Image Features. Work on project 1. All is Vanity, by C. Allan Gilbert,

Transcription

1 Image Features Work on project 1 All is Vanity, by C. Allan Gilbert,

2 Feature extrac*on: Corners and blobs c

3 Mo*va*on: Automa*c panoramas Credit: Ma9 Brown

4 Why extract features? Mo*va*on: panorama s*tching We have two images how do we combine them?

5 Why extract features? Mo*va*on: panorama s*tching We have two images how do we combine them? Step 1: extract features Step 2: match features

6 Why extract features? Mo*va*on: panorama s*tching We have two images how do we combine them? Step 1: extract features Step 2: match features Step 3: align images

7 Harder still? NASA Mars Rover images

8 Answer below (look for tiny colored squares ) NASA Mars Rover images with SIFT feature matches Figure by Noah Snavely

9 More motivation Feature points are used for: Image alignment (e.g., panoramas) 3D reconstruction Motion tracking Object recognition Indexing and database retrieval Robot navigation many others

10 Ideal property Suppose your feature detector always extracts 100 features from an image Ideal property of your detector?

11 Challenges: Invariance Find features that are invariant to transformations geometric invariance: translation, rotation, scale photometric invariance: brightness, exposure,

12 Two Problems for Features Feature detection

13 Two Problems for Features Feature detection Feature descriptor

14 Two Problems for Features Feature detection Feature descriptor

15 What makes a good feature? Zoom-in demo

16 Want uniqueness Look for unusual image regions Lead to unambiguous matches in other images How to define unusual?

17 Local measures of uniqueness Consider a small window of pixels Where are features good and bad? Slide adapted from Darya Frolova, Denis Simakov, Weizmann Institute.

18 Local measures of uniqueness Consider a small window of pixels Where are features good and bad? Slide adapted from Darya Frolova, Denis Simakov, Weizmann Institute.

19 Feature detection Uniqueness = How does it change when shifted by a small amount? flat region: no change in all directions edge : no change along the edge direction corner : significant change in all directions Slide adapted from Darya Frolova, Denis Simakov, Weizmann Institute.

20 Feature detection Define E(u,v) = amount of change when you shift the window by (u,v) E(u,v) is small for all shifts E(u,v) is small for some shifts E(u,v) is small for no shifts We want to be

21 Feature detection: the math Consider shifting the window W by (u,v) how do the pixels in W change? compare each pixel before and after by Sum of the Squared Differences (SSD) this defines an SSD error E(u,v): W

22 Small motion assumption Taylor Series expansion of I: If the motion (u,v) is small, then first order approx is good Plugging this into the formula on the previous slide

23 Feature detection: the math Consider shifting the window W by (u,v) how do the pixels in W change? compare each pixel before and after by summing up the squared differences this defines an error of E(u,v): W

24 Feature detection: the math This can be rewritten:

25 Feature detection: the math This can be rewritten: Which [u v] maximizes E(u,v)? Which [u v] minimizes E(u,v)?

26 Feature detection: the math This can be rewritten: Which [u v] maximizes E(u,v)? Which [u v] minimizes E(u,v)?

27 Feature detection: the math This can be rewritten: x - x + Eigenvector Eigenvector x + x - with the largest eigen value? with the smallest eigen value?

28 Quick eigenvalue/eigenvector review The eigenvectors of a matrix A are the vectors x that satisfy: The scalar λ is the eigenvalue corresponding to x The eigenvalues are found by solving: In our case, A = H is a 2x2 matrix, so we have The solution:

29 Feature detection Local measure of feature uniqueness E(u,v) = amount of change when you shift the window by (u,v) E(u,v) is small for all shifts We want E(u,v) is small for some shifts to be large = E(u,v) is small for no shifts

30 Eigenvalues of H

31 Feature detection summary Here s what you do Compute the gradient at each point in the image Create the H matrix from the entries in the gradient Compute the eigenvalues. Find points with large response (λ - > threshold) Choose those points where λ - is a local maximum as features

