Image Processing II

Signal Processing in Computer Vision 복습. 카메라 팀에서는 이게 기본이구나. Radiometry, Stereo Depth 물어볼 줄 알았었는데. 아니었음. 이 짱짱 교수님 강의 정리해봄: https://www.youtube.com/watch?v=m_11ntjkn4k&list=PL2zRqk16wsdorCSZ5GWZQr1EMWXs2TDeu&index=8

Fourier Transform.

Do you know Fourier Transform? What are the properties of Fourier Transform?

If you have a periodic function, it can be expressed as a sum of sinusoids without loss of information. It allows a signal in spatial domain to spectral (frequency, fourier) domain. In other words, it expresses a signal with frequency representation (amplitude and phase).

Some summary:

• Forward Fourier Transform (FT) reprsents a signal f(x) in terms of Amplitudes and Phases of its Constituent Sinusoids
• Inverse Fourier Transform (IFT) computes the signal (Fx) from the Amplitudes and Phases of its Constituent Sinusoids
• F(u) holds Amplitudes and Phases of the Constituent Sinusoids

Properties of Fourier Transform

Linearity: Linear combination of two functions in spatial domain can be kept the same in their frequency domain

Scaling: Inverse scaling in frequency domain (strecting in spatial domain will become shrinking in frequency domain and vice versa)

Shifting: 뭐라 설명해야하지.. 흠.. 다시 Forward FT를 할 필요는 없고 상수만 곱해주면됨. 

Differentiation: 밑에 나와있음. 편함. multiply (i2𝛑u)^n

Convolution Theorem

What is a convolution operationa nd how is it related to Fourier Transform?

Convolution of two functions is integral of two functions' product.

Convolution of two functions in Spatial Domain is a product of Fourier Transform functions in Frequency Domain. So you can do convolution by 1) Forward Fourier Transform 2) Product 3) Inverse Fourier Transform. Why is this important? Convolution of very big kernels and masks can be done much more efficiently in a frequency domain (there are very efficient FT & IFT algorithms). 

Also Convolution's Fourier Transform is easier to interpret what actually filters do. Ex) Low pass filtering of Gaussian Kernel

Sampling Theorem

Reduction of a continuous time signal to a discrete time signal. Signals are continuous in the world, but the systems understand in a discrete manner, so we need this. In image processing, camera's image sensor converts (captures) a continuous optical image to a discrete digital image. But how do you sample so that you capture important information without undesirable noise and artifacts? How "dense" should the samples be? If it’s undersampled, there’s an Aliasing issue, where significant information is lost. (could introduce new frequency and artifact) -> sampling theorem

Why sampling is important

How do we sample?

We multiply Sha Function (Impulse Train) to a Continuous Signal. Fourier Transform of Shah Function (Impulse Train) is also Shah Function (Impulse Train).

What is sampling (multiplication of Shah Function to a signal) doing in Frequency Domain?

It is convolving the Shah Function with the input signal in Frequency Domain. I basically since Shah Function is Impulse Train and delta function is an identity function, you are copying the original function.

From this Fourier Analysis, we can notice that "A signal can be represented in its samples and can be recovered back when sampling frequency is greater or equal to twice of maximum frequency component (Umax) present in the signal."

What is Nyquist Theorem?

Nyquist frequency is 1/2x0. If the maximum frequency component is less than Nyquist frequency, you can recover back from the samples. <-> Aliasing occurs when a scene image (signal) has frequencies above the image sensor's Nyquist Frequency.

How to combat Aliasing?

You can clip the signal above the Nyquist frequency. Effectively, "blur" the scene before sampling.

Use box filtering or anti-aliasing optical filter