Fourier Transform

https://www.bilibili.com/video/BV1aW4y1y7Hs/?spm_id_from=333.337.search-card.all.click&vd_source=7cf01a4cc731f5d6335bce33c114b60c

在图像处理中，信号指的是图像中的像素值。

在图像处理中，信号频率通常指的是图像中的空间频率。空间频率表示图像中像素值的变化率，即一个图像中不同位置上的像素值如何随空间位置的变化而变化。

  1. 低空间频率：低频分量代表图像中较大区域的平滑变化，例如背景或均匀纹理。这些分量对应于图像中的整体趋势。
  2. 中等空间频率：中等频率分量代表中等大小的特征和纹理，如一些物体的轮廓或边缘。它们通常对应于物体之间的界限和边界。
  3. 高空间频率：高频分量对应于小细节、细小的特征、纹理和边缘。它们表示图像中的急剧变化，通常包含图像的细节信息。

In the spatial frequency domain representation of a Fourier-transformed image, the center of the graph typically represents low spatial frequencies. As you move away from the center towards the edges of the graph, the spatial frequencies increase, and you encounter higher spatial frequencies.

the center often corresponds to the background or low-frequency components of the image. As you move away from the center towards the edges of the graph, you encounter higher spatial frequencies, which correspond to rapidly changing or finer details in the image.

Back Propogation

https://www.zhihu.com/question/27239198

activation function

https://www.youtube.com/watch?v=s-V7gKrsels

https://www.youtube.com/watch?v=68BZ5f7P94E&t=327s

complex image, phase information

Representing an image in the complex number domain is not the typical way images are handled in standard image processing or computer vision applications. However, there are certain situations or specific applications where complex numbers might be used to represent image data.

One common application where complex numbers come into play is in the field of signal processing, particularly when dealing with Fourier Transforms. The Fourier Transform is a mathematical operation that decomposes a signal (which can be an image) into its constituent frequencies. In this context, the use of complex numbers helps capture both the amplitude and phase information of different frequency components.

Here’s a simplified explanation:

  1. Real and Imaginary Parts: In the context of the Fourier Transform, the real part of a complex number represents the amplitude or intensity of a particular frequency component, while the imaginary part represents the phase information.
  2. Phase Information: The phase information is crucial in understanding the spatial relationships and patterns in an image. It tells us about the arrangement of pixels and the transitions between light and dark areas.
  3. Complex Images in Fourier Space: When you perform a Fourier Transform on an image, you get a complex-valued representation. The complex numbers in this representation encode both the amplitude and phase of different spatial frequencies present in the image.

It’s worth noting that in most practical image processing applications, you’ll work with the standard pixel values between 0 and 255, as you mentioned. The use of complex numbers in the Fourier Transform is more of an intermediate step in certain types of analysis rather than the standard representation of an image.

why use Fourier transform

Fourier Transform can be applied to the image to convert it into the frequency domain. The transformed image can then be manipulated in the frequency domain to achieve encryption. Encryption techniques may involve modifying the amplitude or phase information of specific frequency components. This process can make it more challenging for unauthorized users to understand or reconstruct the original image without the proper decryption key.

Transient Image(but we are not going to use this)

Regular Imaging: Imagine taking a photo with a regular camera. It captures a single moment in time, like a snapshot. But what if you wanted to see how things change over a very short time, like a blink of an eye?

Transient Imaging to the Rescue: Transient imaging is like having a special camera that can capture not just one moment but a sequence of very quick moments, almost like a super-fast video. This helps us see how light moves and changes in a scene over a tiny amount of time.

How It Works:

  1. Time-of-Flight Magic: Transient imaging often uses a trick called “time-of-flight.” This means the camera measures how long it takes for light to travel to objects and bounce back. By knowing the time, we can figure out distances.
  2. Ultrafast Speeds: Imagine taking pictures so fast that you can catch even the quickest movements of light. Transient imaging uses super-fast cameras and laser pulses to do just that.

Cool Things It Can Do:

  • Depth Perception: It helps us figure out how far away things are in a scene.
  • Seeing Through Stuff: Transient imaging can sometimes see through objects by capturing light that bounces off hidden things.

Example: Imagine you’re in a room with your eyes closed, and someone quickly turns on and off a flashlight. Regular imaging is like taking a photo with your eyes closed – you won’t see the flashlight. But if you could open your eyes and close them super quickly, like in a sequence, you’d “see” the light turning on and off. That’s a bit like what transient imaging does with light in a scene.

So, transient imaging is like having a super-speedy camera that can reveal the hidden dance of light, helping us understand things happening in the blink of an eye!

