Computer Vision: Filtering


 Phebe Eaton
 2 years ago
 Views:
Transcription
1 Computer Vision: Filtering Raquel Urtasun TTI Chicago Jan 10, 2013 Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
2 Today s lecture... Image formation Image Filtering Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
3 Readings Chapter 2 and 3 of Rich Szeliski s book Available online here Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
4 How is an image created? The image formation process that produced a particular image depends on lighting conditions scene geometry surface properties camera optics [Source: R. Szeliski] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
5 Image formation Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
6 [Source: A. Efros] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82 What is an image?!"#"$%&'(%)*+%' 0*1&&'23456'37'$*6*'"7'$"6'4&%66',*'./*'
7 From photons to RGB values Sample the 2D space on a regular grid. Quantize each sample, i.e., the photons arriving at each active cell are integrated and then digitized. [Source: D. Hoiem] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
8 What is an image? A grid (matrix) of intensity values #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #% % #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ &$ &$ &$ #$$ #$$ #$$ #$$ #$$ #$$!" #$$ #$$ &$ '$ '$ &$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ '( )#& )*$ )&$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ )#& )*$ )&$ )&$ )&$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ )#& )*$ #%% #%% )&$ )&$ '$ #$$ #$$ #$$ #$$ #$$ )#& )*$ #%% #%% )&$ )&$ '$ *& #$$ #$$ #$$ #$$ )#& )*$ )*$ )&$ )#& )#& '$ *& #$$ #$$ #$$ #$$ &* )#& )#& )#& '$ '$ '$ *& #$$ #$$ #$$ #$$ #$$ &* &* &* &* &* &* #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ #$$ Common to use one byte per value: 0=black, 255=white) [Source: N. Snavely] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
9 What is an image? We can think of a (grayscale) image as a function f : R 2 R giving the intensity at position (x, y) f (x, y) x y A digital image is a discrete (sampled, quantized) version of this function [Source: N. Snavely] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
10 Image Transformations As with any function, we can apply operators to an image!g (x,y) = f (x,y) + 20!g (x,y) = f (x,y) We ll talk about special kinds of operators, correlation and convolution (linear filtering) [Source: N. Snavely] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
11 Filtering Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
12 Question: Noise reduction Given a camera and a still scene, how can you reduce noise? Take lots of images and average them! Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
13 Question: Noise reduction Given a camera and a still scene, how can you reduce noise? Take lots of images and average them! Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
14 Question: Noise reduction Given a camera and a still scene, how can you reduce noise? Take lots of images and average them! What s the next best thing? [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
15 Question: Noise reduction Given a camera and a still scene, how can you reduce noise? Take lots of images and average them! What s the next best thing? [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
16 Image filtering Modify the pixels in an image based on some function of a local neighborhood of each pixel #'" $" #"!" &"!" #" #" %" 5).0"678*9)8" %" ()*+,".+/0"1+2+" 3)1401".+/0"1+2+" [Source: L. Zhang] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
17 Applications of Filtering Enhance an image, e.g., denoise, resize. Extract information, e.g., texture, edges. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
18 Applications of Filtering Enhance an image, e.g., denoise, resize. Extract information, e.g., texture, edges. Detect patterns, e.g., template matching. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
19 Applications of Filtering Enhance an image, e.g., denoise, resize. Extract information, e.g., texture, edges. Detect patterns, e.g., template matching. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
20 Noise reduction Simplest thing: replace each pixel by the average of its neighbors. This assumes that neighboring pixels are similar, and the noise to be independent from pixel to pixel. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
21 Noise reduction Simplest thing: replace each pixel by the average of its neighbors. This assumes that neighboring pixels are similar, and the noise to be independent from pixel to pixel. [Source: S. Marschner] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
22 Noise reduction Simplest thing: replace each pixel by the average of its neighbors. This assumes that neighboring pixels are similar, and the noise to be independent from pixel to pixel. [Source: S. Marschner] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
23 Noise reduction Simplest thing: replace each pixel by the average of its neighbors This assumes that neighboring pixels are similar, and the noise to be independent from pixel to pixel. Moving average in 1D: [1, 1, 1, 1, 1]/5 [Source: S. Marschner] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
24 Noise reduction Simpler thing: replace each pixel by the average of its neighbors This assumes that neighboring pixels are similar, and the noise to be independent from pixel to pixel. Nonuniform weights [1, 4, 6, 4, 1] / 16 [Source: S. Marschner] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
25 Moving Average in 2D [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
26 Moving Average in 2D [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
27 Moving Average in 2D [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
28 Moving Average in 2D [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
29 Moving Average in 2D [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
30 Moving Average in 2D [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
31 Linear Filtering: Correlation Involves weighted combinations of pixels in small neighborhoods. The output pixels value is determined as a weighted sum of input pixel values g(i, j) = k,l f (i + k, j + l)h(k, l) Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
32 Linear Filtering: Correlation Involves weighted combinations of pixels in small neighborhoods. The output pixels value is determined as a weighted sum of input pixel values g(i, j) = k,l f (i + k, j + l)h(k, l) The entries of the weight kernel or mask h(k, l) are often called the filter coefficients. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
33 Linear Filtering: Correlation Involves weighted combinations of pixels in small neighborhoods. The output pixels value is determined as a weighted sum of input pixel values g(i, j) = k,l f (i + k, j + l)h(k, l) The entries of the weight kernel or mask h(k, l) are often called the filter coefficients. This operator is the correlation operator g = f h Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
34 Linear Filtering: Correlation Involves weighted combinations of pixels in small neighborhoods. The output pixels value is determined as a weighted sum of input pixel values g(i, j) = k,l f (i + k, j + l)h(k, l) The entries of the weight kernel or mask h(k, l) are often called the filter coefficients. This operator is the correlation operator g = f h Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
35 Smoothing by averaging What if the filter size was 5 x 5 instead of 3 x 3? [Source: K. Graumann] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
36 Gaussian filter What if we want nearest neighboring pixels to have the most influence on the output? Removes highfrequency components from the image (lowpass filter). [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
37 Smoothing with a Gaussian [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
38 Mean vs Gaussian Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
39 Gaussian filter: Parameters Size of kernel or mask: Gaussian function has infinite support, but discrete filters use finite kernels. [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
40 Gaussian filter: Parameters Variance of the Gaussian: determines extent of smoothing. [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
