详细讲解在图像处理中傅里叶变换，并帮助我理解记忆---**Textual Information:**
线性系统的基本理论与运算
1. 线性系统与非线性系统
* 设系统的特性可表示成对输入图像进行T运算，并令 f₁(x,y)、f₂(x,y)与 T[f₁(x,y)]、T[f₂(x,y)] 分别代表两对不同的输入和输出图像，则当系统满足:
T[f₁(x,y)]+T[f₂(x,y)] = T[f₁(x,y)+f₂(x,y)]
关系时，称系统具有叠加性。当系统满足:
T[kf₁(x,y)] = kT[f₁(x,y)]
关系时，称系统具有齐次性。
同时满足叠加性和齐次性的系统称为线性系统。
图像处理与应用 77
线性系统的基本理论与运算
2. 二维线性移不变系统
当系统的单位脉冲输入为 $\delta(x-\alpha, y-\beta)$，即输入的单位脉冲函数延迟了 $\alpha$、$\beta$ 单位时，输出为 $\text{h}(x-\alpha, x-\beta)$，即输出结果形态不变，仅在位置上延迟了 $\alpha$、$\beta$ 单位，则称这样的系统为移不变系统。
如果一个系统既是线性系统，又是移不变系统，则该系统是线性移不变系统。
图像处理与应用
78
Here is the extracted content from the image:
**Title:**
线性系统的基本理论与运算
**Content:**
对于一个二维线性移不变系统h(x,y)，设其输入为f(x,y)，输出为g(x,y)，线性移不变系统的运算为T,则有：
g(x,y) = T[f(x,y)]
= T[∫_-∞^∞ ∫_-∞^∞ f(α,β)δ(x-α, y-β)dαdβ]
= ∫_-∞^∞ ∫_-∞^∞ T[f(α,β)δ(x-α, y-β)]dαdβ (线性叠加原理)
= ∫_-∞^∞ ∫_-∞^∞ f(α,β)T[δ(x-α, y-β)]dαdβ (齐次性; x,y为变量)
= ∫_-∞^∞ ∫_-∞^∞ f(α,β)h(x-α, y-β)dαdβ (移不变性,卷积表示)
= f(x,y) * h(x,y)
(4.5)
即：线性移不变系统的输出等于系统的输入与系统脉冲响应（点扩展函数）的卷积。
**Footer:**
图像处理与应用
79
**Title:** 线性系统的基本理论与运算 (Basic Theory and Operations of Linear Systems)
**Section Heading:** 同理有：(Similarly, there is:)
**Formula (4.6):**
$g(x,y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x-\alpha, y-\beta)h(\alpha, \beta)d\alpha d\beta$
$= h(x,y) * f(x,y)$
(4.6)
**Section Heading:** 也有：(Also, there is:)
**Formula (4.7):**
$h(x,y) * f(x,y) = f(x,y) * h(x,y)$
(4.7)
**Section Heading:** 所以, 二维线性移不变系统的输入、输出和运算关系可描述为：(Therefore, the input, output, and operation relationship of a 2D linear shift-invariant system can be described as:)
**Chart Description:**
* **Type:** Block Diagram / Flow Diagram
* **Main Elements:**
* A label "$f(x,y)$" on the left, representing the input.
* A right-pointing arrow originating from "$f(x,y)$".
* A rectangular box labeled "$h(x,y)$" in the center, representing the system or filter's impulse response.
* A right-pointing arrow originating from the right side of the "$h(x,y)$" box.
* A label "$g(x,y)=f(x,y)*h(x,y)$" on the right, representing the output as the convolution of the input and the system's impulse response.
* The diagram shows the process flow from input $f(x,y)$ through the system $h(x,y)$ to produce the output $g(x,y)$.
**Footer:** 图像处理与应用 (Image Processing and Applications)
**Page Number:** 80
**Title:** 离散傅立叶变换
**Section Title:** 口一维离散傅里叶变换
**Description:** 设f(x)是在时域上等距离采样的得到的N点离散序列, x是离散实变量, u为离散频率变量, 则离散傅里叶变换对定义为:
**Formulas:**
F(u) = (1 / sqrt(N)) * sum[x=0 to N-1] f(x) * exp[-j * 2π * xu / N], u = 0,1,...,N-1
f(x) = (1 / sqrt(N)) * sum[u=0 to N-1] F(u) * exp[j * 2π * ux / N], x = 0,1,...,N-1
**Definitions/Annotations:** 其中, F(u)为正变换, f(x)=f⁻¹[F(u)]为反变换;
e^(-j * 2π * xu / N) 是正变换核, e^(j * 2π * ux / N) 是反变换核.