Yes, exactly! Transient imaging involves using a camera that can capture a sequence of images extremely quickly, much faster than traditional cameras. These rapid images provide information about how light changes and moves through a scene over a very short period, allowing for unique insights into dynamic processes, depth perception, and even the ability to see through certain objects. So, you can think of transient imaging as a super-fast camera capturing the quick and subtle changes in the way light interacts with the environment.

PSF

Imagine a Flashlight:

  • Think of a small, bright flashlight shining in the dark. In a perfect world, the light from the flashlight would make a perfect, tiny dot on a wall.

Real World isn’t Perfect:

  • In the real world, when you shine that flashlight through a lens or take a picture with a camera, the light doesn’t make a perfect dot. It spreads out a bit, like a soft glow around the dot.

Point Spread Function (PSF):

  • The Point Spread Function (PSF) is like a map that shows how that tiny dot of light spreads out when it goes through a lens or an optical system. It tells us how the light from a point source turns into a bigger shape in the final image.

Why Does it Matter?

  • Understanding the PSF helps us deal with blurriness or fuzziness in images. When we take a photo, we might lose some sharpness because of how the light spreads. Knowing the PSF helps us fix or account for that blurriness.

In Simple Terms:

  • PSF is like the fingerprint of blurriness in pictures. It shows us how a perfect dot of light becomes a little fuzzy spot in the real world of cameras and lenses. Understanding it helps us make images clearer and better!

WHY IT SPREADS OUT IN REAL LIFE?

The spreading out of light, as described by the Point Spread Function (PSF), is a result of various optical phenomena and imperfections in real-world optical systems. Here are a few reasons why light tends to spread out or blur in practical optical systems:

  1. Lens Imperfections:
    • Lenses, which are crucial components in optical systems, are not perfect. They can have imperfections such as spherical aberrations, chromatic aberrations, and distortions. These imperfections cause light rays to converge or diverge slightly, leading to a spread of light.
  2. Diffraction:
    • Diffraction is a phenomenon that occurs when light encounters an obstruction or passes through an aperture. It causes light to spread out as it bends around edges. The size and shape of the aperture or lens affect the degree of diffraction.
  3. Aperture Size:
    • The size of the aperture through which light passes influences the spreading of light. A larger aperture allows more light to enter but can lead to more pronounced diffraction effects, contributing to blurring.
  4. Wave Nature of Light:
    • Light exhibits wave-like behavior. The wave nature of light contributes to phenomena like interference and diffraction, which can result in the spreading out of light as it travels through an optical system.
  5. Aberrations:
    • Various aberrations, including coma and astigmatism, can distort the shape of an image. These aberrations cause light rays from a single point to converge at slightly different locations, contributing to the spread of light.

Understanding and characterizing the PSF help scientists, engineers, and researchers in the fields of optics and image processing to account for and mitigate the blurring effects introduced by these optical phenomena. Techniques like deconvolution can be applied to reverse or reduce the impact of the PSF, improving the sharpness and clarity of images.

convolve PSF with an image

When you convolve an image with its Point Spread Function (PSF), the output is a new image that simulates the blurring or spreading effects that the optical system would introduce. In other words, the convolution with the PSF models how a sharp, idealized image would appear after going through a real-world optical system.

When we say “convolve with PSF” in the context of image processing, we are referring to the process of convolving an image with its Point Spread Function (PSF). This operation is a fundamental step in understanding and modeling the blurring introduced by an optical system.

Here’s a simplified explanation of the process:

  1. Image and PSF:
    • An image captured by an optical system is affected by various factors like lens imperfections, diffraction, and aberrations. The PSF represents how light from a point source is spread out in the image due to these factors.
  2. Convolution:
    • Convolution is a mathematical operation that combines two functions to produce a third. In the context of image processing, convolution with the PSF is performed by sliding the PSF over each pixel of the image and calculating the weighted sum of pixel values based on the PSF at each position.
  3. Blurring Effect:
    • The convolution operation effectively simulates the blurring effect caused by the optical system. The result of the convolution is a new image that incorporates the blurring characteristics of the PSF.
  4. Deconvolution:
    • In some cases, deconvolution is applied to reverse the blurring effect introduced by the convolution. Deconvolution techniques attempt to estimate and undo the impact of the PSF, restoring the image closer to its original state.

The convolution with the PSF is crucial for tasks such as image restoration, deblurring, and understanding the behavior of imaging systems. It is particularly relevant in situations where achieving a sharp and accurate representation of the original scene is essential, such as in medical imaging or astronomy.

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