32 Feature detection summary Here s what you do Compute the gradient at each point in the image Create the H matrix from the entries in the gradient Compute the eigenvalues. Find points with large response (λ - > threshold) Choose those points where λ - is a local maximum as features Called non-local max suppression

33 The Harris operator λ - is a variant of the Harris operator for feature detection f Flat

34 The Harris operator λ - is a variant of the Harris operator for feature detection f Flat

35 The Harris operator λ - is a variant of the Harris operator for feature detection f Flat?

36 The Harris operator λ - is a variant of the Harris operator for feature detection f Flat Edge

37 The Harris operator λ - is a variant of the Harris operator for feature detection f Flat Edge?

38 The Harris operator λ - is a variant of the Harris operator for feature detection f Flat Edge Corner

39 The Harris operator λ - is a variant of the Harris operator for feature detection f Flat Edge Corner?

40 The Harris operator λ - is a variant of the Harris operator for feature detection f Flat Edge Corner Strong corner

41 The Harris operator λ - is a variant of the Harris operator for feature detection The trace is the sum of the diagonals, i.e., trace(h) = h 11 + h 22 Very similar to λ - but less expensive (no square root) Called the Harris Corner Detector or Harris Operator Lots of other detectors, this is one of the most popular

42 The Harris operator Harris operator

43 Harris detector example

44 f value (red high, blue low)

45 Threshold (f > value)

46 Find local maxima of f

47 Harris features (in red)

48 Invariance Suppose you rotate the image by some angle Will you still pick up the same features? What if you change the brightness? Scale?

49 Scale invariant detection Suppose you re looking for corners Key idea: find scale that gives local maximum of f f is a local maximum in both position and scale

50 Lindeberg et et al, al., 1996 Slide from Tinne Tuytelaars

51

52

53

54

55

56

57 Two Problems for Features Feature detection Feature descriptor

58 Feature descriptors We know how to detect good points Next question: How to match them??

59 Feature descriptors We know how to detect good points Next question: How to match them?? Lots of possibilities (this is a popular research area) Simple option: Sum of squared differences State of the art approach: SIFT David Lowe, UBC

60 The most popular descriptor SIFT 60

61 Scale Invariant Feature Transform Histograms of gradient directions over spatial regions

62 Scale Invariant Feature Transform Basic idea: Take 16x16 square window around detected feature Compute edge orientation (angle of the gradient - 90 ) for each pixel Throw out weak edges (threshold gradient magnitude) Create histogram of surviving edge orientations 0 2π angle histogram Adapted from slide by David Lowe

63 SIFT descriptor Full version Divide the 16x16 window into a 4x4 grid of cells (2x2 case shown below) Compute an orientation histogram for each cell 16 cells * 8 orientations = 128 dimensional descriptor Adapted from slide by David Lowe

64 Properties of SIFT Extraordinarily robust matching technique Handle viewpoint changes Up to about 60 degree out of plane rotation Handle significant illumination changes in illumination Sometimes even day vs. night (below) Fast and efficient can run in real time Lots of code available

65 Invariance (for descriptors) Compare two images I 1 and I 2 I 2 may be a transformed version of I 1 What kinds of transformations are likely in practice?

66 Invariance (for descriptors) Compare two images I 1 and I 2 I 2 may be a transformed version of I 1 What kinds of transformations are likely in practice? In practice Limited 3D rotations (up to about 60 degrees) Limited affine transformations (some are fully affine invariant) Limited illumination/contrast changes

67 Rotation invariance Find dominant orientation This is given by x +, the eigenvector of H corresponding to λ + λ + is the larger eigenvalue Rotate the window according to this angle Figure by Matthew Brown

68 Scale Invariance Automatic scale selection Detect at multiple scales 68

69 Scale Invariance Automatic scale selection Detect at multiple scales 69

70 Scale Invariance Automatic scale selection Detect at multiple scales 70

71 Next feature matching 71