41 Gaussian filter: Parameters [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
42 Is this the most general Gaussian? No, the most general form for x R d N (x; µ, Σ) = ( 1 exp 1 ) (2π) d/2 Σ 1/2 2 (x µ)t Σ 1 (x µ) But the simplified version is typically use for filtering. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
43 Properties of the Smoothing All values are positive. They all sum to 1. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
44 Properties of the Smoothing All values are positive. They all sum to 1. Amount of smoothing proportional to mask size. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
45 Properties of the Smoothing All values are positive. They all sum to 1. Amount of smoothing proportional to mask size. Remove highfrequency components; lowpass filter. [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
46 Properties of the Smoothing All values are positive. They all sum to 1. Amount of smoothing proportional to mask size. Remove highfrequency components; lowpass filter. [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
47 Example of Correlation What is the result of filtering the impulse signal (image) F with the arbitrary kernel H? [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
48 Convolution Convolution operator g(i, j) = k,l f (i k, j l)h(k, l) = k,l f (k, l)h(i k, j l) = f h and h is then called the impulse response function. Equivalent to flip the filter in both dimensions (bottom to top, right to left) and apply crosscorrelation. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
49 Convolution Convolution operator g(i, j) = k,l f (i k, j l)h(k, l) = k,l f (k, l)h(i k, j l) = f h and h is then called the impulse response function. Equivalent to flip the filter in both dimensions (bottom to top, right to left) and apply crosscorrelation. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
50 Matrix form Correlation and convolution can both be written as a matrixvector multiply, if we first convert the twodimensional images f (i, j) and g(i, j) into rasterordered vectors f and g with H a sparse matrix. g = Hf Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
51 Correlation vs Convolution Convolution g(i, j) = k,l f (i k, j l)h(k, l) G = H F Correlation g(i, j) = k,l f (i + k, j + l)h(k, l) G = H F For a Gaussian or box filter, how will the outputs differ? If the input is an impulse signal, how will the outputs differ? h δ?, and h δ? Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
52 Example What s the result? [Source: D. Lowe] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
53 Example What s the result? [Source: D. Lowe] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
54 Example What s the result? [Source: D. Lowe] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
55 Example What s the result? [Source: D. Lowe] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
56 Example What s the result? * #" #" #" #" $" #" #" #" #"!"!"!" !!"!"!" %"!"!"!" Original! [Source: D. Lowe] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
57 Example What s the result? * #" #" #" #" $" #" #" #" #" !!"!"!"!"!"!"!"!"!" 0" Original!!"#$%&'(')*+,&$* %&''()*+&*(,"(.(,/* [Source: D. Lowe] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
58 Sharpening [Source: D. Lowe] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
59 Gaussian Filter Convolution with itself is another Gaussian *!" Convolving twice with Gaussian kernel of width σ is the same as convolving once with kernel of width σ 2 [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
60 Sharpening revisited What does blurring take away?!" #"!"#$#%&'( )*!!+,.(/0102(.+&#'( Let s add it back #"$"!" *$+,+'#) (&.#+)!"#$%&'&() [Source: S. Lazebnik] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
61 Sharpening!"#$%&'&() #$%&'&() [Source: N. Snavely] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
62 Optical Convolution Camera Shake!" * Figure: Fergus, et al., SIGGRAPH 2006 Blur in outoffocus regions of an image. Figure: Bokeh: Click for more info [Source: N. Snavely] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
63 Correlation vs Convolution The convolution is both commutative and associative. The Fourier transform of two convolved images is the product of their individual Fourier transforms. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
64 Correlation vs Convolution The convolution is both commutative and associative. The Fourier transform of two convolved images is the product of their individual Fourier transforms. Both correlation and convolution are linear shiftinvariant (LSI) operators, which obey both the superposition principle and the shift invariance principle h (f 0 + f 1 ) = h f o + h f 1 if g(i, j) = f (i + k, j + l) (h g)(i, j) = (h f )(i + k, j + l) which means that shifting a signal commutes with applying the operator. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
65 Correlation vs Convolution The convolution is both commutative and associative. The Fourier transform of two convolved images is the product of their individual Fourier transforms. Both correlation and convolution are linear shiftinvariant (LSI) operators, which obey both the superposition principle and the shift invariance principle h (f 0 + f 1 ) = h f o + h f 1 if g(i, j) = f (i + k, j + l) (h g)(i, j) = (h f )(i + k, j + l) which means that shifting a signal commutes with applying the operator. Is the same as saying that the effect of the operator is the same everywhere. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
66 Correlation vs Convolution The convolution is both commutative and associative. The Fourier transform of two convolved images is the product of their individual Fourier transforms. Both correlation and convolution are linear shiftinvariant (LSI) operators, which obey both the superposition principle and the shift invariance principle h (f 0 + f 1 ) = h f o + h f 1 if g(i, j) = f (i + k, j + l) (h g)(i, j) = (h f )(i + k, j + l) which means that shifting a signal commutes with applying the operator. Is the same as saying that the effect of the operator is the same everywhere. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
67 Boundary Effects The results of filtering the image in this form will lead to a darkening of the corner pixels. The original image is effectively being padded with 0 values wherever the convolution kernel extends beyond the original image boundaries. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
68 Boundary Effects The results of filtering the image in this form will lead to a darkening of the corner pixels. The original image is effectively being padded with 0 values wherever the convolution kernel extends beyond the original image boundaries. A number of alternative padding or extension modes have been developed. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
69 Boundary Effects The results of filtering the image in this form will lead to a darkening of the corner pixels. The original image is effectively being padded with 0 values wherever the convolution kernel extends beyond the original image boundaries. A number of alternative padding or extension modes have been developed. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
70 Boundary Effects The results of filtering the image in this form will lead to a darkening of the corner pixels. The original image is effectively being padded with 0 values wherever the convolution kernel extends beyond the original image boundaries. A number of alternative padding or extension modes have been developed. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
71 Separable Filters The process of performing a convolution requires K 2 operations per pixel, where K is the size (width or height) of the convolution kernel. In many cases, this operation can be speed up by first performing a 1D horizontal convolution followed by a 1D vertical convolution, requiring 2K operations. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
72 Separable Filters The process of performing a convolution requires K 2 operations per pixel, where K is the size (width or height) of the convolution kernel. In many cases, this operation can be speed up by first performing a 1D horizontal convolution followed by a 1D vertical convolution, requiring 2K operations. If his is possible, then the convolution kernel is called separable. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
