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CSCE 641 Computer Graphics: Image Processing. Jinxiang Chai. Review. 2D Fourier Transform Nyquist sampling theory - Sampling Rate ≥ 2 * max frequency in the signal Antialiasing - prefiltering Gaussian pyramid - Mipmap. Review. 2D Fourier Transform Nyquist sampling theory

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## CSCE 641 Computer Graphics: Image Processing

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**CSCE 641 Computer Graphics:Image Processing**Jinxiang Chai**Review**2D Fourier Transform Nyquist sampling theory - Sampling Rate ≥ 2 * max frequency in the signal Antialiasing - prefiltering Gaussian pyramid - Mipmap**Review**2D Fourier Transform Nyquist sampling theory - Sampling Rate ≥ 2 * max frequency in the signal Antialiasing - prefiltering Gaussian pyramid - Mipmap**Review: 2D Fourier Transform**Fourier Transform: Inverse Fourier transform:**f(x)**F(u) x fmax u -fmax Fs(u) fs(x) x fmax u -fmax … … -2T -T 0 T 2T Review: Nyquist Sampling Rate Fourier transform Inverse Fourier transform Sampling Rate ≥ 2 * max frequency in the signal!**fmax**u -fmax Review: Antialiasing Increase the sampling rate to above twice the highest frequency Introduce an anti-aliasing filter to reduce fmax Fs(u) fmax u -fmax**fmax**u -fmax Review: Antialiasing Increase the sampling rate to above twice the highest frequency Introduce an anti-aliasing filter to reduce fmax • - Sinc filter • Mean filter • - Triangular filter • - Gaussian filter Fs(u) fmax u -fmax**Review: Image Pyramids**Known as a Gaussian Pyramid • MipMap (Williams, 1983)**Outline**Image Processing - Median filtering - Bilateral filtering - Edge detection - Corner detection**Filtering**input Gaussian filter**Median Filter**• For each neighbor in image, sliding the window • Sort pixel values • Set the center pixel to the median**Median Filter**input Gaussian filter Median filter**Median Filter Examples**input Median 7X7**Median Filter Examples**Median 11X11 Median 3X3**Median Filter Examples**Straight edges kept Median 11X11 Median 3X3 Sharp features lost**Median Filter Properties**Can remove outliers (peppers and salts) Window size controls size of structure Preserve some details but sharp corners and edges might get lost**Comparison of Mean, Gaussian, and Median**original Mean with 6 pixels**Comparison of Mean, Gaussian, and Median**original Gaussian with 6 pixels**Comparison of Mean, Gaussian, and Median**original Median with 6 pixels**Common Problems**Mean: blurs image, removes simple noise, no details are preserved**Common Problems**Mean: blurs image, removes simple noise, no details are preserved Gaussian: blurs image, preserves details only for small σ.**Common Problems**Mean: blurs image, removes simple noise, no details are preserved Gaussian: blurs image, preserves details only for small σ. Median: preserves some details, good at removing strong noise**Common Problems**Mean: blurs image, removes simple noise, no details are preserved Gaussian: blurs image, preserves details only for small σ. Median: preserves some details, good at removing strong noise Can we find a filter that not only smooths regions but preserves edges?**Common Problems**Mean: blurs image, removes simple noise, no details are preserved Gaussian: blurs image, preserves details only for small σ. Median: preserves some details, good at removing strong noise Can we find a filter that not only smooths regions but preserves edges? - yes, bilateral filter**Outline**Image Processing - Median filtering - Bilateral filtering - Edge detection - Corner detection**What Is Bilateral Filter?**Bilateral - Affecting or undertaken by two sides equally Property: - Convolution filter - Smooth image but preserve edges - Operates in the domain and the range of image**Bilateral Filter Example**Gaussian filter Bilateral filter**Bilateral Filter Example**Gaussian filter Bilateral filter**1D Graphical Example**Center Sample u I(p) Neighborhood p It is clear that in weighting this neighborhood, we would like to preserve the step**The Weights**I(p) p**Filtered Values**I(p) Filtered value p**Edges Are Smoothed**I(p) Filtered value p**What Causes the Problem?**I(p) Filtered value p**What Causes the Problem?**Same weights for these two pixels!! I(p) Filtered value p**The Weights**I(p) p**Bilateral Filtering**Denoise Feature preserving Bilateral filter Normalization**Kernel Properties**• Per each sample, we can define a ‘Kernel’ that averages its neighborhood • This kernel changes from sample to sample! • The sum of the kernel entries is 1 due to the normalization, • The center entry in the kernel is the largest, • Subject to the above, the kernel can take any form (as opposed to filters which are monotonically decreasing).**Filter Parameters**As proposed by Tomasi and Manduchi, the filter is controlled by 3 parameters: The filter can be applied for several iterations in order to further strengthen its edge-preserving smoothing N(u) – The neighbor size of the filter support, c – The variance of the spatial distances, s – The variance of the value distances,**Bilateral Filter**input**Bilateral Filter**input**Bilateral Filter**input Wc**Bilateral Filter**input Wc Ws**Bilateral Filter**input Ws*Wc Wc Ws**Bilateral Filter**input Output Ws*Wc Wc Ws**Bilateral Filter Results**Original**Bilateral Filter Results**σc = 3, σs = 3**Bilateral Filter Results**σc = 6, σs = 3**Bilateral Filter Results**σc = 12, σs = 3**Bilateral Filter Results**σc = 12, σs = 6**Bilateral Filter Results**σc = 15, σs = 8

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