of different bands(波段间的algebra，P180).
Applications
• Indices are used extensively in mineral exploration and vegetation analysis to bring out small differences between various rock types and vegetation classes.
• Indices can also be used to minimize shadow effects in satellite and aircraft multispectral images. Black and white images of individual indices or a color combination of three ratios may be generated.
Integer Scaling Considerations
由于波段比值可能变动很大，计算结果重新定标取整问题在这里特别突出。
Index Examples （P182-183）
（回顾第一章的Figure 1-6图，帮助理解植被指数等。）
Hyperspectral Image Processing
Hyperspectral image processing is, in many respects, simply an extension of the techniques used for multispectral data sets; indeed, there is no set number of bands beyond which a data set is hyperspectral. Thus, many of the techniques or algorithms currently used for multispectral data sets are logically applicable. What is of relevance in evaluating these data sets is not the number of bands per se, but the spectral bandwidth of the bands (channels). As the bandwidths get smaller, it becomes possible to view the data set as an absorption spectrum rather than a collection of discontinuous bands. Analysis of the data in this fashion is termed imaging spectrometry.
A hyperspectral image data set is recognized as a three-dimensional pixel array（Figure 65）（6-23）.
A data set with narrow contiguous bands can be plotted as a continuous spectrum and compared to a library of known spectra. A serious complication in using this approach is assuring that all spectra are corrected to the same background.
At present, it is possible to obtain spectral libraries of common materials. The JPL and USGS mineral spectra libraries are included in ERDAS IMAGINE. These are laboratory-measured reflectance spectra of reference minerals, often of high purity and defined particle size. The spectrometer is commonly purged（净化） with pure nitrogen to avoid absorbance by atmospheric gases. Conversely, the remote sensor records an image after the sunlight has passed through the atmosphere (twice) with variable and unknown amounts of water vapor, CO2. The unknown atmospheric absorbances superimposed upon the Earth"s surface reflectances makes comparison to laboratory spectra or spectra taken with a different atmosphere inexact. Indeed, it has been shown that atmospheric composition can vary within a single scene. This complicates the use of spectral signatures even within one scene.
A number of approaches have been advanced to help compensate for this atmospheric contamination of the spectra.
Fourier Analysis
Image enhancement techniques can be divided into two basic categories: point and neighborhood. Point techniques enhance the pixel based only on its value, with no concern for the values of neighboring pixels. These techniques include contrast stretches (nonadaptive), classification, and level slices. Neighborhood techniques enhance a pixel based on the values of surrounding pixels. As a result, these techniques require the processing of a possibly large number of pixels for each output pixel. The most common way of implementing these enhancements is via a moving window convolution. However, as the size of the moving window increases, the number of requisite calculations becomes enormous. An enhancement that requires a convolution operation in the spatial domain can be implemented as a simple multiplication in frequency space—a much faster calculation.
In ERDAS IMAGINE, the FFT is used to convert a raster image from the spatial (normal) domain into a frequency domain image. The FFT calculation converts the image into a series of two-dimensional sine waves of various frequencies. The Fourier image itself cannot be easily viewed, but the magnitude of the image can be calculated, which can then be displayed either in the Viewer or in the FFT Editor. Analysts can edit the Fourier image to reduce noise or remove periodic features, such as striping. Once the Fourier image is edited, it is then transformed back into the spatial domain by using an IFFT. The result is an enhanced version of the original image.
The basic premise（前提） behind a Fourier transform is that any one-dimensional function, f(x) (which might be a row of pixels), can be represented by a Fourier series consisting of some combination of sine and cosine terms and their associated coefficients.
176页上的图72
A Fourier transform is a linear transformation that allows calculation of the coefficients necessary for the sine and cosine terms to adequately represent the image. This theory is used extensively in electronics and signal processing. Therefore, DFT has been developed. Because of the computational load in calculating the values for all the sine and cosine terms along with the coefficient multiplications, a highly efficient version of the DFT was developed and called the FFT.
DFT ——177上的公式，FFT 快速算法
e-j2π(ax+by) = cos2π(ax+by) -jsin2π(ax+by)
The raster image generated by the FFT calculation is not an optimum image for viewing or editing. Each pixel of a fourier image is a complex number (i.e., it has two components: real and imaginary). For display as a single image, these components are combined in a root-sum of squares operation（傅利叶变换的模作为像元值来显示 u,v图像）. Also, since the dynamic range of Fourier spectra vastly exceeds the range of a typical display device, the Fourier Magnitude calculation involves a logarithmic function.
Finally, a Fourier image is symmetric about the origin (u, v = 0, 0). If the origin is plotted at the upper left corner, the symmetry is more difficult to see than if the origin is at the center of the image. Therefore, in the Fourier magnitude image, the origin is shifted to the center of the raster array.