Spatial Filtering Fundamentals

4/28/2008 Spatial filtering fundamentals by  Gleb  V. Tcheslavski:  [email  protected] lamar. edu http://ee. lamar. edu/gleb/dip/index. htm Spring 2008 ELEN 4304/5365 DIP 1 Mechanics of spatial filtering Considering frequency domain filtering, the effect of LPF applied to an image is to blur (smooth) it. Similar smoothing effect can be achieved by using spatial filters (spatial masks, kernels, templates, or windows). We discussed that a spatial filter consists of a neighborhood and a pre-defined operation performed on the image pixels defining the neighborhood.The result of filtering – a new pixel with coordinated of the neighborhood’s center and the value defined by the operation. g y p If the operation is linear, the filter is said to be a linear spatial filter. Spring 2008 ELEN 4304/5365 DIP 2 1 4/28/2008 Mechanics of spatial filtering Assuming a 3 x 3 neighborhood, at any point (x,y) in the image, the response of the spatial filter is g ( x, y ) = w(? 1, ? 1 ) f ( x ? 1, y ? 1) + w(? 1, 0) f ( x ? 1, y ) + †¦ + w(0, 0) f ( x, y ) + †¦ + w(1,1) f ( x + 1, y + 1) Filter coefficient Pixel intensity In general: g ( x, y ) = s =? a t =? b ? ? w(s, t ) f ( x + s, y + t ) a bSpring 2008 ELEN 4304/5365 DIP 3 Mechanics of spatial filtering Here a mask size is m x n. m = 2a + 1 n = 2b + 1 Where a and b are some integers. For a 3 x 3 mask Spring 2008 ELEN 4304/5365 DIP 4 2 4/28/2008 Spatial correlation and convolution Correlation is a process of moving the filter mask over the image and computing the sum of products at each location as previously described. Convolution is the same except that the filter is first rotated by 1800. For a 1D case, we first zeropad f by m-1 zeros on each size. We compute a sum of products in both cases†¦ Spring 2008 ELEN 4304/5365 DIP 5 Spatial correlation and convolutionCorrelation is a function of displacement of the filter. A function containing a single 1 with the rest being zeros is g g g called a d iscrete unit impulse. Correlation of a function with a discrete unit impulse yields a rotated version of a function at the location of the impulse. To perform a convolution, we need to pre-rotate the filter by 1800 and perform the same operation as in correlation. Spring 2008 ELEN 4304/5365 DIP 6 3 4/28/2008 Spatial correlation and convolution In a 2D case, for a filter of size m x n, we pad the image with m-1 rows of zeros at the top and bottom and n-1 columns of zeros on the left and right.For convolution, we pre-rotate the mask and perform the sliding sum of products. Spring 2008 ELEN 4304/5365 DIP 7 Spatial correlation and convolution Correlation of a filter w(x,y) of size m x n with an image f(x,y) is w( x, y ) f ( x, y ) = s =? a t =? b ? ? w(s, t ) f ( x + s, y + t ) ? ? w(s, t ) f ( x ? s, y ? t ) a b a b Convolution of a filter w(x,y) of size m x n with an image f(x,y) is w( x, y ) ? f ( x, y) = s =? a t =? b Spring 2008 ELEN 4304/5365 DIP 8 4 4/28/2008 Vector representatio n of linear filtering It is convenient sometimes to represent a sum of products asR = ? wk zk = w T z k =1 Filter coeffs Image intensities mn For example, for a 3 x 3 filter: p , R = ? wk zk = w T z k =1 Spring 2008 ELEN 4304/5365 DIP 9 9 Generating spatial filter masks Generating an m x n linear spatial filter requires specification of mn mask coefficients. These coefficients are selected based on what the filter is supposed to do keeping in mind that all we can do with linear filtering is to implement a sum of products. Assuming that we need to replace the pixels in an image with the average pixel intensities of a 3Ãâ€"3 neighborhood centered on those pixels.If zi are the intensities, the average is R= 9 1 9 ? zi 9 i =1 Which is: R = ? wi zi = w T z; i =1 ELEN 4304/5365 DIP wi = 1 9 10 Spring 2008 5 4/28/2008 Smoothing spatial filters Smoothing filters are used for blurring and noise reduction. Blurring may be implemented in preprocessing tasks to remove small details from an ima ge prior to large object extraction. The output of a smoothing (averaging or lowpass) linear spatial filter is the average of the pixels contained in the neighborhood of the filter mask.By replacing the value of every pixel in an image by the average of the intensity levels in