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# 计算机代写|计算机视觉代写Computer Vision代考|CS766 SPATIO-TEMPORAL SPECTRAL MODELS

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## 计算机代写|计算机视觉代写Computer Vision代考|SPATIO-TEMPORAL SPECTRAL MODELS

It is possible to detect and measure image motion purely by Fourier means. This approach exploits the fact that motion creates a covariance in the spatial and temporal spectra of the time-varying image $I(x, y, t)$, whose three-dimensional (spatio-temporal) Fourier transform is defined:
$$F\left(\omega_x, \omega_y, \omega_t\right)=\int_X \int_Y \int_T I(x, y, t) e^{-i\left(\omega_x x+\omega_y y+\omega_t t\right)} d x d y d t$$

In other words, rigid image motion has a 3D spectral consequence: the local 3D spatio-temporal spectrum, rather than filling up 3-space $\left(\omega_x, \omega_y, \omega_t\right)$, collapses onto a 2D inclined plane which includes the origin. Motion detection then occurs just by filtering the image sequence in space and in time, and observing that tuned spatio-temporal filters whose center frequencies are co-planar in this 3-space are activated together. This is a consequence of the SPECTRAL CO-PLANARITY THEOREM:
Theorem: Translational image motion of velocity $\overrightarrow{\mathbf{v}}$ has a 3D spatiotemporal Fourier spectrum that is non-zero only on an inclined plane through the origin of frequency-space. Spherical coordinates of the unit normal to this spectral plane correspond to the speed and direction of motion.
Let $I(x, y, t)$ be a continuous image in space and time.
Let $F\left(\omega_x, \omega_y, \omega_t\right)$ be its 3D spatio-temporal Fourier transform:
$$F\left(\omega_x, \omega_y, \omega_t\right)=\int_X \int_Y \int_T I(x, y, t) e^{-i\left(\omega_x x+\omega_y y+\omega_t t\right)} d x d y d t .$$
Let $\overrightarrow{\mathbf{v}}=\left(v_x, v_y\right)$ be the local image velocity.
Uniform motion $\overrightarrow{\mathbf{v}}$ implies that for all time shifts $t_o$,
$$I(x, y, t)=I\left(x-v_x t_o, y-v_y t_o, t-t_o\right) .$$
Taking the 3D spatio-temporal Fourier transform of both sides, and applying the shift theorem, gives
$$F\left(\omega_x, \omega_y, \omega_t\right)=e^{-i\left(\omega_x v_x t_o+\omega_y v_y t_o+\omega_t t_o\right)} F\left(\omega_x, \omega_y, \omega_t\right)$$

## 计算机代写|计算机视觉代写Computer Vision代考|Lambertian and specular surfaces. Reflectance maps

How can we infer information about the surface reflectance properties of objects from raw measurements of image brightness? This is a more recondite matter than it might first appear, because of the many complex factors which determine how (and where) objects scatter light.
Some definitions of surface type and properties:

• Surface albedo refers to the fraction of the illuminant that is re-emitted from the surface in all directions, in total. Thus, albedo corresponds moreor-less to “greyness.”

The amount of light reflected is the product of two factors: the albedo of the surface, times a geometric factor that depends on angle.

• A Lambertian surface is “pure matte.” It reflects light equally well in all directions.

Examples of Lambertian surfaces include snow, non-glossy paper, pingpong balls, magnesium oxide, projection screens, $\ldots$

A Lambertian surface looks equally bright from all directions; the amount of light reflected depends only on the angle of incidence.

## 计算机代写|计算机视觉代写Computer Vision代考|SPATIO-TEMPORAL SPECTRAL MODELS

$$F\left(\omega_x, \omega_y, \omega_t\right)=\int_X \int_Y \int_T I(x, y, t) e^{-i\left(\omega_x x+\omega_y y+\omega_t t\right)} d x d y d t$$

$$F\left(\omega_x, \omega_y, \omega_t\right)=\int_X \int_Y \int_T I(x, y, t) e^{-i\left(\omega_x x+\omega_y y+\omega_t t\right)} d x d y d t .$$

$$I(x, y, t)=I\left(x-v_x t_o, y-v_y t_o, t-t_o\right) .$$

$$F\left(\omega_x, \omega_y, \omega_t\right)=e^{-i\left(\omega_x v_x t_o+\omega_y v_y f_o+\omega_t t_o\right)} F\left(\omega_x, \omega_y, \omega_t\right)$$

## 计算机代写|计算机视兴代奇Computer Vision代考|Lambertian and specular surfaces. Reflectance maps

• 表面反照率是指从表面向所有方向重新发射的光源总量的分数。因此，反照率或多或少对应于“灰度”。
反射的光量是两个因嗉的乘积：表面的反照率乘以取决于角度的几何因嗉。
• 朗伯表面是”纯哑光”。它在所有方向上都能㑡好地反射光线。
朗伯表面的例子包括雪、无光泽纸、乒乓球、華化镁、投影屏幕、 $\ldots$
朗伯表面从各个方向看起来都同样明亮; 反射的光量仅取决于入射角。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。