**Footer:** 图像处理与应用 81
```plain text
离散傅立叶变换
一维离散傅里叶变换
根据欧拉公式 $e^{\pm ix} = \cos x \pm i \sin x$ 有
$\exp(-j2\pi ux) = \cos 2\pi ux - j \sin 2\pi ux$
所以, $F(u)$ 一般是复数, 并可以写成
$F(u) = R(u) + jI(u)$
且 $|F(u)| = \sqrt{R^2(u) + I^2(u)}$ , $\varphi(u) = \arctan \left[\frac{I(u)}{R(u)}\right]$
其中, $|F(u)|$ 称为 $f(x)$ 的傅里叶谱, 反映了 $f(x)$ 的幅频特性; $\varphi(u)$ 称为相角, 反映了 $f(x)$ 的相频特性。
图像处理与应用 82
```
**Title:**
离散傅立叶变换
**Section Header:**
二维离散傅里叶变换
**Description:**
设f(x,y)是在空间域上等间隔采样的得到的M×N的二维离散信号, x和y是离散实变量, u和v为离散频率变量, 则二维离散傅里叶变换对一般地定义:
**Mathematical Formulas:**
1. F(u,v) = √(1/MN) * Σ(x=0 to M-1) Σ(y=0 to N-1) f(x,y)exp[-j2π(xu/M + yv/N)]
(u = 0, 1, ..., M-1; v = 0, 1, ..., N-1)
2. f(x,y) = √(1/MN) * Σ(u=0 to M-1) Σ(v=0 to N-1) F(u,v)exp[j2π(ux/M + vy/N)]
(x = 0, 1, ..., M-1; y = 0, 1, ..., N-1)
**Footer:**
图像处理与应用
83
**Title:**
离散傅立叶变换
**Heading:**
二维离散傅里叶变换
**Introduction 1:**
与一维时的情况类似，可将二维离散傅里叶变换的频谱和相角定义为：
**Formula 1:**
$P(u, v) = |F(u, v)|^2 = R^2(u, v) + I^2(u, v)$ (4.18a)
**Formula 2:**
$\phi(u, v) = \arctan[I(u, v) / R(u, v)]$ (4.18b)
**Introduction 2:**
将二维离散傅里叶变换的频谱的平方定义为 $f(x, y)$ 的功率谱，记为：
**Formula 3:**
$P(u, v) = |F(u, v)|^2 = R^2(u, v) + I^2(u, v)$ (4.19)
**Conclusion:**
反映了二维离散信号的能量在空间频率域上的分布情况。
**Footer:**
图像处理与应用 84
离散傅里叶变换
□图像傅里叶变换的意义
(1) 简化计算，也即傅里叶变换可将空间域中复杂的卷积运算转化为频率域中简单的乘积运算。
(2) 对于某些在空间域中难以处理或处理起来比较复杂的问题，利用傅里叶变换把用空间域表示的图像映射到频率域，再利用频率域滤波或频率域分析方法对其进行处理和分析，然后再把其在频率域处理和分析的结果变换回空间域，从而可达到简化处理和分析的目的。
(3) 某些只能在频率域处理的特定应用需求，比如在频率域进行图像特征提取、数据压缩、纹理分析、水印嵌入等。
图像处理与应用 85
图像傅里叶变换频谱分析
1、图像傅里叶频谱关于(M/2, N/2)的对称性
设 f(x,y) 是一幅大小为 M×N 的图像，根据离散傅里叶变换的周期性
公式: F(u,v) = F(u+mM, v+nN)
有: |F(u,v)| = |F(u+M, v+N)| (4.33)
再根据离散傅里叶变换的共轭对称性式:
|F(u,v)| = |F(-u,-v)|
就可得: |F(u,v)| = |F(M-u, N-v)| (4.34)
图像处理与应用 86
**Textual Information:**
* **Title:** 图像傅里叶变换频谱分析
* **Section Title:** 1、图像傅里叶频谱关于(M/2, N/2)的对称性
* **Mathematical Formula:** |F(u,v)|=|F(M-u,N-v)|
* **Relevant Text:** 根据 (4. 34)，对于 u = 0:
* **Equations and Text for u=0:**
* 当 v = 0时: |F(0,0)| = |F(M, N)|
* 当 v = 1时: |F(0,1)| = |F(M, N-1)|
* 当 v = 2时: |F(0,2)| = |F(M, N-2)|
* Vertical ellipsis (indicating continuation)
* 当 v = N/2时: |F(0, N/2)| = |F(M, N/2)|
* **Footer:** 图像处理与应用 87
**Chart/Diagram Description:**
* **Type:** 2D coordinate system representing a frequency domain, divided into quadrants.
