IMAGE ENHANCEMENT II (CONVOLUTION)

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Melvin Goodwin
6 years ago
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1 MOTIVATION Recorded images often exhibit problems such as: blurry noisy Image enhancement aims to improve visual quality Cosmetic processing Usually empirical techniques, with ad hoc parameters ( whatever works ) Distinct from restoration, which is processing designed to recover true object using blur and noise reduction on a degraded image ECE/OPTI533 Digital Image Processing class notes 65 Dr. Robert A. Schowengerdt 2003

2 TWO-COMPONENT IMAGE MODEL Write any image as the sum of two components, image ( mn, ) = LP ( m, n ) + HP ( m, n ) The Low-Pass (LP) component contains the large area variations, which determine the local-to-global contrast ( macro-contrast ) of the image The High-Pass (HP) component contains the small area variations, which determine the sharpness and neighborhood contrast ( micro-contrast ) Extract the LP component by convolution with a Low-Pass Filter (LPF) Weighted moving average of input pixels The corresponding HP component is equal to the original image minus the LP component Fourier filters can also be used to extract image components ECE/OPTI533 Digital Image Processing class notes 66 Dr. Robert A. Schowengerdt 2003

3 2-D LP and HP examples image(m,n) LP(m,n) = HP(m,n) 3 x 3 LPF 7 x 7 LPF ECE/OPTI533 Digital Image Processing class notes 67 Dr. Robert A. Schowengerdt 2003

4 Convolution Filter for HP Component Linearity of convolution implies LP and HP filters add image ( mn, ) = LP ( m, n ) + HP ( m, n ) = = = LPF imag emn (, ) + HPF imag emn (, ) [ LPF + HPF ] imag emn (, ) δ ( mn, ) imag emn (, ) 3 x 3 example n n = LPF + n HPF m /9x /9x m m ECE/OPTI533 Digital Image Processing class notes 68 Dr. Robert A. Schowengerdt 2003

5 -D LP and HP examples input 20 input 00 x3 LP 00 x7 LP index 50 input 00 x3 HP DN DN input x7 HP 50 DN DN index index index ECE/OPTI533 Digital Image Processing class notes 69 Dr. Robert A. Schowengerdt 2003

6 General Filter Properties LPFs preserve large features remove small features Example histograms original DN sum of filter weights = preserve image mean, reduce variance HPFs preserve small features DN remove large features sum of filter weights = 0 zero image mean, greatly reduce variance DN number of pixels number of pixels number of pixels LP-component HP-component ECE/OPTI533 Digital Image Processing class notes 70 Dr. Robert A. Schowengerdt 2003

7 High-Boost Filter (HBF) Boosts amplitude of the high spatial frequency components (details) ( unsharp masking darkroom technique) Parametric example HB ( m, n ; K ) = original ( m, n ) + K HP ( m, n) = ( K + )original m, n ( ) K LP m n (, ), K > 0 3 x 3 convolution filter weights for K =, 2 and 3 K = K = 2 K = Larger K increases high-frequency boost, but also amplifies noise ECE/OPTI533 Digital Image Processing class notes 7 Dr. Robert A. Schowengerdt 2003

8 Application to photographic image original K = noise amplification trade-off with sharpening K = 2 K = 3 ECE/OPTI533 Digital Image Processing class notes 72 Dr. Robert A. Schowengerdt 2003

9 HBFs increase DN range Must clip GL at 0 and 255 to maximize contrast original 6 < DN < 23 ECE/OPTI533 Digital Image Processing class notes 73 Dr. Robert A. Schowengerdt 2003

10 HB with K = -53 < DN < 30 min-max stretch clipped ECE/OPTI533 Digital Image Processing class notes 74 Dr. Robert A. Schowengerdt 2003

11 HB with K = 2-32 < DN < 47 min-max stretch clipped ECE/OPTI533 Digital Image Processing class notes 75 Dr. Robert A. Schowengerdt 2003

12 HB with K = 3-28 < DN < 525 min-max stretch clipped ECE/OPTI533 Digital Image Processing class notes 76 Dr. Robert A. Schowengerdt 2003

13 HB original = K x HP HB with K = 3 unstretched histogram-equalized ECE/OPTI533 Digital Image Processing class notes 77 Dr. Robert A. Schowengerdt 2003

