Filtering in frequency domain

Transcription

1 Filtering in frequency domain FFT FFT = Inverse FFT

2 Filtering in frequency domain Can be faster than filtering in spatial domain (for large filters) Can help understand effect of filter Algorithm: 1. Convert image and filter to fft (fft2 in matlab) 2. Pointwise-multiply ffts 3. Convert result to spatial domain with ifft2

3 Filtering in spatial domain * =

4 Filtering in spatial domain Slide filter over image and take dot product at each position Remember linearity (for linear filters) Examples 2D: [1 0 0 ; ; 0 0 1]/4

5 Sinc Filter What is the spatial representation of the hard cutoff in the frequency domain? Frequency Domain Spatial Domain

6 LPF HPF frequency domain spatial domain

7 Filtering with the FFT in Matlab (from previous lecture) im = double(imread('baboon1.bmp'))/255; im = rgb2gray(im); % im should be a gray-scale floating point image figure; imshow(im); [imh, imw] = size(im); hs = 50; % filter half-size fil = fspecial('gaussian', hs*2+1, 10); figure; bar3(fil); fftsize = 1024; % should be order of 2 (for speed) and include padding im_fft = fft2(im, fftsize, fftsize); % 1) fft im with padding figure(); imagesc(log(abs(fftshift(im_fft)))), axis image, colormap jet fil_fft = fft2(fil, fftsize, fftsize); image % 2) fft fil, pad to same size as figure(); imagesc(log(abs(fftshift(fil_fft)))), axis image, colormap jet im_fil_fft = im_fft.* fil_fft; % 3) multiply fft images im_fil = ifft2(im_fil_fft); % 4) inverse fft2 im_fil = im_fil(1+hs:size(im,1)+hs, 1+hs:size(im, 2)+hs); % 5) remove padding figure; imshow(im_fil);

8 image/video enhancement image/video restoration (contrast, sharpness, color) (noise reduction) - remove irrelevant information - increase spatio-temporal correlation - improve compression

9 Restoration vs. Enhancement Restoration: Undo or invert a mathematical model of the degradation to obtain the ideal image. Some serious image degradations cannot be accounted for by simple filtering techniques unless the degradation is correctly modeled and addressed in a principled way. Enhancement: Produce a more pleasing image without using a particular model of the degradation, but possibly using a model of human perception. contrast improvement, sharpening, etc.

10 Image Restoration: An Example The power of knowing and using the degradation model (From Gonzalez & Woods, Digital Image Processing)

11 Noise: random unwanted portion of signal/data without meaning; an unwanted by-product of other activities. Image noise: random variation of brightness or color in images; usually an aspect of electronic noise. Gaussian noise: caused primarily by thermal noise in amplifier circuitry of camera Salt and Pepper noise: caused by analog-todigital converter errors, bit errors in transmission Shot noise: variation in the number of photons sensed at a given exposure level Film grain: due to the grain of photographic film

12 Salt and pepper noise: contains random occurrences of black and white pixels Impulse noise: contains random occurrences of white pixels Gaussian noise: variations in intensity drawn from a Gaussian normal distribution CMSC 733 Computer Processing of Pictorial Information

13 Noise Characterization The level of noise is often measured in terms of SNR. It may be impossible to distinguish genuine image details from variations of the image intensity due to noise. Often both will be characterized by statistical models. Noise models: Additive/multiplicative, signaldependent/signal-independent, white/colored, Gaussian/other distribution, etc. A simple additive model: g(x,y,t) = f(x,y,t) + w(x,y,t)

14 Noise Filters linear filters - mean, average, box filters - Gaussian filters order-statistic filters - median filters multiresolution filters

15 Noise Reduction Gaussian Filter Additive Gaussian Noise

16 Reducing Gaussian noise Smoothing with larger standard deviations suppresses noise, but also blurs the image Source: S. Lazebnik

17 Reducing salt-and-pepper noise by Gaussian smoothing 3x3 5x5 7x7

18 Alternative idea: Median filtering A median filter operates over a window by selecting the median intensity in the window Is median filtering linear? Source: K. Grauman

19 Median filter What advantage does median filtering have over Gaussian filtering? Robustness to outliers Source: K. Grauman

20 Median filter Salt-and-pepper noise Median filtered MATLAB: medfilt2(image, [h w]) Source: M. Hebert

21 Median vs. Gaussian filtering 3x3 5x5 7x7 Gaussian Median

22 Other non-linear filters Weighted median (pixels further from center count less) Clipped mean (average, ignoring few brightest and darkest pixels) Bilateral filtering (weight by spatial distance and intensity difference) Bilateral filtering Image:

