CITS 4402 Computer Vision
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1 CITS 4402 Computer Vision Prof Ajmal Mian Adj/A/Prof Mehdi Ravanbakhsh, CEO at Mapizy ( and InFarm ( Lecture 04 Greyscale Image Analysis
2 Lecture 03 Summary Images as 2-D signals Linear and non-linear filters Fourier series Fourier Transform Discrete Fourier Transform 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 2
3 Image Enhancement Aim: Make images easier to interpret for the human eye Generate better input for other image processing techniques There are two main categories of techniques Spatial domain methods which operate directly on pixels Frequency domain methods which operate on the Fourier transform of the image 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 3
4 Single Pixel Manipulation The value of g(x, y) depends directly on the value of f x, y This is a greyscale transformation or simply mapping Common mapping functions: Identity transformation Thresholding 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 4
5 Common Mapping Functions lightening increase contrast 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 5
6 Common Mapping Functions enhance contrast in dark regions enhance contrast in light regions 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 6
7 (Hawkes Bay, NZ) 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 7
8 Sample MATLAB Code The Matlab code for enhancing the Hawkes Bay image on the previous slide is extremely simple! im = imread('unequalized_hawkes_bay_nz.jpg'); Imshow(im); % after obesrving the intensity histogram of the input image... f = 0:255; % input intensity g(1:100) = linspace(0,20,100); g(101:200) = linspace(21,240,100); g(201:256) = linspace(241,255,56); figure, plot(f,g,'b-'); % generate output image newim = g(im+1); newim = uint8(newim); % convert to uint8 figure, imshow(newim); 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 8
9 Histogram Equalization stretch out the histogram to produce a more uniform distribution Image data squashed into a small range of grey values Ideally we want the image data to spread uniformly over all grey values 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 9
10 Histogram Equalization For digital images, we have a discrete formulation. Let n k number of pixels with grey level k N total number of pixels Then the probability of obtaining grey level k in input image f is: P f f k = n k N The transformation is: g k = T f k = Note: the values of g k will have to be scaled up by 255 and rounded to the nearest integer. This discretization of g k means that the transformed image will not have a perfectly uniform distribution Indeed, the total number of distinct grey levels is reduced! k i=0 ni N 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 10
11 Histogram Equalization (cont.) An example: suppose that we have a image with 8 grey levels: f i n i n i /N g i (ideally) /7=0 1/ /7=0.14 2/ /7=0.28 3/ /7=0.42 4/ /7=0.57 5/ /7=0.71 6/ /7= /7=1 We want the cumulative distribution N=4096 of the output image to look as much like a 45 line as possible. 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 11
12 Histogram Equalization (cont.) f i n i n i /N g i (ideally) /7=0 1/ /7=0.14 2/ /7=0.28 3/ /7=0.42 4/ /7=0.57 5/ /7=0.71 6/ /7= /7=1 Thus intensity mapping is: InputIntensity OutputIntensity 0 1/7 1/7 3/7 2/7 5/7 3/7 6/7 4/7 6/7 5/7 1 6/ Input image has 8 grey levels Output image has only 5 grey levels 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 12
13 Histogram Equalization (cont.) f i n i n i /N g i n i of g i / / / / / / Thus intensity mapping is: InputIntensity OutputIntensity 0 1/7 1/7 3/7 2/7 5/7 3/7 6/7 4/7 6/7 5/7 1 6/ Input image has 8 grey levels Output image has only 5 grey levels 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 13
14 Histogram Equalization (cont.) f i n i n i /N g i n i of g i / / / / / / /19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 14
15 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 15
16 A very noisy image 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 16
17 Using Neighborhood Pixels So far we modified values of single pixels What if we want to take neighborhood into account? A common application of linear filtering is image smoothing using an averaging filter or averaging mask or averaging kernel Each point in the smoothed image g(x, y) is obtained from the average pixel value in a neighbourhood of (x, y) in the input image The averaging filter is also known as the box filter. Each pixel under the mask is multiplied by 1/9, summed, and the result is placed in the output image The mask is successively moved across the image. That is, we convolve the image with the mask /19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 17