the neighborhood defined by a filter mask, the resulting image will have reduced â€Å"sharp† transitions in intensities. Since random noise typically corresponds to such transitions, we can achieve denoising. Spring 2008 ELEN 4304/5365 DIP 11 Smoothing spatial filters However, edges (characterized by sharp intensity transitions) will be blurred. Examples of such masks: 1) A box filter – spatial averaging filter 3Ãâ€"3; 2) Weighted average filter – attempt to reduce blurring: g a g ( x, y ) = s =? a t =? b ? ? (s, t ) f ( x + s, y + t ) s =? a t =? b b ? ? w(s, t ) 12 a b Spring 2008 ELEN 4304/5365 DIP 6 4/28/2008 Smoothing spatial filters The effect of filter size. The original 500Ãâ€"5 00 image And the results of smoothing with a square averaging filter of sizes m = 3, 5, 9, 15, 25, and 35 pixels. Spring 2008 ELEN 4304/5365 DIP 13 Smoothing spatial filters Frequently, blurring is desired for ease of object detection: an original Hubble image, the result of applying a 15Ãâ€"15 averaging mask to it and the result of thresholding with a threshold of 25% of the highest intensity. Spring 2008 ELEN 4304/5365 DIP 14 7 4/28/2008Order-statistic (nonlinear) filters Order-statistic filter are nonlinear spatial filters whose response is based on ordering (Ranking) the pixels in the neighborhood and then replacing the value of the center pixel by the value determined by the ranking result. The median filters are quite effective against the impulse noise (salt-and-pepper noise). The median of a set of values is such that half the values in the set are greater than the median and half is lesser than it: Ex: the 3Ãâ€"3 neighborhood has values (10, 20, 20, 20,15, 20, 100, 25, 20 ). These values are ranked as (10, 15, 20, 20, 20, 20, 20, 25, 100).The median will be 20. There are also max and min filters. Spring 2008 ELEN 4304/5365 DIP 15 Order-statistic (nonlinear) filters Original image with salt-andpepper noise Spring 2008 Noise reduction with a 3Ãâ€"3 averaging mask ELEN 4304/5365 DIP Noise reduction with a 3Ãâ€"3 median mask 16 8 4/28/2008 Sharpening spatial filters: foundations The main objective of sharpening is to highlight transitions in intensity. Since averaging is analogous to spatial integration, we y g g g p g can assume that sharpening is analogous to differentiation in space. The derivatives of a digital function are defined in differences.The first derivative must be: 1) Zero in areas of constant intensity; 2) Non-zero at the onset and end of an intensity step or ramp; 3) Non-zero along ramps of constant slope. The second derivative must be: 1) Zero in areas of constant intensity; 2) Non-zero at the onset and end of an intensity step or ram p; 3) Zero along ramps of constant slope. Spring 2008 ELEN 4304/5365 DIP 17 Sharpening spatial filters: foundations The first-order derivative: ?f = f ( x + 1) ? f ( x) ? x The second-order derivative: ?2 f = f ( x + 1) + f ( x ? 1) ? 2 f ( x) ? x 2 It can be verified that these definitions satisfy the conditions for derivatives.Spring 2008 ELEN 4304/5365 DIP 18 9 4/28/2008 Sharpening spatial filters: foundations The circles indicate the onset or end of intensity transitions. The sign of the second derivative changes at the onset and end of a step of ramp. The second derivative enhances fine details much better than the first derivative. This is suitable for sharpening. Spring 2008 ELEN 4304/5365 DIP 19 Using the second derivative for image sharpening – the Laplacian We consider isotropic filters – the response is independent of the direction of the discontinuity in the image Such filters are image. rotation invariant.The simplest isotropic derivative operator is the L aplacian: ?2 f ? 2 f ? f = 2 + 2 ? x ? y 2 Therefore: ? 2 f = f ( x + 1, y ) + f ( x ? 1, y ) + f ( x, y + 1) + f ( x, y ? 1) ? 