Basic Steps for Filtering in the Frequency Domain

1. Basic Steps for Filtering in the Frequency Domain Computer Vision Lecture 15: Region Detection

2. Noisy image Noise-cleaned image Fourier spectrum Noise Removal Computer Vision Lecture 15: Region Detection

3. Low Pass Filtering Original Low Pass Butterworth 50% cutoff diameter 10 (left) and 25 Computer Vision Lecture 15: Region Detection

4. High Pass Filtering Original High Pass Butterworth 50% cutoff diameter 10 (left) and 25 Computer Vision Lecture 15: Region Detection

5. Motion Blurring Filter Aerial photo blurred by motion and its spectrum The blur vector and its spectrum Computer Vision Lecture 15: Region Detection

6. Motion Blurring Filter The result of dividing the original spectrum by the motion spectrum and then retransforming Computer Vision Lecture 15: Region Detection

7. Convolution Theorem • Let 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. Computer Vision Lecture 15: Region Detection

8. 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. Computer Vision Lecture 15: Region Detection

9. Region Detection • There 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. Computer Vision Lecture 15: Region Detection

10. 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. Computer Vision Lecture 15: Region Detection

11. 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. Computer Vision Lecture 15: Region Detection

12. 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. Computer Vision Lecture 15: Region Detection

13. Region Detection • A 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, • 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. Computer Vision Lecture 15: Region Detection

14. 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. Computer Vision Lecture 15: Region Detection

15. 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. Computer Vision Lecture 15: Region Detection

16. The Split-and-Merge Algorithm • Sample image to be segmented with  = 1 Computer Vision Lecture 15: Region Detection

17. The Split-and-Merge Algorithm • First split Computer Vision Lecture 15: Region Detection

18. The Split-and-Merge Algorithm • Second split Computer Vision Lecture 15: Region Detection

19. The Split-and-Merge Algorithm • Third split Computer Vision Lecture 15: Region Detection

20. The Split-and-Merge Algorithm • Merge Computer Vision Lecture 15: Region Detection

21. The Split-and-Merge Algorithm • Final result Computer Vision Lecture 15: Region Detection