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Image-sharpening spatial filters

Dr. Huidae Cho
Institute for Environmental and Spatial Analysis
University of North Georgia

1   Sharpening

Smoothing/blurring is a process of “convolution” or “integration.”

Naturally, sharpening is a process of “derivation.”

2   Requirements of derivatives

First-order derivatives

  1. 0 in flat segments
  2. nonzero at the onset or end of a gray-level change
  3. nonzero along ramps

Second-order derivatives

  1. 0 in flat segments $\because \frac{\partial f}{\partial x}=0$
  2. nonzero at the onset or end of a gray-level change $\because \frac{\partial f}{\partial x}$ changes
  3. 0 along ramps of a constant slope $\because \frac{\partial f}{\partial x}$ is constant

$\partial x=1$

3   Derivatives for digital images

\[\frac{\partial f}{\partial x}=f(x+1)-f(x)\]

\[\begin{split} \frac{\partial^2 f}{\partial x^2}&=\frac{\partial}{\partial x}f(x)-\frac{\partial}{\partial x}f(x-1)\\ &=f(x+1)+f(x-1)-2f(x) \end{split}\]

4   Gray-level profile

gray-level-profile

5   Responses

First derivatives

  • Generally thicker edges
  • Stronger response to steps

Second derivatives

  • Stronger response to fine details
  • Double responses to steps

6   Isotropy

We are interested in “isotropic” filters, which are rotation-invariant.

That is, Rotate(Filter(Image)) = Filter(Rotate(Image)).

7   Simpler notations for a subimage

simpler-notations

8   First derivatives (gradient)

The gradient $\mathbf{\nabla f}$ is a vector. \[\mathbf{\nabla f}= \begin{bmatrix} G_x\\ G_y \end{bmatrix} =\begin{bmatrix} \frac{\partial f}{\partial x}\\ \frac{\partial f}{\partial y} \end{bmatrix}\]

We use the magnitude of the gradient in digital image processing. \[\begin{split} \nabla f=\left|\mathbf{\nabla f}\right| &=\left[G_x^2+G_y^2\right]^{1/2}\\ &=\left[\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2\right]^{1/2} \end{split}\]

However, its computation for individual pixels is computationally expensive, so it is approximated as: \[\nabla f\approx|G_x|+|G_y|\]

8.1   Simplest approximation

\[G_x=z_8-z_5\] \[G_y=z_6-z_5\]

8.2   Roberts cross-gradient operators

\[G_x=z_9-z_5\] \[G_y=z_8-z_6\]

8.3   Sobel operators

An extension of Roberts cross-gradient operators

\[\begin{split} G_x&=(z_7-z_1)+2(z_8-z_2)+(z_9-z_3)\\ &=(z_7+2z_8+z_9)-(z_1+2z_2+z_3) \end{split}\]

\[\begin{split} G_y&=(z_3-z_1)+2(z_6-z_4)+(z_9-z_7)\\ &=(z_3+2z_6+z_9)-(z_1+2z_4+z_7) \end{split}\]

\[\nabla f\approx|G_x|+|G_y|\]

Multiplier $2$ for giving more importance to the current pixel

8.4   Filters for Sobel operators

sobel-x

sobel-y

9   Edge extraction exercise

  1. First derivative using the Sobel filter
    • Input: map.tif
    • Output: first_deriv.tif
  2. Gray-level slice
    • Input: first_deriv.tif
    • Lower gray level: 200
    • Upper gray level: 1334
    • New gray level: 255
    • Binary: Checked
    • Output: edges.tif
  3. Raster calculator
    • Int("edges.tif")
    • Output: int_edges.tif
  4. Thin
    • Input: int_edges.tif
    • Output: thin_edges.tif
    • Shape for corners: Sharp
    • Maximum thickness: 30 ($10\times\text{cell size}$)
  5. Raster to polyline
    • Input: thin_edges.tif
    • Output: edges.shp

10   Second derivatives (Laplacian)

\[\nabla^2 f=\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}\]

\[\frac{\partial^2 f}{\partial x^2}=z_8+z_2-2z_5\]

\[\frac{\partial^2 f}{\partial y^2}=z_6+z_4-2z_5\]

\[\nabla^2 f=z_2+z_4+z_6+z_8-4z_5\]

10.1   Filters for the Laplacian

laplacian-4

laplacian-8

10.2   Sharpening

\[g(x,y)=f(x,y)-\nabla^2 f(x,y)\]

11   Sharpening exercise 1

  1. Second derivative using the Laplacian filter
    • Input: map.tif
    • Output: second_deriv.tif
  2. Subtract
    • Input 1: map.tif
    • Input 2: second_deriv.tif
    • Output: sharpened_manual.tif

What happened?

Why?

12   Sharpening exercise 2

  1. Rescale gray levels
    • Input: second_deriv.tif
    • Lower gray level: -2040
    • Upper gray level: 2040
    • Output: rescaled_second_deriv.tif
  2. Subtract
    • Input 1: map.tif
    • Input 2: rescaled_second_deriv.tif
    • Output: sharpened_better_manual.tif

13   Sharpening exercise 3

  1. Raster calculator
    • "map.tif" - "second_deriv.tif" / 16
    • Output: sharpened_better_manual_2.tif

14   Unsharp masking

\[f_\text{s}(x,y)=f(x,y)-\bar{f}(x,y)\] where $\bar{f}(x,y)$ is a blurred $f(x,y)$

15   High-boost filtering

A generalized unsharp masking \[\begin{split} f_\text{hb}(x,y)&=A\cdot f(x,y)-\bar{f}(x,y)\\ &=(A-1)f(x,y)+f(x,y)-\bar{f}(x,y)\\ &=(A-1)f(x,y)+f_\text{s}(x,y) \end{split}\]

If we use the Laplacian sharpening for $f_\text{s}(x,y)$,

\[\begin{split} f_\text{hb}(x,y)&=(A-1)f(x,y)+f_\text{s}(x,y)\\ &=(A-1)f(x,y)+f(x,y)-\nabla^2 f(x,y)\\ &=A\cdot f(x,y)-\nabla^2 f(x,y). \end{split}\]

When $A=1$, $f_\text{hb}(x,y)$ is the Laplacian sharpening.

16   Homework 6

Develop a 3-by-3 filter that can sharpen the input image in one pass. The output image need not be gray-level rescaled. In other words, the range of its pixel value doesn’t have to be between 0 and 255. Fill this blank filter with your nine coefficients.

homework-6

Hints

  • This one-pass filter is a combination of the diagonal Laplacian filter and sharpening exercise 3.
  • Think about the identity and diagonal Laplacian filters.
  • You can only run the Convolute tool once to sharpen the image. Again, don’t worry about its output range.
  • Basically, you need to create a parameter value for “Filter mask values separated by a space” in the Convolute tool. That is nine numbers separated by a space.
  • Coefficients are all real numbers. Use up to four significant figures.