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# Basic Steps for Filtering in the Frequency Domain

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Basic Steps for Filtering in the Frequency Domain. Noisy image. Noise-cleaned image. Fourier spectrum. Noise Removal. Low Pass Filtering. Original. Low Pass Butterworth 50% cutoff diameter 10 (left) and 25. High Pass Filtering. Original. High Pass Butterworth - PowerPoint PPT Presentation
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November 5, 2013 Computer Vision Lecture 15: Region 1 Basic Steps for Filtering in the Frequency Domain
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November 5, 2013 Computer Vision Lecture 15: Region Detection

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Basic Steps for Filtering in the Frequency Domain

November 5, 2013 Computer Vision Lecture 15: Region Detection

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Noise Removal

Fourier spectrum

Noisy image

Noise-cleaned image

November 5, 2013 Computer Vision Lecture 15: Region Detection

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Low Pass FilteringOriginal

Low Pass Butterworth 50% cutoff diameter 10 (left) and 25

November 5, 2013 Computer Vision Lecture 15: Region Detection

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High Pass FilteringOriginal

High Pass Butterworth 50% cutoff diameter 10 (left) and 25

November 5, 2013 Computer Vision Lecture 15: Region Detection

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Motion Blurring Filter

Aerial photo blurred by motion and its spectrum

The blur vector andits spectrum

November 5, 2013 Computer Vision Lecture 15: Region Detection

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Motion Blurring Filter

The result of dividing the original spectrum by the motion spectrum and then retransforming

November 5, 2013 Computer Vision Lecture 15: Region Detection

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Convolution TheoremLet F {.} denote the application of the Fourier transform and * denote convolution (as usual). Then we have:

F {(f*h)(x, y)} = F(u, v) H(u, v) and

F {f(x,y) h(x, y)} = (F*H)(u, v),

where F and H are the Fourier transformed images f and h, respectively.This means that instead of computing the convolution directly, we can Fourier transform f and h, multiply them, and then transform them back. In other words, a convolution in the space domain corresponds to a multiplication in the frequency domain, and vice versa.

November 5, 2013 Computer Vision Lecture 15: Region Detection

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Demo Website

I highly recommend taking a look at this website:

http://users.ecs.soton.ac.uk/msn/book/new_demo/

It has nice interactive demonstrations of the Fourier transform, the Hough transform, edge detection, and many other useful operations.

November 5, 2013 Computer Vision Lecture 15: Region Detection

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Region DetectionThere are two basic – and often complementary – approaches to segmenting an image into individual objects or parts of objects: region-based segmentation and boundary estimation.Region-based segmentation is based on region detection, which we will discuss in this lecture.Boundary estimation is based on edge detection, which we already discussed earlier.

November 5, 2013 Computer Vision Lecture 15: Region Detection

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Region Detection

We have already seen the simplest kind of region detection.It is the labeling of connected components in binary images.Of course, in general, region detection is not that simple. Successful region detection through component labeling requires that we can determine an intensity threshold in such a way that all objects consist of 1-pixels and do not touch each other.

November 5, 2013 Computer Vision Lecture 15: Region Detection

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Region Detection

We will develop methods that can do a better job at finding regions in real-world images.In our discussion we will first address the question of how to segment an image into regions.Afterwards, we will look at different ways to represent the regions that we detected.

November 5, 2013 Computer Vision Lecture 15: Region Detection

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Region Detection

How shall we define regions?The basic idea is that within the same region the intensity, texture, or other features do not change abruptly.Between adjacent regions we do find such a change in at least one feature.Let us now formalize the idea of partitioning an image into a set of regions.

November 5, 2013 Computer Vision Lecture 15: Region Detection

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Region DetectionA partition S divides an image I into a set of n regions Ri. Regions are sets of connected pixels meeting three requirements:

• The union of regions includes all pixels in the image,IR

n

ii

1

• Each region Ri is homogeneous, i.e., satisfies a homogeneity predicate P so that P(Ri) = True.

• The union of two adjacent regions Ri and Rj never satisfies the homogeneity predicate, i.e., P(Ri Rj) = False.

November 5, 2013 Computer Vision Lecture 15: Region Detection

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Region Detection

The homogeneity predicate could be defined as, for example, the maximum difference in intensity values between two pixels being no greater than a some threshold .

Usually, however, the predicate will be more complex and include other features such as texture.

Also, the parameters of the predicate such as may be adapted to the properties of the image.

Let us take a look at the split-and-merge algorithm of image segmentation.

November 5, 2013 Computer Vision Lecture 15: Region Detection

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The Split-and-Merge Algorithm

First, we perform splitting:• At the start of the algorithm, the entire image is

considered as the candidate region.• If the candidate region does not meet the

homogeneity criterion, we split it into four smaller candidate regions.

• This is repeated until there are no candidate regions to be split anymore.

Then, we perform merging:• Check all pairs of neighboring regions and merge

them if it does not violate the homogeneity criterion.

November 5, 2013 Computer Vision Lecture 15: Region Detection

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The Split-and-Merge Algorithm

Sample image to be segmented with = 1

1 1 1 1 1 1 1 21 1 1 1 1 1 1 03 1 4 9 9 8 1 01 1 8 8 8 4 1 01 1 6 6 6 3 1 01 1 5 6 6 3 1 01 1 5 6 6 2 1 01 1 1 1 1 1 0 0

November 5, 2013 Computer Vision Lecture 15: Region Detection

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The Split-and-Merge Algorithm

First split

1 1 1 1 1 1 1 21 1 1 1 1 1 1 03 1 4 9 9 8 1 01 1 8 8 8 4 1 01 1 6 6 6 3 1 01 1 5 6 6 3 1 01 1 5 6 6 2 1 01 1 1 1 1 1 0 0

November 5, 2013 Computer Vision Lecture 15: Region Detection

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The Split-and-Merge Algorithm

Second split

1 1 1 1 1 1 1 21 1 1 1 1 1 1 03 1 4 9 9 8 1 01 1 8 8 8 4 1 01 1 6 6 6 3 1 01 1 5 6 6 3 1 01 1 5 6 6 2 1 01 1 1 1 1 1 0 0

November 5, 2013 Computer Vision Lecture 15: Region Detection

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The Split-and-Merge Algorithm

Third split

1 1 1 1 1 1 1 21 1 1 1 1 1 1 03 1 4 9 9 8 1 01 1 8 8 8 4 1 01 1 6 6 6 3 1 01 1 5 6 6 3 1 01 1 5 6 6 2 1 01 1 1 1 1 1 0 0

November 5, 2013 Computer Vision Lecture 15: Region Detection

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The Split-and-Merge Algorithm

Merge

1 1 1 1 1 1 1 21 1 1 1 1 1 1 03 1 4 9 9 8 1 01 1 8 8 8 4 1 01 1 6 6 6 3 1 01 1 5 6 6 3 1 01 1 5 6 6 2 1 01 1 1 1 1 1 0 0

November 5, 2013 Computer Vision Lecture 15: Region Detection

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The Split-and-Merge Algorithm

Final result

1 1 1 1 1 1 1 21 1 1 1 1 1 1 03 1 4 9 9 8 1 01 1 8 8 8 4 1 01 1 6 6 6 3 1 01 1 5 6 6 3 1 01 1 5 6 6 2 1 01 1 1 1 1 1 0 0

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