More Linear Processing - Lecture Notes | CAP 5415, Study notes of Computer Science

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CAP 5415: Computer Vision

Lecture 2

More Linear Processing

Today

• We’ll revisit convolutions and Fourier Transforms
• Today is the day to get a solid understanding
• Don’t let me go on if you don’t understand something

Convolution Revisited

• Consider this set-up

Laser Pointer

Wall

Convolution Revisited

• You see a dot

Laser Pointer

Wall

Kernel

• The kernel is sometimes referred to as a point-spread function
• Relates how an input at one point affects our measurements
• Laser beam -> Frosted Glass -> Blurry dot

Convolution Revisited

• What if there were many laser pointers
• How would we calculate what we see?

Wall

Cloth or Frosted Glass

Resulting Image Input Image^

Point Spread Function

Back to the equation

• Remember K relates how an input at one point affects the rest of the image
• This equation tells us how to compute the result

Practical Example

• The value at the center is 1i + 2h + 3g + … + 7c +8b+9a

a b c d e f

* g h i

More General Linear

Transformations

• Now that we can filter images, is there a good way to analyze what the filtering is doing?

On to the Fourier Transform

• Think of this transform as a function

Transform^ Fourier

256x256 Real-ValuedImage 256x256 Complex-ValuedTransformed Image

A simple example of the value of

transformations

• Let's consider some simple data
• (Handout)

A Short Linear Algebra Review

• Can also think of this in terms of orthogonal projection
• In 2D, we can express a vector in terms of the sum of two orthogonal vectors

v - c b 1

c b 1

v

Another way of thinking of a dot-

product

• So, if the vector has length 1 (unit length), we can think of the dot product measuring how much of v lies along b 1

v - c b 1

c b 1

v

Links to the Fourier Transform

• The handout examined points in 2D
• We can think of images as points in N -D
  • Raster-scan images into a vector
• The Fourier Transform will essentially re- project the image onto a new basis
• Will give a different way of looking at the image.