23 Compression How is it that a 4MP image can be compressed to a few hundred KB without a noticeable change?

24 Lossy Image Compression (JPEG) Block-based Discrete Cosine Transform (DCT) Slides: Efros

25 Using DCT in JPEG The first coefficient B(0,0) is the DC component, the average intensity The top-left coeffs represent low frequencies, the bottom right high frequencies

26 Image compression using DCT Quantize More coarsely for high frequencies (which also tend to have smaller values) Many quantized high frequency values will be zero Encode Filter responses Can decode with inverse dct Quantization table Quantized values

27 JPEG Compression Summary 1. Convert image to YCrCb 2. Subsample color by factor of 2 People have bad resolution for color 3. Split into blocks (8x8, typically), subtract For each block a. Compute DCT coefficients b. Coarsely quantize Many high frequency components will become zero c. Encode (e.g., with Huffman coding)

28 Detecting Features in an Image - Edges - Lines - Points - Patches Source: D. Lowe

29 Edge detection Goal: Identify sudden changes (discontinuities) in an image Intuitively, most semantic and shape information from the image can be encoded in the edges More compact than pixels Ideal: artist s line drawing (but artist is also using object-level knowledge) Source: D. Lowe

30 Why do we care about edges? Extract information, recognize objects Recover geometry and viewpoint Vanishing line Vanishing point Vertical vanishing point (at infinity) Vanishing point

31 Origin of Edges surface normal discontinuity depth discontinuity surface color discontinuity illumination discontinuity Edges are caused by a variety of factors Source: Steve Seitz

32 Closeup of edges Source: D. Hoiem

33 Closeup of edges Source: D. Hoiem

34 Closeup of edges Source: D. Hoiem

35 Closeup of edges Source: D. Hoiem

36 Characterizing edges An edge is a place of rapid change in the image intensity function image intensity function (along horizontal scanline) first derivative edges correspond to extrema of derivative

37 Edge Detection with Gradient Operators separable

38 Other filters Sobel Vertical Edge (absolute value)

39 Other filters Sobel Horizontal Edge (absolute value)

40 Intensity profile Source: D. Hoiem

41 With a little Gaussian noise Gradient Source: D. Hoiem

42 Effects of noise Consider a single row or column of the image Plotting intensity as a function of position gives a signal Where is the edge? Source: S. Seitz

43 Effects of noise Difference filters respond strongly to noise Image noise results in pixels that look very different from their neighbors Generally, the larger the noise the stronger the response What can we do about it? Source: D. Forsyth

44 Solution: smooth first f g f * g d dx ( f g) To find edges, look for peaks in ( f g) d dx Source: S. Seitz

45 Derivative theorem of convolution Differentiation is convolution, and convolution is associative: d d ( f g) = f g dx dx This saves us one operation: f d dx g f d dx g Source: S. Seitz

46 Derivative of Gaussian filter * [1-1] =

47 Tradeoff between smoothing and localization 1 pixel 3 pixels 7 pixels Smoothed derivative removes noise, but blurs edge. Also finds edges at different scales. Source: D. Forsyth

48 Implementation issues The gradient magnitude is large along a thick trail or ridge, so how do we identify the actual edge points? How do we link the edge points to form curves? Source: D. Forsyth

49 Designing an edge detector Criteria for a good edge detector: Good detection: the optimal detector should find all real edges, ignoring noise or other artifacts Good localization the edges detected must be as close as possible to the true edges the detector must return one point only for each true edge point Cues of edge detection Differences in color, intensity, or texture across the boundary Continuity and closure High-level knowledge Source: L. Fei-Fei

50 Slide Texas Instruments Inc, Gradient Operators The gradient of the image I(x,y) at location (x,y), is the vector: The magnitude of the gradient: The direction of the gradient vector: ( ) ( ) = = y y x I x y x I G G I y x,, [ ] 2 2 G x G y I I + = = ( ) = y x G G y x 1 tan, θ

51 The Meaning of the Gradient It represents the direction of the strongest variation in intensity Vertical Horizontal Generic Edge Strength: Edge Direction: I θ = G x ( x, y) = 0 I θ = G ( x, y) y = π 2 I = θ 2 2 [ G + G ] G 1 y ( x, y) = tan Gx x y The direction of the edge at location (x,y) is perpendicular to the gradient vector at that point 2007 Texas Instruments Inc, Slide 51