* **Axes:**
* Horizontal axis is labeled 'v', pointing to the right, with tick marks and labels at 0, N/2, and N.
* Vertical axis is labeled 'u', pointing downwards, with tick marks and labels at 0, M/2, and M.
* The origin (0,0) of the coordinate system is at the top-left corner of the plotted region.
* **Domain:** A rectangular region defined by 0 <= v <= N and 0 <= u <= M. The vertices are at diagram coordinates (v, u): (0,0), (N,0), (0,M), and (N,M). The bottom-right corner is at (N,M).
* **Divisions:** The rectangle is divided into four sub-regions (quadrants) by a vertical dashed line at v=N/2 and a horizontal dashed line at u=M/2. The intersection of these lines is at the center of the rectangle, diagram coordinate (N/2, M/2).
* **Quadrants:** The four quadrants are labeled A, B, C, and D:
* A: Top-left quadrant, spanning 0 <= v <= N/2 and 0 <= u <= M/2.
* B: Top-right quadrant, spanning N/2 <= v <= N and 0 <= u <= M/2.
* C: Bottom-left quadrant, spanning 0 <= v <= N/2 and M/2 <= u <= M.
* D: Bottom-right quadrant, spanning N/2 <= v <= N and M/2 <= u <= M.
* **Labels within Diagram:**
* The label (M/2, N/2) is placed inside quadrant A.
* The label (M/2, N) is placed to the right of the vertical dashed line (v=N/2) and approximately at the horizontal level u=M/2.
* The label (M, N/2) is placed below the horizontal dashed line (u=M/2) and approximately at the vertical level v=N/2.
* The label (M, N) is placed near the bottom-right corner of the rectangle.
* Note: The placement and format of these coordinate labels (X,Y) may be inconsistent with the v (horizontal), u (vertical) axis labeling of the diagram, or with standard (u,v) frequency conventions.
* **Arrows:**
* A red horizontal arrow points from v=0 to v=N/2 along the v-axis.
* A red horizontal arrow points from v=N to v=N/2 along the v-axis.
* A red horizontal arrow points from v=N/2 to v=N just below the bottom edge of the rectangle (at u=M level).
* A red horizontal arrow points from v=N to v=N/2 just below the bottom edge of the rectangle (at u=M level).
Here is the extraction of the content from the image:
**Title:**
图像傅里叶变换频谱分析
(Image Fourier Transform Spectrum Analysis)
**Question:**
1、图像傅里叶频谱关于(M/2, N/2)的对称性
(1. Symmetry of the image Fourier spectrum about (M/2, N/2))
**Formula:**
$|F(u, v)| = |F(M-u, N-v)|$
**Text Below Formula:**
同理, 对于v=0:
(Similarly, for v=0:)
当u=0时: $|F(0, 0)| = |F(M, N)|$
(When u=0: $|F(0, 0)| = |F(M, N)|$)
当u=1时: $|F(1, 0)| = |F(M-1, N)|$
(When u=1: $|F(1, 0)| = |F(M-1, N)|$)
当u=2时: $|F(2, 0)| = |F(M-2, N)|$
(When u=2: $|F(2, 0)| = |F(M-2, N)|$)
...
当u=M/2时: $|F(M/2, 0)| = |F(M/2, N)|$
(When u=M/2: $|F(M/2, 0)| = |F(M/2, N)|$)
由此可得:
(From this, we can conclude:)
频谱图A区与D区和B区与C区关于坐标(M/2, N/2)对称。
(Spectrum map area A and area D, and area B and area C are symmetric about the coordinate (M/2, N/2).)
**Diagram Description:**
* **Type:** 2D Coordinate System / Grid Diagram.
* **Coordinate Axes:**
* Horizontal axis labeled 'v', pointing right. Origin at the bottom left corner of the main square.
* Vertical axis labeled 'u', pointing down. Origin at the bottom left corner of the main square.
* **Scale/Labels on Axes:**
* v-axis has points labeled 0, N/2, N.
* u-axis has points labeled 0, M/2, M.
* **Main Square:** A square defined by the points (0,0), (N,0), (N,M), (0,M).
* **Subdivisions:** The square is divided into four smaller quadrants by dashed lines intersecting at (N/2, M/2).
* **Quadrants/Regions:**
* Top-left: Labeled 'A'. Corner coordinates: (0,0), (N/2,0), (N/2,M/2), (0,M/2).