14 Directional HPFs Extract details in a given direction Directional st and 2nd derivatives Convolution filter weights direction m n m=n azimuthal st derivative sin α 0 -sinα-cosα cos α 2nd derivative Can also implement directional LPFs to remove details in a given direction ECE/OPTI533 Digital Image Processing class notes 78 Dr. Robert A. Schowengerdt 2003

15 Example directional HPF results derivative direction ECE/OPTI533 Digital Image Processing class notes 79 Dr. Robert A. Schowengerdt 2003

16 Cascaded Linear Filters Example Sequentially applied convolution filters can be cascaded into a single filter h /9x n m Example with two filters g = ( f h ) h 2 = = f ( h h 2 ) f h net where the equivalent filter is: h net = h h 2 linear not circular convolution h 2 h net /9x /9x n n m m symmetric ECE/OPTI533 Digital Image Processing class notes 80 Dr. Robert A. Schowengerdt 2003

17 The Box-Filter Algorithm Very efficient recursive algorithm for certain configurations of window weights Box-Filter Algorithm (along rows) generate column sum of weighted input pixels add column sums to obtain current output pixel (leftmost output pixel in row only) move window one pixel along row subtract leftmost column sum and add new rightmost column sum to obtain current output pixel repeat steps 3 and 4 until done To apply along rows, the weights within each window row must be constant (but do not have to be equal from row-to-row) ECE/OPTI533 Digital Image Processing class notes 8 Dr. Robert A. Schowengerdt 2003

18 Box-filter algorithm (3 x 3 example) column sums C m C m2 C m3 C m C m2 C m3 C m4 at beginning of each output row m k C m = f k k C m2 = f k 2 k C m3 = f k 3 (, )hm ( k, ) (, )hm ( k, 2) (, )hm ( k, 3) gm2 (, ) = C m + C m2 + C m3 at next pixel in output row m gm3 (, ) = C m2 + C m3 + C m4 = gm2 (, ) C m + C m4 ECE/OPTI533 Digital Image Processing class notes 82 Dr. Robert A. Schowengerdt 2003

19 Number of operations/output pixel 3 x 3 filter 3 multiplies (C m4 ) 2 adds (C m4 ) 2 adds (pixel recursion) 7 total/output pixel w x w2 filter w multiplies (w - ) adds 2 adds 2w + total/output pixel Number of operations grows linearly with number of rows in filter Can also be implemented in row direction by maintaining stack of w most-recently processed rows in memory For large images, speed is nearly independent of filter size ECE/OPTI533 Digital Image Processing class notes 83 Dr. Robert A. Schowengerdt 2003

20 Extension of box-filter algorithm Some filters can be written as combinations of box filters, e.g. high-pass = original image - low-pass or approximated by cascaded box filters, e.g. Gaussian LPF LPF LPF Example decomposition into box filters n n n n h net /35x = h + h 2 + h 3 /25x m /9x m m m symmetric symmetric symmetric ECE/OPTI533 Digital Image Processing class notes 84 Dr. Robert A. Schowengerdt 2003

21 LINEAR VERSUS CIRCULAR CONVOLUTION border region for 3 x 3 filter Linear Strong practical reasons to maintain same size output image as input image (N x N) Must fill empty rows and columns around border of output image Several possible solutions (aka tricks ): extend input image with zeros or other values (nearest pixel or image mean); truncate after linear convolution to N x N repeat the nearest valid output pixels to N x N reduce the window size near border Not equal to convolution using Discrete Fourier Transforms Non-physical result in border region ECE/OPTI533 Digital Image Processing class notes 85 Dr. Robert A. Schowengerdt 2003

22 Circular (periodic) 2-D periodic extension of image Assume image and filter are each M x N Modify defining equation, g cir mn, M ( ) = f k l k = 0 N l = 0 (, )h c m k (, n l) where h c ( mn, ) = hm ( modulom, n modulon ) Equivalent to: linear convolution of W x W filter with periodic extension of M x N image extract center M x N result for output image Equal to convolution using Discrete Fourier Transforms Non-physical result in border region ECE/OPTI533 Digital Image Processing class notes 86 Dr. Robert A. Schowengerdt 2003

23 Example with 3 x 3 filter ECE/OPTI533 Digital Image Processing class notes 87 Dr. Robert A. Schowengerdt 2003

Empirical Mean and Variance!

Empirical Mean and Variance! Global Image Properties! Global image properties refer to an image as a whole rather than components. Computation of global image properties is often required for image enhancement, preceding image analysis.!