18 Example of 2-D Convolution Convolution with box filters of size 1,5,15,3,9,35 (reading order) 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 18
19 Image filtering g.,. g.,. = 1 9 f [.,.] h[.,.] h[ m, n] g[ k, l] f [ m k, n l] k, l 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 19 Credit: S. Seitz
20 Image filtering g.,. g.,. = 1 9 f [.,.] h[.,.] h[ m, n] g[ k, l] f [ m k, n l] k, l 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 20 Credit: S. Seitz
21 Image filtering g.,. g.,. = 1 9 f [.,.] h[.,.] h[ m, n] g[ k, l] f [ m k, n l] k, l 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 21 Credit: S. Seitz
22 Image filtering g.,. g.,. = 1 9 f [.,.] h[.,.] h[ m, n] g[ k, l] f [ m k, n l] k, l 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 22 Credit: S. Seitz
23 Image filtering g.,. g.,. = 1 9 f [.,.] h[.,.] h[ m, n] g[ k, l] f [ m k, n l] k, l 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 23 Credit: S. Seitz
24 Image filtering g.,. g.,. = 1 9 f [.,.] h[.,.] ? h[ m, n] g[ k, l] f [ m k, n l] k, l 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 24 Credit: S. Seitz
25 Image filtering g.,. g.,. = 1 9 f [.,.] h[.,.] ? h[ m, n] g[ k, l] f [ m k, n l] k, l 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 25 Credit: S. Seitz
26 Image filtering g.,. g.,. = 1 9 f [.,.] h[.,.] h[ m, n] g[ k, l] f [ m k, n l] k, l 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 26 Credit: S. Seitz
27 Practice with Linear Filters ? Original 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 27
28 Practice with Linear Filters Original Filtered (no change) 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 28
29 Practice with Linear Filters ? Original 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 29
30 Practice with Linear Filters Original Shifted left By 1 pixel 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 30
31 Practice with Linear Filters ? Original (Note that filter sums to 1) 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 31
32 Practice with Linear Filters Original Sharpening filter - Accentuates differences with local average 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 32
33 Sharpening 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 33 Source: D. Lowe
34 Other Averaging Filters One expects the value of a pixel to be more closely related to the values of pixels close to it than to those further away Accordingly it is usual to weight the pixels near the centre of the mask more strongly than those at the edge 1? /19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 34
35 Other Averaging Filters One expects the value of a pixel to be more closely related to the values of pixels close to it than to those further away Accordingly it is usual to weight the pixels near the centre of the mask more strongly than those at the edge /19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 35
36 Important filter: Gaussian Gaussian Filter x 5, = 1 σ 2 is also known as the width of the kernel Use Matlab s fspecial function to create a Gaussian filter. 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis Slide credit: The University Christopher of Western Australia Rasmussen 36
37 Smoothing with box filter Example Box Filter Smoothing 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 37
38 Smoothing with Gaussian filter Example Gaussian Smoothing 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 38
39 Key Properties of Linear Filters Linearity: filter(f 1 + f 2 ) = filter(f 1 ) + filter(f 2 ) Shift invariance: same behavior regardless of pixel location filter(shift(f)) = shift(filter(f)) Any linear, shift-invariant operator can be represented as a convolution 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 39 Source: S. Lazebnik
40 Key Properties of Linear Filters Commutative: a * b = b * a Conceptually no difference between filter and signal Associative: a * (b * c) = (a * b) * c Often apply several filters one after another: (((a * b 1 ) * b 2 ) * b 3 ) This is equivalent to applying one filter: a * (b 1 * b 2 * b 3 ) Distributes over addition: a * (b + c) = (a * b) + (a * c) Scalars factor out: ka * b = a * kb = k (a * b) Identity: unit impulse e = [0, 0, 1, 0, 0], a * e = a 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 40 Source: S. Lazebnik
41 Gaussian Filters Linear filters Remove high-frequency components from the image (low-pass filter) Images become more smooth Convolution of a Gaussian with a Gaussian is another Gaussian So can smooth with small-width kernel, repeat, and get same result as larger-width kernel would have Convolving two times with Gaussian kernel of width σ is same as convolving once with kernel of width σ 2 Separable kernels Factors into product of two 1D Gaussians 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 41