73 Separable Filters The process of performing a convolution requires K 2 operations per pixel, where K is the size (width or height) of the convolution kernel. In many cases, this operation can be speed up by first performing a 1D horizontal convolution followed by a 1D vertical convolution, requiring 2K operations. If his is possible, then the convolution kernel is called separable. And it is the outer product of two kernels K = vh T Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
74 Separable Filters The process of performing a convolution requires K 2 operations per pixel, where K is the size (width or height) of the convolution kernel. In many cases, this operation can be speed up by first performing a 1D horizontal convolution followed by a 1D vertical convolution, requiring 2K operations. If his is possible, then the convolution kernel is called separable. And it is the outer product of two kernels K = vh T Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
75 Let s play a game... Is this separable? If yes, what s the separable version? Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
76 Let s play a game... Is this separable? If yes, what s the separable version? What does this filter do? Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
77 Let s play a game... Is this separable? If yes, what s the separable version? Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
78 Let s play a game... Is this separable? If yes, what s the separable version? What does this filter do? Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
79 Let s play a game... Is this separable? If yes, what s the separable version? Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
80 Let s play a game... Is this separable? If yes, what s the separable version? What does this filter do? Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
81 Let s play a game... Is this separable? If yes, what s the separable version? Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
82 Let s play a game... Is this separable? If yes, what s the separable version? What does this filter do? Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
83 Let s play a game... Is this separable? If yes, what s the separable version? Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
84 Let s play a game... Is this separable? If yes, what s the separable version? What does this filter do? Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
85 How can we tell if a given kernel K is indeed separable? Inspection... this is what we were doing. Looking at the analytic form of it. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
86 How can we tell if a given kernel K is indeed separable? Inspection... this is what we were doing. Looking at the analytic form of it. Look at the singular value decomposition (SVD), and if only one singular value is nonzero, then it is separable K = UΣV T = i σ i u i v T i with Σ = diag(σ i ). Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
87 How can we tell if a given kernel K is indeed separable? Inspection... this is what we were doing. Looking at the analytic form of it. Look at the singular value decomposition (SVD), and if only one singular value is nonzero, then it is separable K = UΣV T = i σ i u i v T i with Σ = diag(σ i ). σ1 u 1 and σ 1 v T 1 are the vertical and horizontal kernels. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
88 How can we tell if a given kernel K is indeed separable? Inspection... this is what we were doing. Looking at the analytic form of it. Look at the singular value decomposition (SVD), and if only one singular value is nonzero, then it is separable K = UΣV T = i σ i u i v T i with Σ = diag(σ i ). σ1 u 1 and σ 1 v T 1 are the vertical and horizontal kernels. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
89 Application of filtering: Template matching Filters as templates: filters look like the effects they are intended to find. Use normalized crosscorrelation score to find a given pattern (template) in the image. Normalization needed to control for relative brightnesses. [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
90 Template matching [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
91 More complex Scenes Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
92 Let s talk about Edge Detection Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
93 Filtering: Edge detection Map image from 2d array of pixels to a set of curves or line segments or contours. More compact than pixels. Look for strong gradients, postprocess. Figure: [Shotton et al. PAMI, 07] [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
94 Origin of edges Edges are caused by a variety of factors surface normal discontinuity depth discontinuity surface color discontinuity illumination discontinuity [Source: N. Snavely] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
95 What causes an edge? [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
96 Looking more locally... [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
97 Images as functions Edges look like steep cliffs [Source: N. Snavely] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
98 Characterizing Edges An edge is a place of rapid change in the image intensity function. [Source: S. Lazebnik] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
99 How to Implement Derivatives with Convolution How can we differentiate a digital image F[x,y]? Option 1: reconstruct a continuous image f, then compute the partial derivative as f (x, y) x = lim ɛ 0 f (x + ɛ, y) f (x) ɛ Option 2: take discrete derivative (finite difference) f (x, y) x f [x + 1, y] f [x] 1 Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
100 How to Implement Derivatives with Convolution How can we differentiate a digital image F[x,y]? Option 1: reconstruct a continuous image f, then compute the partial derivative as f (x, y) x = lim ɛ 0 f (x + ɛ, y) f (x) ɛ Option 2: take discrete derivative (finite difference) f (x, y) x f [x + 1, y] f [x] 1 What would be the filter to implement this using convolution? Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
101 How to Implement Derivatives with Convolution How can we differentiate a digital image F[x,y]? Option 1: reconstruct a continuous image f, then compute the partial derivative as f (x, y) x = lim ɛ 0 f (x + ɛ, y) f (x) ɛ Option 2: take discrete derivative (finite difference) f (x, y) x f [x + 1, y] f [x] 1 What would be the filter to implement this using convolution?!"!" [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
102 How to Implement Derivatives with Convolution How can we differentiate a digital image F[x,y]? Option 1: reconstruct a continuous image f, then compute the partial derivative as f (x, y) x = lim ɛ 0 f (x + ɛ, y) f (x) ɛ Option 2: take discrete derivative (finite difference) f (x, y) x f [x + 1, y] f [x] 1 What would be the filter to implement this using convolution?!"!" [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
103 Partial derivatives of an image Figure: Using correlation filters [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
104 Finite Difference Filters [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
105 Image Gradient The gradient of an image f = [ ] f x, f y The gradient points in the direction of most rapid change in intensity Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
106 Image Gradient The gradient of an image f = [ ] f x, f y The gradient points in the direction of most rapid change in intensity The gradient direction (orientation of edge normal) is given by: ( f θ = tan 1 y / f ) x Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
107 Image Gradient The gradient of an image f = [ ] f x, f y The gradient points in the direction of most rapid change in intensity The gradient direction (orientation of edge normal) is given by: ( f θ = tan 1 y / f ) x The edge strength is given by the magnitude f = ( f x )2 + ( f y )2 [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