* Top-right: Labeled 'B'. Corner coordinates: (N/2,0), (N,0), (N,M/2), (N/2,M/2). Top-right corner labeled (N, M/2) is incorrect, should be (N,0) at the top right. Top-right corner of B is (N, M/2), top-right corner of main square is (N,0) on the v-axis end, and (N,M) at the very end of both axes intersection. The coordinates shown (M/2, N/2) at the center are incorrect labels; the center is at (N/2, M/2). The top right corner of the whole square is (N,0) according to the labels on the axes, but (N,M) based on the coordinate description. The labels (M/2, N/2) next to the center point and (M, N) at the bottom right corner are labels for u,v coordinates in (u,v) format, which seems swapped compared to the (N/2, M/2) label for the center point and the (N,M) label at the bottom right corner. Let's interpret based on the u and v axes directions and labels.
* The diagram shows u axis downwards and v axis to the right, with origin (0,0) at the top left.
* v axis labels: 0, N/2, N.
* u axis labels: 0, M/2, M.
* Center point is labeled (N/2, M/2). This seems to be (v, u) format based on the diagram's axes.
* Region A: Top-left, u from 0 to M/2, v from 0 to N/2.
* Region B: Top-right, u from 0 to M/2, v from N/2 to N.
* Region C: Bottom-left, u from M/2 to M, v from 0 to N/2. Corner labeled (M, N/2).
* Region D: Bottom-right, u from M/2 to M, v from N/2 to N. Corner labeled (M, N).
* **Arrows:**
* Red arrow pointing right from N/2 on the v-axis, above region B.
* Purple arrow pointing up from M on the u-axis, right of region D.
* Red arrow pointing left from M on the u-axis (incorrect axis), below region C. This red arrow should likely be indicating horizontal symmetry, maybe on the v axis, but it's positioned below the square pointing left.
* Purple arrow pointing down from N on the v-axis (incorrect axis), left of region A. This purple arrow should likely be indicating vertical symmetry, maybe on the u axis, but it's positioned left of the square pointing down.
* The arrows' intended meaning for symmetry is unclear from their placement alone, but the text states symmetry about (M/2, N/2).
**Images at the Bottom:**
There are four small images arranged horizontally. Below the left two, the label "(a) 图像 (b) 图像的原频谱图" (Image (a) (b) Original Image Spectrum Map) appears. Below the right two, the same label appears "(a) 图像 (b) 图像的原频谱图". It seems the label should be "(a) 图像" for the left image and "(b) 图像的原频谱图" for the right image in each pair.
* **Image 1 (leftmost):** Titled "(a) 图像". Appears to be an image with diagonal white stripes on a black background.
* **Image 2 (second from left):** Titled "(b) 图像的原频谱图". Appears to be the spectrum of Image 1. Shows bright points arranged in a diagonal pattern around the center, against a dark background.
* **Image 3 (third from left):** Titled "(a) 图像". Appears to be a black image with four white squares arranged in a 2x2 grid.
* **Image 4 (rightmost):** Titled "(b) 图像的原频谱图". Appears to be the spectrum of Image 3. Shows a cross shape (bright horizontal and vertical lines) with bright spots at the center and ends, against a dark background, overlaid with a grid pattern.
**Title:** 图像傅里叶变换频域分析 (Image Fourier Transform Frequency Domain Analysis)
**Section Title:** 2、图像傅里叶频谱特性及其频谱图 (2. Image Fourier Spectrum Characteristics and its Spectrum Images)
**Diagram 1 (Top Left):**
* Type: Coordinate system with a rectangle and diagonal lines.
* Coordinate Axes: X-axis (horizontal, right), Y-axis (vertical, up). Origin at bottom left (0).
* Rectangle: Defined by corners (0,0) and (M, N). Sides aligned with axes.
* Content: Parallel diagonal lines within the rectangle.
* Labels: Axes X, Y. Origin 0. Top right corner (M, N). Marks M on Y-axis, N on X-axis.
**Diagram 2 (Top Middle Left):**
* Type: Coordinate system with a rectangle and polygonal lines.
* Coordinate Axes: u-axis (horizontal, right), v-axis (vertical, up). Origin at bottom left (0).
* Rectangle: Defined by corners (0,0) and (M, N). Sides aligned with axes.
* Content: Polygonal lines within the rectangle.
* Labels: Axes u, v. Origin 0. Top right corner (M, N). Marks M on v-axis, N on u-axis.
**Diagram 3 (Bottom Left):**
* Type: Coordinate system with a rectangle divided into quadrants, polygonal lines, and a dashed square.
