Problem Session #6. EE368/CS232 Digital Image Processing

Transcription

1 Problem Session #6 EE368/CS232 Digital Image Processing

2 . Robustness of Harris Keypoints to Rotation and Scaling Part A: Apply a Harris corner detector and threshold the cornerness response so that about Harris corners are detected. Please submit a result showing the detected Harris corners superimposed on the original image, and report the threshold that you choose. Describe which objects or regions in the image seem to generate large numbers of Harris corners. Hint: Use the MATLAB functions cornermetric and imregionalmax.

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4 Part B: Rotate the image in increments of 5 degrees, from degrees to 36 degrees. For each rotated image, compute Harris corners using the same settings that you chose in part (a). Then, compute repeatability as follows. Set number of feature matches to be. For each Harris corner at [x, y] in the original image: Predict the position [x r, y r ] where [x, y] should appear in the rotated image. Search for a nearby Harris corner in the rotated image (coordinates [x o, y o ]) satisfying x o - x r < 2 and y o - y r < 2. If such an [x o, y o ] is found, increment number of feature matches by. Compute repeatability as (number of feature matches) / (number of Harris corners in the original image). Plot repeatability against rotation angle and comment on the Harris corner detector s robustness against rotation. imgrot = imrotate(img, angle, bicubic );

5 Part B: A detected keypoint [x, y] in original image. Anti-clockwise rotation Predicted keypoint [x, y ] in the rotated image. Predicted keypoint coordinates Center of rotated image Rotation matrix Detected keypoint in the original image Center of original image

6 Robustness to Rotation: Cover-.jpg Robustness to Rotation: Cover-2.jpg Rotation Angle (degrees) Rotation Angle (degrees)

7 Part C: Conduct an experiment analogous to part (b), but, instead of rotating the image, resize the image by the scaling factors m, m, m 2,, m 8, where m =.2. Compute Harris corners for each resized image. Please perform bicubic interpolation with MATLAB function imresize. By comparing the Harris corners of the original image and the corresponding Harris corners of the resized image, compute and plot repeatability against scaling factor (log scale may be most appropriate), and comment on the Harris corner detector s robustness against scale changes. imgscale = imresize(img, scale, bicubic ); Robustness to Scaling: Cover-.jpg Robustness to Scaling: Cover-2.jpg Rescaling Factor Rescaling Factor

8 2. Robustness of SIFT Keypoints to Rotation and Scaling Part A: Apply a SIFT keypoint detector and adjust the peak and edge thresholds so that about SIFT keypoints are detected. Please submit a result showing the detected SIFT keypoints superimposed on the original image, and report the thresholds you choose. Describe which objects or regions in the image seem to generate large numbers of SIFT keypoints. % Set up MATLAB interface to VLFeat library run('/path/to/library/vlfeat-.9.7/toolbox/vl_setup.m'); % Extract SIFT keypoints on grayscale image peakthresh = 4; edgethresh = ; [fc, dc] = vl_sift(single(gray_img),... 'PeakThresh', peakthresh,... 'EdgeThresh', edgethresh);

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11 Original Image: Cover-.jpg

12 Original Image: Cover-2.jpg

13 Part B: Using the procedure defined in Problem 2, plot repeatability versus rotation angle (in increments of 5 degrees, from degrees to 36 degrees). Comment on the SIFT keypoint detector s robustness against rotation and compare against the corresponding result for the Harris keypoint detector. SIFT Keypoints

14 SIFT Keypoints Robustness to Rotation: Cover-.jpg Robustness to Rotation: Cover-2.jpg.8.8 Harris Keypoints Rotation Angle (degrees) Rotation Angle (degrees)

15 Part C: Using the same procedure defined in Problem 2, plot repeatability versus scaling factor (with the scaling factors m, m, m 2,, m 8, where m =.2). Comment on the SIFT keypoint detector s robustness against scale changes and compare against the corresponding result for the Harris keypoint detector. Robustness to Scaling: Cover-.jpg Robustness to Scaling: Cover-2.jpg.8.8 SIFT Keypoints Rescaling Factor Rescaling Factor

16 Robustness to Scaling: Cover-.jpg Robustness to Scaling: Cover-2.jpg.8.8 SIFT Keypoints Rescaling Factor Rescaling Factor Robustness to Scaling: Cover-.jpg Robustness to Scaling: Cover-2.jpg.8.8 Harris Keypoints Rescaling Factor Rescaling Factor

17 3. Scale Selection for Determinant-of-Hessian Consider a continuous-space version of the determinant-of-hessian (DoH) keypoint detector. For an input image, the DoH response at scale t is defined as follows: ( ) 2 det H t ( xy, ) = f t ( xyf, ) t ( xy, ) - f t ( xy, ) xx yy xy 2 t t fxx ( x, y) = g ( x, y) * f( x, y) 2 x 2 t t fyy ( x, y) = g ( x, y) * f( x, y) 2 y 2 t t fxy ( x, y) = g ( x, y) * f( x, y) yx 2 2 æ (, ) exp x + y ö g t x y = ç - 2pt è 2t ø

18 Part A: Assume the input image is a Gaussian blob of scale t > : f (x, y) = 2πt g t (x, y). From the above definitions, derive and report an expression for the following objective functions: t H ( t ) = det H (,) A t H ( t) = tdet H (,) B H t t 2 t C () = det H (,) t t t+ t ( )* = p ( )* = p t g x, y f( x, y) 2 t g x, y g ( x, y) 2 t g ( x, y) ( ) = 2πt 2 f t xx (x, y) = 2 g t ( x, y) f (x, y) x 2 t 2 f yy (x, y) = 2πt y g t+t 2 ( x, y) f t xy (x, y) = f t 2 yx (x, y) = 2πt x y g t+t det H t (,) = f xx t t (,) f yy t (,) f xy (,) ( ) 2 x 2 g t+t ( x, y) ( x, y)

19 Part B: Derive and report the scale values t A *, t B *, and t C * that maximize H A (t), H B (t), and H C (t), respectively. Also report the maximal values. Please interpret the results. t A * =... t B * =... Are H A, H B, and H c suitable for finding the scale t? t C * =... H A t A * H B t B * H C t C * ( ) =... ( ) =... ( ) =... Do maximum values of H A, H B, and H c depend on the scale t?