108 Image Gradient The gradient of an image f = [ ] f x, f y The gradient points in the direction of most rapid change in intensity The gradient direction (orientation of edge normal) is given by: ( f θ = tan 1 y / f ) x The edge strength is given by the magnitude f = ( f x )2 + ( f y )2 [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
109 Image Gradient [Source: S. Lazebnik] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
110 Effects of noise Consider a single row or column of the image. Plotting intensity as a function of position gives a signal.!"#$%&#'()*&#+,.& [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
111 Effects of noise Smooth first, and look for picks in x (h f ). [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
112 Derivative theorem of convolution Differentiation property of convolution It saves one operation h (h f ) = ( x x ) f = h ( f x ) [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
113 2D Edge Detection Filters Gaussian h σ (x, y) = 1 2πσ exp u 2 2 +v 2 2σ 2 Derivative of Gaussian (x) x h σ(u, v) [Source: N. Snavely] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
114 Derivative of Gaussians [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
115 Laplacian of Gaussians Edge by detecting zerocrossings of bottom graph [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
116 2D Edge Filtering with 2 the Laplacian operator 2 f = 2 f x + 2 f 2 y 2 [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
117 Effect of σ on derivatives The detected structures differ depending on the Gaussian s scale parameter: Larger values: larger scale edges detected. Smaller values: finer features detected. [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
118 Derivatives Use opposite signs to get response in regions of high contrast. They sum to 0 so that there is no response in constant regions. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
119 Derivatives Use opposite signs to get response in regions of high contrast. They sum to 0 so that there is no response in constant regions. High absolute value at points of high contrast. [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
120 Derivatives Use opposite signs to get response in regions of high contrast. They sum to 0 so that there is no response in constant regions. High absolute value at points of high contrast. [Source: K. Grauman] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
121 Bandpass filters The Sobel and corner filters are bandpass and oriented filters. More sophisticated filters can be obtained by convolving with a Gaussian filter G(x, y, σ) = 1 ( 2πσ 2 exp x 2 + y 2 ) 2σ 2 and taking the first or second derivatives. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
122 Bandpass filters The Sobel and corner filters are bandpass and oriented filters. More sophisticated filters can be obtained by convolving with a Gaussian filter G(x, y, σ) = 1 ( 2πσ 2 exp x 2 + y 2 ) 2σ 2 and taking the first or second derivatives. These filters are bandpass filters: they filter low and high frequencies. Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
123 Bandpass filters The Sobel and corner filters are bandpass and oriented filters. More sophisticated filters can be obtained by convolving with a Gaussian filter G(x, y, σ) = 1 ( 2πσ 2 exp x 2 + y 2 ) 2σ 2 and taking the first or second derivatives. These filters are bandpass filters: they filter low and high frequencies. The second derivative of a twodimensional image is the laplacian operator 2 f = 2 f x f y 2 Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
124 Bandpass filters The Sobel and corner filters are bandpass and oriented filters. More sophisticated filters can be obtained by convolving with a Gaussian filter G(x, y, σ) = 1 ( 2πσ 2 exp x 2 + y 2 ) 2σ 2 and taking the first or second derivatives. These filters are bandpass filters: they filter low and high frequencies. The second derivative of a twodimensional image is the laplacian operator 2 f = 2 f x f y 2 Blurring an image with a Gaussian and then taking its Laplacian is equivalent to convolving directly with the Laplacian of Gaussian (LoG) filter, Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
125 Bandpass filters The Sobel and corner filters are bandpass and oriented filters. More sophisticated filters can be obtained by convolving with a Gaussian filter G(x, y, σ) = 1 ( 2πσ 2 exp x 2 + y 2 ) 2σ 2 and taking the first or second derivatives. These filters are bandpass filters: they filter low and high frequencies. The second derivative of a twodimensional image is the laplacian operator 2 f = 2 f x f y 2 Blurring an image with a Gaussian and then taking its Laplacian is equivalent to convolving directly with the Laplacian of Gaussian (LoG) filter, Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
126 Bandpass filters The directional or oriented filter can obtained by smoothing with a Gaussian (or some other filter) and then taking a directional derivative u = u u (G f ) = u (G f ) = ( u G) f with u = (cos θ, sin θ). The Sobel operator is a simple approximation of this: Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
127 Practical Example [Source: N. Snavely] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
128 Finding Edges Figure: Gradient magnitude [Source: N. Snavely] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
129 Finding Edges!"#$#%&'%("#%#)*#+% Figure: Gradient magnitude [Source: N. Snavely] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
130 NonMaxima Suppression Figure: Gradient magnitude Check if pixel is local maximum along gradient direction: requires interpolation [Source: N. Snavely] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
131 Finding Edges Figure: Thresholding [Source: N. Snavely] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
132 Finding Edges Figure: Thinning: Nonmaxima suppression [Source: N. Snavely] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
133 Canny Edge Detector Matlab: edge(image, canny ) 1 Filter image with derivative of Gaussian 2 Find magnitude and orientation of gradient 3 Nonmaximum suppression 4 Linking and thresholding (hysteresis): Define two thresholds: low and high Use the high threshold to start edge curves and the low threshold to continue them [Source: D. Lowe and L. FeiFei] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
134 Canny edge detector Still one of the most widely used edge detectors in computer vision J. Canny, A Computational Approach To Edge Detection, IEEE Trans. Pattern Analysis and Machine Intelligence, 8: , Depends on several parameters: σ of the blur and the thresholds Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
135 Canny edge detector large σ detects largescale edges small σ detects fine edges original Canny with Canny with [Source: S. Seitz] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
136 Scale Space (Witkin 83) first derivative peaks larger Gaussian filtered signal Properties of scale space (w/ Gaussian smoothing) edge position may shift with increasing scale (σ) two edges may merge with increasing scale an edge may not split into two with increasing scale [Source: N. Snavely] Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
137 Next class... more on filtering and image features Raquel Urtasun (TTIC) Computer Vision Jan 10, / 82
Convolution. 1D Formula: 2D Formula: Example on the web: http://www.jhu.edu/~signals/convolve/
Basic Filters (7) Convolution/correlation/Linear filtering Gaussian filters Smoothing and noise reduction First derivatives of Gaussian Second derivative of Gaussian: Laplacian Oriented Gaussian filters
More informationSpatial filtering. Computer Vision & Digital Image Processing. Spatial filter masks. Spatial filters (examples) Smoothing filters
Spatial filtering Computer Vision & Digital Image Processing Spatial Filtering Use of spatial masks for filtering is called spatial filtering May be linear or nonlinear Linear filters Lowpass: attenuate
More informationWhat is an Edge? Computer Vision Week 4. How to detect edges? What is an Edge? Edge Detection techniques. Edge Detection techniques.
What is an Edge? Computer Vision Week 4 Edge Detection Linear filtering; pyramids, wavelets Interest Operators surface normal discontinuity depth discontinuity surface color discontinuity illumination
More informationEdge detection. (Trucco, Chapt 4 AND Jain et al., Chapt 5) Edges are significant local changes of intensity in an image.