* Coordinate Axes: u-axis (horizontal, right), v-axis (vertical, up). Origin at bottom left (0,0).
* Rectangle: A larger rectangle is divided into four quadrants by lines passing through (M/2, N/2).
* Content: Polygonal lines distributed across the quadrants, similar pattern in opposite quadrants.
* Labels: Axes u, v. Origin (0,0). Point (M/2, N/2) is labeled.
* Shapes: Dashed square centered around (M/2, N/2).
**Diagram 4 (Bottom Middle Left):**
* Type: Coordinate system with a dashed square and polygonal lines.
* Coordinate Axes: u-axis (horizontal, right), v-axis (vertical, down). Origin at the center.
* Shapes: Dashed square centered at the origin.
* Content: Polygonal lines within the dashed square, similar to the central part of Diagram 3.
* Labels: Axes u, v with arrows indicating direction.
**Frequency Spectrum Image 1 (Top Middle Right):**
* Type: Grayscale image (Frequency Spectrum).
* Content: Shows a bright cross shape (horizontal and vertical high-intensity lines) centered within a darker region, with some diagonal patterns and corner concentrations.
**Frequency Spectrum Image 2 (Top Right):**
* Type: Grayscale image (Frequency Spectrum).
* Content: Shows a bright grid-like structure of horizontal and vertical lines. The origin seems to be in the top-left corner. This image likely corresponds to the spectrum of the diagonal pattern in Diagram 1.
**Text Label below Images 1 & 2:**
原频谱图 - 原点在(0,0)时的频谱图 (Original Spectrum Image - Spectrum Image when the origin is at (0,0))
**Frequency Spectrum Image 3 (Bottom Middle Right):**
* Type: Grayscale image (Frequency Spectrum).
* Content: Looks similar to Image 1 but the bright cross shape and other features are centered in the image. This is likely the centered version of Image 1's spectrum.
**Frequency Spectrum Image 4 (Bottom Right):**
* Type: Grayscale image (Frequency Spectrum).
* Content: Looks similar to Image 2 but the bright grid-like structure is centered in the image. This is likely the centered version of Image 2's spectrum.
**Text Label below Images 3 & 4:**
频谱图 - 原点平移到(M/2, N/2)后的频谱图 (Spectrum Image - Spectrum Image after the origin is shifted to (M/2, N/2))
**Footer:**
图像处理与应用 (Image Processing and Applications)
89
图像傅里叶变换频谱分析
2、图像傅里叶频谱特性及其频谱图
对于式(4.43): $f(x, y)\exp \left[j2\pi \left(\frac{u_0x}{M} + \frac{v_0y}{N}\right)\right] \leftrightarrow F(u - u_0, v - v_0)$
当$u_0 = M/2$, $v_0 = N/2$时, 有
$\exp \left[j2\pi \left(\frac{u_0x}{M} + \frac{v_0y}{N}\right)\right] = \exp \left[j2\pi \left(\frac{M}{2}\frac{x}{M} + \frac{N}{2}\frac{y}{N}\right)\right]$
$e^{j\pi(x+y)} = (e^{j\pi})^{(x+y)} = (\cos\pi + j\sin\pi)^{(x+y)} = (-1)^{(x+y)}$
也即 $f(x, y)(-1)^{(x+y)} \leftrightarrow F(u - \frac{M}{2}, v - \frac{N}{2})$
也就是说, 图4.7的频谱图(a)和(b)实质上是函数 $f(x,y)(-1)^{(x+y)}$ 的傅里叶频谱图。
**Image Text Extraction:**
**Title:** 图像傅里叶变换频率分析 (Image Fourier Transform Frequency Analysis)
**Section 3:**
3、傅里叶变换在图像处理中的应用 (Applications of Fourier Transform in Image Processing)
**基本思路是:** (Basic Idea is:)
先用 $(-1)^{(x+y)}$ 乘以图像得 $(-1)^{(x+y)} f(x,y)$; 然后对其进行傅里叶正变换得到原点在 $(M/2, N/2)$ 之处的 $F(u,v)$; 接着根据图像的频率特性, 利用有关的低通频率滤波器, 或高通频率滤波器等, 对其进行滤波处理; 再将处理的结果进行傅里叶反变换; 最后给反变换的结果再乘以 $(-1)^{(x+y)}$ 就可以得到最终的结果。
**典型的应用有:** (Typical applications include:)
去除图像噪声、图像数据压缩、图像识别、图像重构和图像描述等。 (Removing image noise, image data compression, image recognition, image reconstruction, and image description, etc.)
**Footer:**
图像处理与应用 (Image Processing and Applications)
91