42 Separability of the Gaussian filter Gaussian Filters 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 42
43 Practical Matters What about near the edge? The filter window falls off the edge of the image Need to extrapolate Methods: clip filter (black) wrap around copy edge reflect across edge 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 43
44 Q? Practical Matters Methods (MATLAB): clip filter (black): imfilter(f, g, 0) wrap around: imfilter(f, g, circular ) copy edge: imfilter(f, g, replicate ) reflect across edge: imfilter(f, g, symmetric ) 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 44
45 Practical Matters MATLAB: filter2(g, f, shape) shape = full : output size is sum of sizes of f and g shape = same : output size is same as f shape = valid : output size is difference of sizes of f and g g g g g g g f f f g g g g g full same valid g 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 45
46 Low-Pass Filtering Removing all high spatial frequencies from a signal to retain only low spatial frequencies is called low-pass filtering.? Old Spectrum New Spectrum Low-Pass Filtered Image 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 46
47 High-Pass Filtering Removing all low spatial frequencies from a signal to retain only high spatial frequencies is called high-pass filtering.? Old Spectrum New Spectrum High-Pass Filtered Image 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 47
48 Low-Pass Filtering An Example 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 48
49 Low-Pass Filtering An Example 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 49
50 High-Pass Filtering An Example 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 50
51 High-Pass Filtering An Example 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 51
52 Filtering in the Spatial Domain Low-pass filtering -> convolve the image with a box / Gaussian filter High-pass filtering ->? 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 52
53 Filtering in the Spatial Domain Low-pass filtering -> convolve the image with a box / Gaussian filter High-pass filtering -> -1/9-1/9-1/9-1/9 8/9-1/9-1/9-1/9-1/9 Since the sum of the weights is 0, the resulting signal will have a 0 DC value (i.e., the average value or the coefficient of the zero frequency component). To display the image you will need to take the absolute values. 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 53
54 Non-Linear Filtering Neighbourhood averaging or Gaussian smoothing will tend to blur edges because the high frequency components are attenuated An alternative approach is to use median filtering. Here we set the grey level to be the median of the pixel values in the neighbourhood Example: pixel values in 3 3 neighbourhood median value Sort the values Pixels with outlying values are forced to become more like their neighbours 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 54
55 Median Filtering An Example Median filter removes outliers Median filter smooths the image without blurring the edges 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 55
56 High-Boost Filtering Here we take the original image and boost the high frequency components. Can think of HighPass = Original LowPass. Thus HighBoost = b*original LowPass = (b-1)*original + Original LowPass = (b-1)*original + HighPass b is the boosting factor. When b=1, HighBoost = HighPass High-boost filtering useful for emphasizing high frequencies while retaining some low frequency components of the original image to aid in the interpretation of the image 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 56
57 High Boost Filtering (cont.) How can we perform high-boost filtering in the spatial domain? -1/9-1/9-1/9-1/9 w/9-1/9-1/9-1/9-1/9 where w = 9*b - 1 Original High Boosted (intermediate result) High Boosted (contrast stretching) 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 57
58 Homomorphic Filtering The brightness of an image point f(x, y) is a function of the illumination at that point and the reflectance of the object at that point, i.e., f x, y = i x, y r(x, y) illumination 0 i x, y < reflectance 0 r x, y 1 It is the reflectance that tells us information about the scene. We want to reduce the influence of illumination. Assumptions: 1. Illumination variations vary with low spatial frequency 2. Features of interest are the result of different reflectance properties of objects across the image and these vary with high(er) spatial frequency 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 58