Edge detection (Trucco, Chapt 4 AND Jain et al., Chapt 5) Definition of edges Edges are significant local changes of intensity in an image. Edges typically occur on the boundary between two different
More informationLecture 2: 2D Fourier transforms and applications
Lecture 2: 2D Fourier transforms and applications B14 Image Analysis Michaelmas 2014 A. Zisserman Fourier transforms and spatial frequencies in 2D Definition and meaning The Convolution Theorem Applications
More informationComputational Foundations of Cognitive Science
Computational Foundations of Cognitive Science Lecture 15: Convolutions and Kernels Frank Keller School of Informatics University of Edinburgh keller@inf.ed.ac.uk February 23, 2010 Frank Keller Computational
More informationjorge s. marques image processing
image processing images images: what are they? what is shown in this image? What is this? what is an image images describe the evolution of physical variables (intensity, color, reflectance, condutivity)
More informationToday s Topics. Lecture 11: LoG and DoG Filters. Recall: First Derivative Filters. SecondDerivative Filters
Today s Topics Lecture : LoG and DoG Filters Laplacian of Gaussian (LoG) Filter  useful for finding edges  also useful for finding blobs! approimation using Difference of Gaussian (DoG) Recall: First
More informationConvolution, Noise and Filters
T H E U N I V E R S I T Y of T E X A S Convolution, Noise and Filters Philip Baldwin, Ph.D. Department of Biochemistry Response to an Entire Signal The response of a system with impulse response h(t) to
More information What is a feature?  Image processing essentials  Edge detection (Sobel & Canny)  Hough transform  Some images
Seminar: Feature extraction by André Aichert I Feature detection  What is a feature?  Image processing essentials  Edge detection (Sobel & Canny)  Hough transform  Some images II An Entropybased
More informationBIL Computer Vision Mar 5, 2014
BIL 719  Computer Vision Mar 5, 2014 Image pyramids Aykut Erdem Dept. of Computer Engineering Hacettepe University Image Scaling This image is too big to fit on the screen. How can we reduce it? How to
More informationCorrelation and Convolution Class Notes for CMSC 426, Fall 2005 David Jacobs
Correlation and Convolution Class otes for CMSC 46, Fall 5 David Jacobs Introduction Correlation and Convolution are basic operations that we will perform to extract information from images. They are in
More informationDigital Image Processing Using Matlab. Haris PapasaikaHanusch Institute of Geodesy and Photogrammetry, ETH Zurich
Haris PapasaikaHanusch Institute of Geodesy and Photogrammetry, ETH Zurich haris@geod.baug.ethz.ch Images and Digital Images A digital image differs from a photo in that the values are all discrete. Usually
More informationImage Segmentation Preview Segmentation subdivides an image to regions or objects Two basic properties of intensity values Discontinuity Edge detection Similarity Thresholding Region growing/splitting/merging
More informationBildverarbeitung und Mustererkennung Image Processing and Pattern Recognition
Bildverarbeitung und Mustererkennung Image Processing and Pattern Recognition 1. Image PreProcessing  Pixel Brightness Transformation  Geometric Transformation  Image Denoising 1 1. Image PreProcessing
More informationDigital Imaging and Multimedia. Filters. Ahmed Elgammal Dept. of Computer Science Rutgers University
Digital Imaging and Multimedia Filters Ahmed Elgammal Dept. of Computer Science Rutgers University Outlines What are Filters Linear Filters Convolution operation Properties of Linear Filters Application
More informationImage Filtering & Edge Detection
What is imae filterin? Imae Filterin & Ede Detection Modify the pixels in an imae based on some function of a local neihborhood of the pixels. Some function Readin: Chapter 7 and 8, F&P Linear Functions
More informationImage Processing. Alpha Channel. Blending. Image Compositing. Blending Errors. Blending in OpenGL
CSCI 42 Computer Graphics Lecture 22 Image Processing Blending Display Color Models Filters Dithering [Ch 7.13, 8.118.12] Jernej Barbic University of Southern California Alpha Channel Frame buffer Simple
More informationLinear Filtering Part II
Linear Filtering Part II Selim Aksoy Department of Computer Engineering Bilkent University saksoy@cs.bilkent.edu.tr Fourier theory Jean Baptiste Joseph Fourier had a crazy idea: Any periodic function can
More informationPicture Perfect: The Mathematics of JPEG Compression
Picture Perfect: The Mathematics of JPEG Compression May 19, 2011 1 2 3 in 2D Sampling and the DCT in 2D 2D Compression Images Outline A typical color image might be 600 by 800 pixels. Images Outline A
More informationDigital image processing
Digital image processing The twodimensional discrete Fourier transform and applications: image filtering in the frequency domain Introduction Frequency domain filtering modifies the brightness values
More informationREAL TIME TRAFFIC LIGHT CONTROL USING IMAGE PROCESSING
REAL TIME TRAFFIC LIGHT CONTROL USING IMAGE PROCESSING Ms.PALLAVI CHOUDEKAR Ajay Kumar Garg Engineering College, Department of electrical and electronics Ms.SAYANTI BANERJEE Ajay Kumar Garg Engineering
More informationImplementation of Canny Edge Detector of color images on CELL/B.E. Architecture.
Implementation of Canny Edge Detector of color images on CELL/B.E. Architecture. Chirag Gupta,Sumod Mohan K cgupta@clemson.edu, sumodm@clemson.edu Abstract In this project we propose a method to improve
More informationImage Processing with. ImageJ. Biology. Imaging
Image Processing with ImageJ 1. Spatial filters Outlines background correction image denoising edges detection 2. Fourier domain filtering correction of periodic artefacts 3. Binary operations masks morphological
More informationCanny Edge Detection
Canny Edge Detection 09gr820 March 23, 2009 1 Introduction The purpose of edge detection in general is to significantly reduce the amount of data in an image, while preserving the structural properties
More informationQUALITY TESTING OF WATER PUMP PULLEY USING IMAGE PROCESSING
QUALITY TESTING OF WATER PUMP PULLEY USING IMAGE PROCESSING MRS. A H. TIRMARE 1, MS.R.N.KULKARNI 2, MR. A R. BHOSALE 3 MR. C.S. MORE 4 MR.A.G.NIMBALKAR 5 1, 2 Assistant professor Bharati Vidyapeeth s college
More informationImage Gradients. Given a discrete image Á Òµ, consider the smoothed continuous image Üµ defined by
Image Gradients Given a discrete image Á Òµ, consider the smoothed continuous image Üµ defined by Üµ Ü ¾ Ö µ Á Òµ Ü ¾ Ö µá µ (1) where Ü ¾ Ö Ô µ Ü ¾ Ý ¾. ½ ¾ ¾ Ö ¾ Ü ¾ ¾ Ö. Here Ü is the 2norm for the
More informationCompressive Sensing. Examples in Image Compression. Lecture 4, July 30, Luiz Velho Eduardo A. B. da Silva Adriana Schulz
Compressive Sensing Examples in Image Compression Lecture 4, July, 09 Luiz Velho Eduardo A. B. da Silva Adriana Schulz Today s Lecture Discuss applications of CS in image compression Evaluate CS efficiency
More informationAdmin stuff. 4 Image Pyramids. Spatial Domain. Projects. Fourier domain 2/26/2008. Fourier as a change of basis
Admin stuff 4 Image Pyramids Change of office hours on Wed 4 th April Mon 3 st March 9.3.3pm (right after class) Change of time/date t of last class Currently Mon 5 th May What about Thursday 8 th May?
More informationLecture 9: Shape Description (Regions)
Lecture 9: Shape Description (Regions) c Bryan S. Morse, Brigham Young University, 1998 2000 Last modified on February 16, 2000 at 4:00 PM Contents 9.1 What Are Descriptors?.........................................