4 f ( x, y ) The Laplacian is a linear operator since derivatives are linear operators. Spring 2008 ELEN 4304/5365 DIP 20 10 4/28/2008 Using the second derivative for image sharpening – the Laplacian The Laplacian can be implemented by these filter masks Since the Laplacian is a derivative operator, its use highlights intensity discontinuities in the image and deemphasize regions with slow varying intensity levels levels.It tends to produce images having grayish edge lines and other discontinuities, and a dark, feature-less background. Spring 2008 ELEN 4304/5365 DIP 21 Using the second derivative for image sharpening – the Laplacian Background features can be preserved together with the sharpening effect of the Laplacian by adding the Laplacian image to the original. If the definition of the Laplacian has a negative central coefficient, the La placian image must be subtracted rather than added to obtain a sharpening result. In general: g ( x, y ) = f ( x, y ) + c 2 f ( x, y ) ? ? ?Output intensity Input intensity -1 – if the center is negative; +1 otherwise Spring 2008 ELEN 4304/5365 DIP 22 11 4/28/2008 Using the second derivative for image sharpening – the Laplacian The Laplacian Laplacian with scaling The original (blurred) image The image sharpened with mask 2 The image sharpened with mask 1 Spring 2008 ELEN 4304/5365 DIP 23 Unsharp masking and highboost filtering An approach used for many years to sharpen images is: 1. Blur the original image; 2. Subtract the blurred image from the original (the result is called the mask): g mask ( x, y ) = f ( x, y ) ? f ( x, y ) Original Blurred image 3.Add the mask to the original: g ( x, y ) = f ( x, y ) + k ? g mask ( x, y ) Here k is a weight. Spring 2008 ELEN 4304/5365 DIP 24 12 4/28/2008 Unsharp masking and highboost filtering When k = 1 – unsharp masking ; k > 1 – highboost filtering; k < 1 – de-emphasize the contribution of a mask. The shown intensity profile can be viewed as a horizontal scan through a vertical edge transition from a dark to li ht t a light region. i This approach is similar to Laplacian method. Spring 2008 ELEN 4304/5365 DIP 25 Unsharp masking and highboost filtering Original ( slightly blurred) image Smoothed with a Gaussian smoothing filter 5Ãâ€"5 Unsharp maskResult of using unshapr mask (k = 1) Result of using highboost filtering with k = 4. 5 Spring 2008 ELEN 4304/5365 DIP 26 13 4/28/2008 Gradient method First derivatives can be implemented for nonlinear image sharpening using the magnitude of the gradient: ? ? f ? g x ? ? ? x ? ? ? f ? grad ( f ) ? ? ? = ? ? ? g y ? ? ? f ? ? ? y ? ? ? The gradient vector points in the direction of the greatest rate of g (x,y). g (length) gradient change of f at location ( y) The magnitude ( g ) of g 2 2 M ( x, y ) = ? f = g x + g y Is the value of rate of cha nge at (x,y) in the direction of gradient. Spring 2008ELEN 4304/5365 DIP 27 Gradient method M(x,y) is an image of the same size as the original and is called the gradient image. Magnitude makes M(x,y) non-linear. It is more s itable in some applications to use: suitable se M ( x, y ) ? g x + g y For an image where z5 represent the pixel f(x,y) and z1 represent the pixel f(x-1,y-1), the simplest (Roberts) definitions for gradients are: M ( x, y ) = ( z9 ? z5 ) + ( z8 ? z6 ) 2 2 M ( x, y ) ? z9 ? z5 + z8 ? z6 However, Roberts cross-gradient operators lead to masks of even sizes, which is inconvenient. ELEN 4304/5365 DIP 28 Spring 2008 14 4/28/2008 Gradient methodThe smallest masks with central symmetry (ones we are interested in) are 3Ãâ€"3. The gradient can be approximated for such masks as following: ?f = ( z7 + 2 z8 + z9 ) ? ( z1 + 2 z2 + z3 ) ? x ? f gy = = ( z3 + 2 z6 + z9 ) ? ( z1 + 2 z4 + z7 ) ? y Therefore, the mask could be: gx = M ( x, y ) ? ( z7 + 2 z8 + z9 ) ? ( z1 + 2 z2 + z3 ) + ( z3 + 2 z6 + z9 ) ? ( z1 + 2 z4 + z7 ) Roberts operators They are Sobel operators. Spring 2008 ELEN 4304/5365 DIP 29 Gradient method The coefficients in all masks shown sum to zero. This indicates that mask will give a zero response in an area of constant intensity as expected of a derivative operator operator.Original image of contact lens Sobel gradient Defect Spring 2008 ELEN 4304/5365 DIP 30 15 4/28/2008 Combining spatial enhancement techniques Frequently, Frequently a combination of several methods is used to enhance an image†¦ 1) Original image – 2) Laplacian – 3) image sharpened by Laplacian – 4) Sobel gradient of the original image – 5) Sobel image smoothed with a 5Ãâ€"5 averaging filter – 6) product of Sobel image with its smoothed version – 7) sharpened image (a sum of the original and 6) – 8) power-law transformation. Spring 2008 ELEN 4304/5365 DIP 31 Spring 2008 ELEN 4304/5365 DIP 32 16