59 Homomorphic Filtering (cont.) Let z x, y = log f x, y = log i x, y + log r x, y In the frequency domain, we have Z ω, ν = I ω, ν + R ω, ν Fourier transform of log(i x, y ) Fourier transform of log(r x, y ) If we apply a high boost filter, then we can suppress the low frequency illumination components and enhance the reflectance components Z ω, ν represents the Fourier Transform of the sum of two images: a low frequency illumination image a high frequency reflectance image 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 59
60 Homomorphic Filtering (cont.) Apply the high-boost filter H ω, ν : S ω, ν = Z ω, ν. H ω, ν Take the inverse FFT: s x, y = F 1 (S ω, ν ) Finally, exponentiate the result to account for taking the log of the original image: g x, y = exp(s x, y ) High-boost f(x, y) log FFT FFT filtering -1 exp g(x, y) 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 60
61 Homomorphic Filtering An Example Original image Filtered image 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 61
62 Smoothing and Sub-sampling In many Computer Vision applications, sub-sampling is often needed, e.g., to build an image pyramid, or simply to reduce the resolution for efficient storage, transmission, processing, 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 62
63 Throw away every other row and column to create a 1/2 size image 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 63
64 Sub-sampling Issues Sub-sampling may be dangerous. Why? Resample the checkerboard by taking one sample at each circle. In the case of the top left board, new representation is reasonable. Top right also yields a reasonable representation. Bottom left is all black (dubious) and bottom right has checks that are too big. 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 64
65 Aliasing in Videos 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 65
66 Aliasing in Graphics 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 66
67 Sampling and Aliasing Top row shows the images, sampled at every second pixel to get the next; bottom row shows the magnitude of frequency spectrum of these images. 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 67
68 Anti-aliasing 1. Sample more often 2. Get rid of all frequencies that are greater than half the new sampling frequency (Nyquist frequency) Will lose information But it s better than aliasing Apply a smoothing filter 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 68
69 Sampling and Aliasing Sampling with smoothing. Top row shows the images. We get the next image by smoothing the image with a Gaussian with sigma 1 pixel, then sampling at every second pixel to get the next; bottom row shows the magnitude spectrum of these images. 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 69
70 Subsampling without Pre-filtering 1/2 1/4 (2x zoom) 1/8 (4x zoom) 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 70
71 Subsampling with Gaussian Pre-filtering Gaussian 1/2 G 1/4 G 1/8 Slide by Steve Seitz 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 71
72 Image Pyramids Key component of many high level computer vision tasks How to create an image pyramid? Represent each image as a layer 1. Convolve layer i with a Gaussian filter 2. Subsample layer i by a factor of two (remove all even-numbered rows and columns) to get layer i + 1 Repeat steps 1 and 2 until a stopping criteria is satisfied 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 72
73 Image Pyramids 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 73
74 Template Matching Finding object of known shape and appearance in an image To identify the object, we have to compare the template image against the source image by sliding it. At each location, we find the matching score between the image and the template 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 74
75 Template Matching Scores Sum of Squared Difference (SSD) in pixel values (Normalized) Correlation coefficient (Normalized) Cross-correlation Template Source Image 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 75
76 Template Matching Scores SSD Normalized Correlation Coefficient Normalized Cross-Correlation 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 76
77 Summary Single Pixel Operations Histogram Equalization Filtering in the Spatial Domain Subsampling and Anti-aliasing Template Matching Acknowledgements: The slides are based on previous lectures by A/Prof Du Huynh and Prof Peter Koveski. Other material has been taken from Wikipedia, computer vision textbook by Forsyth & Ponce, and OpenCV documentation. 3/19/2018 Computer Vision - Lecture 04 - Greyscale Image Analysis The University of Western Australia 77
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