More informationSharpening through spatial filtering
Sharpening through spatial filtering Stefano Ferrari Università degli Studi di Milano stefano.ferrari@unimi.it Elaborazione delle immagini (Image processing I) academic year 2011 2012 Sharpening The term
More informationImage Enhancement in the Frequency Domain
Image Enhancement in the Frequency Domain Jesus J. Caban Outline! Assignment #! Paper Presentation & Schedule! Frequency Domain! Mathematical Morphology %& Assignment #! Questions?! How s OpenCV?! You
More informationOneDimensional Processing for Edge Detection using Hilbert Transform P. Kiran Kumar, Sukhendu Das and B. Yegnanarayana Department of Computer Science and Engineering Indian Institute of Technology, Madras
More information2D Image Processing. Edge and Corner Detection. Prof. Didier Stricker Dr. Alain Pagani
2D Image Processing Edge and Corner Detection Prof. Didier Stricker Dr. Alain Pagani Kaiserlautern University http://ags.cs.unikl.de/ DFKI Deutsches Forschungszentrum für Künstliche Intelligenz http://av.dfki.de
More informationAutomatic 3D Reconstruction via Object Detection and 3D Transformable Model Matching CS 269 Class Project Report
Automatic 3D Reconstruction via Object Detection and 3D Transformable Model Matching CS 69 Class Project Report Junhua Mao and Lunbo Xu University of California, Los Angeles mjhustc@ucla.edu and lunbo
More informationAnalecta Vol. 8, No. 2 ISSN 20647964
EXPERIMENTAL APPLICATIONS OF ARTIFICIAL NEURAL NETWORKS IN ENGINEERING PROCESSING SYSTEM S. Dadvandipour Institute of Information Engineering, University of Miskolc, Egyetemváros, 3515, Miskolc, Hungary,
More informationIMPLEMENTATION OF IMAGE PROCESSING IN REAL TIME CAR PARKING SYSTEM
IMPLEMENTATION OF IMAGE PROCESSING IN REAL TIME CAR PARKING SYSTEM Ms.SAYANTI BANERJEE Ajay Kumar Garg Engineering College, Department of electrical and electronics Ms.PALLAVI CHOUDEKAR Ajay Kumar Garg
More informationEdge Detection Techniques: Evaluations and Comparisons
Applied Mathematical Sciences, Vol. 2, 2008, no. 31, 15071520 Edge Detection Techniques: Evaluations and Comparisons Ehsan Nadernejad Department of Computer Engineering, Faculty of Engineering Mazandaran
More informationMarkov chains and Markov Random Fields (MRFs)
Markov chains and Markov Random Fields (MRFs) 1 Why Markov Models We discuss Markov models now. This is the simplest statistical model in which we don t assume that all variables are independent; we assume
More informationRailway Expansion Joint Gaps and Hooks Detection Using Morphological Processing, Corner Points and Blobs
Railway Expansion Joint Gaps and Hooks Detection Using Morphological Processing, Corner Points and Blobs Samiul Islam ID: 12301053 Rubayat Ahmed Khan ID: 11301026 Supervisor RubelBiswas Department of Computer
More informationZhang Wu Directional LMMSE Image Demosaicking
2014/07/01 v0.5 IPOL article class Published in Image Processing On Line on 2011 09 01. Submitted on 2011 00 00, accepted on 2011 00 00. ISSN 2105 1232 c 2011 IPOL & the authors CC BY NC SA This article
More informationMVA ENS Cachan. Lecture 2: Logistic regression & intro to MIL Iasonas Kokkinos Iasonas.kokkinos@ecp.fr
Machine Learning for Computer Vision 1 MVA ENS Cachan Lecture 2: Logistic regression & intro to MIL Iasonas Kokkinos Iasonas.kokkinos@ecp.fr Department of Applied Mathematics Ecole Centrale Paris Galen
More informationAnother Example: the Hubble Space Telescope
296 DIP Chapter and 2: Introduction and Integral Equations Motivation: Why Inverse Problems? A largescale example, coming from a collaboration with Università degli Studi di Napoli Federico II in Naples.
More informationLectures 6&7: Image Enhancement
Lectures 6&7: Image Enhancement Leena Ikonen Pattern Recognition (MVPR) Lappeenranta University of Technology (LUT) leena.ikonen@lut.fi http://www.it.lut.fi/ip/research/mvpr/ 1 Content Background Spatial
More informationImage Segmentation and Registration
Image Segmentation and Registration Dr. Christine Tanner (tanner@vision.ee.ethz.ch) Computer Vision Laboratory, ETH Zürich Dr. Verena Kaynig, Machine Learning Laboratory, ETH Zürich Outline Segmentation
More informationDigital Image Processing
Digital Image Processing Using MATLAB Second Edition Rafael C. Gonzalez University of Tennessee Richard E. Woods MedData Interactive Steven L. Eddins The MathWorks, Inc. Gatesmark Publishing A Division
More informationEECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines
EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines What is image processing? Image processing is the application of 2D signal processing methods to images Image representation
More informationComputer Vision & Digital Image Processing
Intensity Transformations and Spatial Filtering Basics Computer Vision & Digital Image Processing Intensity Transformations and Spatial Filtering Operations take place in the spatial domain Operate directly
More informationBackground: A CCD consists of an array of individual detector sites, as suggested by the sketch below: Array of independent detector sites
Q: Can I use PixelScope to measure the MTF of a CCD camera? Can I use PixelScope to map the actual sensitive area of a pixel? A: Yes to both. This note explains how. Approach This can be a complicated
More informationApplications of Convolution in Image Processing with MATLAB
University of Washington Applications of Convolution in Image Processing with MATLAB Author: Sung Kim Instructor: Riley Casper August 20, 2013 1 Abstract The article presents a short introduction to image
More informationVisualization and Feature Extraction, FLOW Spring School 2016 Prof. Dr. Tino Weinkauf. Flow Visualization. ImageBased Methods (integrationbased)
Visualization and Feature Extraction, FLOW Spring School 2016 Prof. Dr. Tino Weinkauf Flow Visualization ImageBased Methods (integrationbased) Spot Noise (Jarke van Wijk, Siggraph 1991) Flow Visualization:
More informationBlind Deconvolution of Corrupted Barcode Signals
Blind Deconvolution of Corrupted Barcode Signals Everardo Uribe and Yifan Zhang Advisors: Ernie Esser and Yifei Lou Interdisciplinary Computational and Applied Mathematics Program University of California,
More informationOptical Flow as a property of moving objects used for their registration
Optical Flow as a property of moving objects used for their registration Wolfgang Schulz Computer Vision Course Project York University Email:wschulz@cs.yorku.ca 1. Introduction A soccer game is a real
More informationSGN1158 Introduction to Signal Processing Test. Solutions
SGN1158 Introduction to Signal Processing Test. Solutions 1. Convolve the function ( ) with itself and show that the Fourier transform of the result is the square of the Fourier transform of ( ). (Hints:
More informationAPPLICATIONS AND REVIEW OF FOURIER TRANSFORM/SERIES (Copyright 2001, David T. Sandwell)
1 APPLICATIONS AND REVIEW OF FOURIER TRANSFORM/SERIES (Copyright 2001, David T. Sandwell) (Reference The Fourier Transform and its Application, second edition, R.N. Bracewell, McGrawHill Book Co., New
More informationModule II: Multimedia Data Mining
ALMA MATER STUDIORUM  UNIVERSITÀ DI BOLOGNA Module II: Multimedia Data Mining Laurea Magistrale in Ingegneria Informatica University of Bologna Multimedia Data Retrieval Home page: http://wwwdb.disi.unibo.it/courses/dm/