52 Calculating the Gradient For each pixel the gradient is calculated, based on a 3x3 neighborhood around this pixel. z 1 z 2 z 3 z 4 z 5 z 6 z 7 z 8 z Texas Instruments Inc, Slide 52

53 The Sobel Edge Detector ( z + 2z + z ) ( z + z z ) G y ( z + 2z + z ) ( z + z + z ) G x Texas Instruments Inc, Slide 53

54 The Prewitt Edge Detector ( z + z + z ) ( z + z ) G ( z z z ) ( z z ) y G x z z Texas Instruments Inc, Slide 54

55 The Roberts Edge Detector G x G y z 8 z6 z 9 z 5 The Roberts Edge Detector is in fact a 2x2 operator 2007 Texas Instruments Inc, Slide 55

56 The Canny Method Two Possible Implementations: 1. The image is convolved with a Gaussian filter before gradient evaluation h ( r) = e r 2 2σ 2 r = x 2 + y 2 2. The image is convolved with the gradient of the Gaussian Filter Texas Instruments Inc, Slide 56

57 The Edge Detection Algorithm The gradient is calculated (using any of the four methods described in the previous slides), for each pixel in the picture. If the absolute value exceeds a threshold, the pixel belongs to an edge. The Canny method uses two thresholds, and enables the detection of two edge types: strong and weak edge. If a pixel's magnitude in the gradient image, exceeds the high threshold, then the pixel corresponds to a strong edge. Any pixel connected to a strong edge and having a magnitude greater than the low threshold corresponds to a weak edge Texas Instruments Inc, Slide 57

58 Canny edge detector This is probably the most widely used edge detector in computer vision Theoretical model: step-edges corrupted by additive Gaussian noise Canny has shown that the first derivative of the Gaussian closely approximates the operator that optimizes the product of signal-to-noise ratio and localization J. Canny, A Computational Approach To Edge Detection, IEEE Trans. Pattern Analysis and Machine Intelligence, 8: , Source: L. Fei-Fei

59 Note about Matlab s Canny detector Small errors in implementation Gaussian function not properly normalized First filters with a Gaussian, then a difference of Gaussian (equivalent to filtering with a larger Gaussian and taking difference)

60 Example original image (Lena)

61 Derivative of Gaussian filter x-direction y-direction

62 Compute Gradients (DoG) X-Derivative of Gaussian Y-Derivative of Gaussian Gradient Magnitude

63 Get Orientation at Each Pixel Threshold at minimum level Get orientation theta = atan2(gy, gx)

64 Non-maximum suppression for each orientation At q, we have a maximum if the value is larger than those at both p and at r. Interpolate to get these values. Source: D. Forsyth

65 Edge linking Assume the marked point is an edge point. Then we construct the tangent to the edge curve (which is normal to the gradient at that point) and use this to predict the next points (here either r or s). Source: D. Forsyth

66 Sidebar: Bilinear Interpolation

67 Examples from Sidebar: Interpolation options imx2 = imresize(im, 2, interpolation_type) nearest Copy value from nearest known Very fast but creates blocky edges bilinear Weighted average from four nearest known pixels Fast and reasonable results bicubic (default) Non-linear smoothing over larger area (4x4) Slower, visually appealing, may create negative pixel values

68 Before Non-max Suppression

69 After non-max suppression

70 Before Non-max Suppression

71 After non-max suppression

72 Hysteresis thresholding Threshold at low/high levels to get weak/strong edge pixels Do connected components, starting from strong edge pixels

73 Hysteresis thresholding Check that maximum value of gradient value is sufficiently large drop-outs? use hysteresis use a high threshold to start edge curves and a low threshold to continue them. Source: S. Seitz

74 Final Canny Edges

75 Canny edge detector 1. Filter image with x, y derivatives of Gaussian 2. Find magnitude and orientation of gradient 3. Non-maximum suppression: Thin multi-pixel wide ridges down to single pixel width 4. Thresholding and linking (hysteresis): Define two thresholds: low and high Use the high threshold to start edge curves and the low threshold to continue them MATLAB: edge(image, canny ) Source: D. Lowe, L. Fei-Fei

76 Effect of σ (Gaussian kernel spread/size) original Canny with Canny with The choice of σ depends on desired behavior large σ detects large scale edges small σ detects fine features Source: S. Seitz