ECE Digital Image Processing and Introduction to Computer Vision. Outline

Transcription

1 2/9/7 ECE Digital Image Processing and Introduction to Computer Vision Depart. of ECE, NC State University Instructor: Tianfu (Matt) Wu Spring 207. Recap Outline 2. Sharpening Filtering Illustration Foundations: st and 2 nd derivatives Handling noise: unsharp masking The Laplacian, LoG and DoG Combining spatial enhancement methods

2 2/9/7. Recap, Linear Spatial Filtering Linear spatial filtering can be defined w.r.t. either convolution or crosscorrelation. It is a matter of preference to chose one vs the other. In the literature, it is likely to encounter the terms, convolution filter, convolution mask or convolution kernel. As a rule, these terms are used to denote a spatial filter, and not necessarily that the filter will be used for true convolution. 2. g[x, y] = 302 /0. w s, t f[x + s, y + t], as cross-correlation if image origin at left-top = w s, t f[x s, y t], as convolution if image origin at right-bottom /0.. Recap, Dealing with image boundary w w full w same valid w w w f f f w w w w w w shape = full : output size is sum of sizes of f and w shape = same : output size is same as f shape = valid : output size is difference of sizes of f and w Extrapolate the missing pixel values: clip filter (black), wrap around, copy edge, or reflect across edge, or mean value computed from a training dataset. Source: S. Lazebnik 2

3 2/9/7. Recap, Linear Filters vs Order-Statistic (Nonlinear) Filters Gaussian smoothing Median filtering: robustness to outliers Source: K. Grauman 2. Sharpening Filters Blur Original Sharpening is to highlight transitions in intensity Source: D. Lowe 3

4 2/9/7 An example 2. Sharpening Filters Source: D. Lowe 2. Sharpening Filters Blur by averaging (integration) Original Sharpening by differentiation Source: D. Lowe 4

5 2/9/7 An Interesting Illustration Hybrid images Gaussian Filter Laplacian Filter A. Oliva, A. Torralba, P.G. Schyns, Hybrid Images, SIGGRAPH 2006 First-order derivatives Foundation Consider a -d function f x, we have, Continuous: 67(9) A>9 7(9) = lim f (x) >9 Discrete: 67(9) f x + f(x), approximated 69 by finite difference Source: DIP 3 rd edition 5

6 2/9/7 First-order derivatives Foundation Consider a 2-d function f x, y, we have, Continuous: F7(9,G) F9 F7(9,G) FG 7 9A>9,G 7(9,G) = lim >9 7 9,GA>G 7(9,G) = lim >G Discrete: F7(9,G) f x +, y f(x, y) F9 f(x, y) f x, y + f(x, y) y First-order derivatives Foundation f ( x, y) x f ( x, y) y - - or - Which shows changes with respect to x? Source: K. Grauman 6

7 2/9/7 First-order derivatives Foundation Other approximations of derivative filters exist Source: K. Grauman Foundation The Gradient of an image f = F7 9,G F9 F7 9,G FG The gradient magnitude: mag f = f x, y x L + f x, y y L 7

8 2/9/7 Foundation Second-order derivatives Consider a -d function f x, we have, Continuous: 6N 7(9) 7 = lim O 9A>9 7 O (9) f (x) 69 N >9 Discrete: 6N 7(9) 69 N f x 2f + f(x + ), approximated by finite difference based on Taylor series expansion (how?) -2 Source: DIP 3 rd edition Foundation Second-order derivatives Source: Robert Collins 8

9 2/9/7 Foundation First- and Second-order derivatives Any definition of a first derivative () must be zero in areas of constant intensity; (2) must be nonzero at the onset of an intensity step or ramp; and (3) must be nonzero along ramps. Any definition of a second derivative () must be zero in constant areas; (2) must be nonzero at the onset and end of an intensity step or ramp; and (3) must be zero along ramps of constant slope. The Laplacian The Laplacian for a function f(x, y) is defined as, L f = L f x L + L f y L Isotropic(rotation invariant): filter response is independent of the direction of the discontinuities in the image to which the filter is applied. Linear The discrete Laplacian: L f = f x +, y + f x, y + f x, y + + f x, y 4f(x, y) 9

10 2/9/7 Different filter kernels for L f The Laplacian isotropic results in increments of 90 isotropic results in increments of 45 Different filter kernels for L f The Laplacian Source: Robert Collins 0

11 2/9/7 c = The Laplacian for Image Sharpening g x, y = f x, y + c[ L f(x, y)] c = + Effects of Noise Source: S. Seitz

12 2/9/7 Unsharp Masking and Highboost Filtering Unsharp Masking and Highboost Filtering = original smoothed (5x5) detail + α = original detail sharpened Source: S. Lazebnik 2

13 2/9/7 Unsharp Masking and Highboost Filtering f + a( f - f * g) = ( + a) f -a f * g = f *(( + a) e - g) image blurred image unit impulse (identity) Source: S. Lazebnik Laplacian of Gaussian f + a( f - f * g) = ( + a) f -a f * g = f *(( + a) e - g) image blurred image unit impulse (identity) unit impulse Gaussian Laplacian of Gaussian Source: S. Lazebnik 3

14 2/9/7 Laplacian of Gaussian (LoG) L f g = f ( L g) Laplacian of Gaussianfiltered image LoG -filtered image g x = 9 N 2πσ L e LZ N -D Gaussian and Derivatives g [ x (x) = σ L 2πσ L e 9 N LZ N g [[ (x) = 2πσ L (xl σ \ σ L)e 9 N LZ N Laplacian of Gaussian (LoG) 2-D LoG g x, y = 9 N AG N 2πσ L e LZ N Derivatives: x-direction y-direction LoG x, y = πσ \ ( xl + y L 2σ L )e 9 N AG N LZ N Mexican Hat 4

15 2/9/7 Laplacian of Gaussian (LoG) Band-pass filter Laplacian of Gaussian (LoG) Efficient Implementation by Difference of Gaussians (DoG) at different scales Why is it efficient? 5

16 2/9/7 Combining Spatial Enhancement Methods A task will often require application of several complementary methods in order to achieve an desirable result. Intensity Transformation (e.g., log and power-law) Histogram equalization and matching Smoothing for denosing Gradient for enhancing edges Sharpening by Laplacian (a) is a nuclear whole body bone scan, used to detect diseases such as bone infection and tumors. Our objective is to enhance this image by sharpening it and by bringing out more of the skeletal detail. (b) Laplacian (c) Sharpened by adding (a) and (b) (d) Sobel gradient of (a) 6

17 2/9/7 (e) Sobel image smoothed with 5 by 5 averaging filter (f) Mask image formed by the product of the sharpened image (c) and (e) (g) Sharpened image obtained by the sum of the original image and (f) (h) Final result obtained by applying a power- law transformation to (g) Summary Sharpening Filtering Illustration Foundations: st and 2nd derivatives Handling noise: unsharp masking The Laplacian, LoG and DoG Combining spatial enhancement methods 7