More informationMaximum and Minimum Values
Jim Lambers MAT 280 Spring Semester 200910 Lecture 8 Notes These notes correspond to Section 11.7 in Stewart and Section 3.3 in Marsden and Tromba. Maximum and Minimum Values In singlevariable calculus,
More informationColorCrack: Identifying Cracks in Glass
ColorCrack: Identifying Cracks in Glass James Max Kanter Massachusetts Institute of Technology 77 Massachusetts Ave Cambridge, MA 02139 kanter@mit.edu Figure 1: ColorCrack automatically identifies cracks
More informationImage Enhancement: Frequency domain methods
Image Enhancement: Frequency domain methods The concept of filtering is easier to visualize in the frequency domain. Therefore, enhancement of image f ( m, n) can be done in the frequency domain, based
More informationDigital Image Processing. Prof. P. K. Biswas. Department of Electronics & Electrical Communication Engineering
Digital Image Processing Prof. P. K. Biswas Department of Electronics & Electrical Communication Engineering Indian Institute of Technology, Kharagpur Lecture  28 Colour Image Processing  III Hello,
More informationImplementation of Image Processing Algorithms on the Graphics Processing Units
Implementation of Image Processing Algorithms on the Graphics Processing Units Natalia Papulovskaya, Kirill Breslavskiy, and Valentin Kashitsin Department of Information Technologies of the Ural Federal
More informationBrightness and geometric transformations
Brightness and geometric transformations Václav Hlaváč Czech Technical University in Prague Center for Machine Perception (bridging groups of the) Czech Institute of Informatics, Robotics and Cybernetics
More informationArmstrong Atlantic State University Engineering Studies MATLAB Marina Image Processing Primer
Armstrong Atlantic State University Engineering Studies MATLAB Marina Image Processing Primer Prerequisites The Image Processing Primer assumes nowledge of the MATLAB IDE, MATLAB help, arithmetic operations,
More informationIMAGE MODIFICATION DEVELOPMENT AND IMPLEMENTATION: A SOFTWARE MODELING USING MATLAB
Available Online at www.ijcsmc.com International Journal of Computer Science and Mobile Computing A Monthly Journal of Computer Science and Information Technology IJCSMC, Vol. 4, Issue. 8, August 2015,
More informationMeanShift Tracking with Random Sampling
1 MeanShift Tracking with Random Sampling Alex Po Leung, Shaogang Gong Department of Computer Science Queen Mary, University of London, London, E1 4NS Abstract In this work, boosting the efficiency of
More informationAssessment. Presenter: Yupu Zhang, Guoliang Jin, Tuo Wang Computer Vision 2008 Fall
Automatic Photo Quality Assessment Presenter: Yupu Zhang, Guoliang Jin, Tuo Wang Computer Vision 2008 Fall Estimating i the photorealism of images: Distinguishing i i paintings from photographs h Florin
More informationA New Image Edge Detection Method using Qualitybased Clustering. Bijay Neupane Zeyar Aung Wei Lee Woon. Technical Report DNA #201201.
A New Image Edge Detection Method using Qualitybased Clustering Bijay Neupane Zeyar Aung Wei Lee Woon Technical Report DNA #201201 April 2012 Data & Network Analytics Research Group (DNA) Computing and
More informationComputer Graphics. Course SS 2007 Antialiasing. computer graphics & visualization
Computer Graphics Course SS 2007 Antialiasing How to avoid spatial aliasing caused by an undersampling of the signal, i.e. the sampling frequency is not high enough to cover all details Supersampling 
More informationComputational Optical Imaging  Optique Numerique.  Deconvolution 
Computational Optical Imaging  Optique Numerique  Deconvolution  Winter 2014 Ivo Ihrke Deconvolution Ivo Ihrke Outline Deconvolution Theory example 1D deconvolution Fourier method Algebraic method
More information1 Notes on Markov Chains
Notes on Markov Chains A Markov chain is a finitestate process that can be characterized completely by two objects the finite set of states Z = {z,, z n } and the transition matrix Π = [π ij i,j=,n that
More informationSachin Patel HOD I.T Department PCST, Indore, India. Parth Bhatt I.T Department, PCST, Indore, India. Ankit Shah CSE Department, KITE, Jaipur, India
Image Enhancement Using Various Interpolation Methods Parth Bhatt I.T Department, PCST, Indore, India Ankit Shah CSE Department, KITE, Jaipur, India Sachin Patel HOD I.T Department PCST, Indore, India
More informationImage Compression. Chapter 6 JORGE REBAZA
Chapter 6 Image Compression JORGE REBAZA One of the central issues in information technology is the representation of data by arrays of bits in the most efficient way possible, a neverending quest for
More information222 The International Arab Journal of Information Technology, Vol. 1, No. 2, July 2004 particular pixels. High pass filter and low pass filter are gen
The International Arab Journal of Information Technology, Vol. 1, No. 2, July 2004 221 A New Approach for Contrast Enhancement Using Sigmoid Function Naglaa Hassan 1&2 and Norio Akamatsu 1 1 Department
More informationChapter 4: TimeDomain Representation for Linear TimeInvariant Systems
Chapter 4: TimeDomain Representation for Linear TimeInvariant Systems In this chapter we will consider several methods for describing relationship between the input and output of linear timeinvariant
More informationAutomated Sudoku Solver. Marty Otzenberger EGGN 510 December 4, 2012
Automated Sudoku Solver Marty Otzenberger EGGN 510 December 4, 2012 Outline Goal Problem Elements Initial Testing Test Images Approaches Final Algorithm Results/Statistics Conclusions Problems/Limits Future
More informationsuggestive contours and abstracted shading Daniel Arias
suggestive contours and abstracted shading Daniel Arias LINES AND SHADES Shape, volume, shades and texture in drawing lines as nonphotorealistic rendering technique Luis Caballero Suggestive contours Many
More informationSegmentation & Clustering
EECS 442 Computer vision Segmentation & Clustering Segmentation in human vision Kmean clustering Meanshift Graphcut Reading: Chapters 14 [FP] Some slides of this lectures are courtesy of prof F. Li,
More informationRobert Collins CSE598C, PSU. Introduction to MeanShift Tracking
Introduction to MeanShift Tracking AppearanceBased Tracking current frame + previous location likelihood over object location appearance model (e.g. image template, or ModeSeeking (e.g. meanshift;
More information3D Scanner using Line Laser. 1. Introduction. 2. Theory
. Introduction 3D Scanner using Line Laser Di Lu Electrical, Computer, and Systems Engineering Rensselaer Polytechnic Institute The goal of 3D reconstruction is to recover the 3D properties of a geometric
More informationImage Enhancement  Frequency Domain
Image Processing  Lesson 7 Image Enhancement  Frequency Domain Low Pass Filter High Pass Filter Band pass Filter Blurring Sharpening Image FFT Filtered FFT 1 Transform Transform Filtered Image filter
More informationAntialiasing. CS 319 Advanced Topics in Computer Graphics John C. Hart
Antialiasing CS 319 Advanced Topics in Computer Graphics John C. Hart Aliasing Aliasing occurs when signals are sampled too infrequently, giving the illusion of a lower frequency signal alias noun (c.
More informationThe Image Deblurring Problem
page 1 Chapter 1 The Image Deblurring Problem You cannot depend on your eyes when your imagination is out of focus. Mark Twain When we use a camera, we want the recorded image to be a faithful representation
More informationMath 56 Homework 4 Michael Downs
Fourier series theory. As in lecture, let be the L 2 norm on (, 2π). (a) Find a unitnorm function orthogonal to f(x) = x on x < 2π. [Hint: do what you d do in vectorland, eg pick some simple new function
More informationLecture 4: Thresholding
Lecture 4: Thresholding c Bryan S. Morse, Brigham Young University, 1998 2000 Last modified on Wednesday, January 12, 2000 at 10:00 AM. Reading SH&B, Section 5.1 4.1 Introduction Segmentation involves
More informationMATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,
MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call
More informationADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S.
ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB Sohail A. Dianat Rochester Institute of Technology, New York, U.S.A. Eli S. Saber Rochester Institute of Technology, New York, U.S.A. (g) CRC Press Taylor
More informationOBJECT TRACKING USING LOGPOLAR TRANSFORMATION
OBJECT TRACKING USING LOGPOLAR TRANSFORMATION A Thesis Submitted to the Gradual Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements
More informationLecture 14. Point Spread Function (PSF)
Lecture 14 Point Spread Function (PSF), Modulation Transfer Function (MTF), Signaltonoise Ratio (SNR), Contrasttonoise Ratio (CNR), and Receiver Operating Curves (ROC) Point Spread Function (PSF) Recollect
More informationDeinterlacing/interpolation of TV signals. Elham Shahinfard Advisors: Prof. M. Ahmadi Prof. M. SidAhmed
Deinterlacing/interpolation of TV signals Elham Shahinfard Advisors: Prof. M. Ahmadi Prof. M. SidAhmed Outline A Review of some terminologies Converting from NTSC to HDTV; What changes need to be considered?
More informationFrequency domain filtering fundamentals
Frequency domain filtering fundamentals by Gleb V. Tcheslavski: gleb@ee.lamar.edu http://ee.lamar.edu/gleb/dip/index.htm Spring 2008 ELEN 4304/5365 DIP 1 Preliminaries For a digital image f(x,y) the basic
More informationDigital Image Processing (CS/ECE 545) Lecture 2: Histograms and Point Operations (Part 1)
Digital Image Processing (CS/ECE 545) Lecture 2: Histograms and Point Operations (Part 1) Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) Histograms Histograms plots how
More informationFeature Tracking and Optical Flow
02/09/12 Feature Tracking and Optical Flow Computer Vision CS 543 / ECE 549 University of Illinois Derek Hoiem Many slides adapted from Lana Lazebnik, Silvio Saverse, who in turn adapted slides from Steve
More informationLinear Systems. Singular and Nonsingular Matrices. Find x 1, x 2, x 3 such that the following three equations hold:
Linear Systems Example: Find x, x, x such that the following three equations hold: x + x + x = 4x + x + x = x + x + x = 6 We can write this using matrixvector notation as 4 {{ A x x x {{ x = 6 {{ b General
More informationFace Recognition using Principle Component Analysis
Face Recognition using Principle Component Analysis Kyungnam Kim Department of Computer Science University of Maryland, College Park MD 20742, USA Summary This is the summary of the basic idea about PCA
More informationCSCI 445 Amin Atrash. Ultrasound, Laser and Vision Sensors. Introduction to Robotics L. Itti & M. J. Mataric
Introduction to Robotics CSCI 445 Amin Atrash Ultrasound, Laser and Vision Sensors Today s Lecture Outline Ultrasound (sonar) Laser rangefinders (ladar, not lidar) Vision Stereo vision Ultrasound/Sonar
More informationAn Iterative Image Registration Technique with an Application to Stereo Vision
An Iterative Image Registration Technique with an Application to Stereo Vision Bruce D. Lucas Takeo Kanade Computer Science Department CarnegieMellon University Pittsburgh, Pennsylvania 15213 Abstract
More informationReal Time Traffic Light Control System (Hardware and Software Implementation)
International Journal of Electronic and Electrical Engineering. ISSN 09742174, Volume 7, Number 5 (2014), pp. 505510 International Research Publication House http://www.irphouse.com Real Time Traffic
More informationMonte Carlo Method: Probability
John (ARC/ICAM) Virginia Tech... Math/CS 4414: The Monte Carlo Method: PROBABILITY http://people.sc.fsu.edu/ jburkardt/presentations/ monte carlo probability.pdf... ARC: Advanced Research Computing ICAM:
More informationComputer Vision  part II
Computer Vision  part II Review of main parts of Section B of the course School of Computer Science & Statistics Trinity College Dublin Dublin 2 Ireland www.scss.tcd.ie Lecture Name Course Name 1 1 2
More informationEnhanced High Resolution Colour Image Using Discrete and Stationary Wavelet Transform
Enhanced High Resolution Colour Image Using Discrete and Stationary Wavelet Transform Sri M.V. Bhavani Sankar 1, Sri K. Suresh Babu 2 1 (Professor of Dept of ECE, DVR & Dr HS MIC college of Technology,
More information