## 5. Image Enhancement by Histogram Manipulation

An image histogram is a representation of the frequency distribution of values in an image. It plots the number of pixels there are containing particular tonal values. Photographers oftentimes use histogram manipulation in order to restore the proper contrast of the images or simply depict a particular mood on the scenery.

If *Px(i)* is the image histogram for pixel value *i*, the *cumulative distribution function* (or CDF) for all pixel values is expressed as

Images are easily modified and best manipulated using cumulative frequencies. **Backprojection** is referred to as the method of manipulating an image’s CDF to conform a desired distribution. This is performed by tracing back the frequency of a particular pixel tone towards the desired CDF and then replace it with its new corresponding value.

For instance we have the image below as an example of a poor-contrast image, with its image histogram and CDF shown on the attached plots.

Figure 2. An example of a dark photograph (in grayscale). Courtesy of M. Beslac. |

Using Scilab and the SIVP toolbox, we can improve the image by implementing histogram manipulation through the backprojection algorithm. First, we use the built-in function imhist() to obtain the image histogram from the grayscale image. We used plot2d3() to allow us to plot our data in vertical bars.

I = imread('fakefree.jpg'); J = rgb2gray(I); [counts, cells] = imhist(J); // defaults to 256 bins scf(0); plot2d3(cells, counts, rect=[0,0, 256, max(counts)]); // rect=[xmin, ymin, xmax, ymax]

We then utilize the function cumsum(), which returns an array corresponding to the cumulative distribution. Note that in our implementation we perform a normalization for the CDF.

cumulative_sum = cumsum(counts); cumsum_max = max(cumulative_sum); cdf = cumulative_sum/cumsum_max; // normalized cumsum scf(1); plot2d(cells, cdf, rect=[0, 0, 256, max(cdf)]);

For better coding practices, we create a self-defined function to solve for the x-values of the traced frequencies on our CDF plot. If for instance we desire to imitate a linear function obeying the equation y = (1/255)*x, we design our function to do an inverse of this relation. (This code snippet of the function may be placed at the beginning of the code or in a separate file.)

function inv_linearCDF = invLinearCDF(y) y = y*255; // denormalization inv_linearCDF = round(y); // linear_cdf = (1/255)*x endfunction

Note that we performed denormalization on the CDF in order to work with actual values, and used the function round() for discrete approximations. Next, we create a second image matrix – a copy of the original – where we will later apply the changes due to our manipulation. Iterating towards each pixel value, we retrace the individual frequencies and replace them with new pixel values on the second image.

K = J; // a copy of the original image for i = 1:256 new_x = invLinearCDF(cdf(i)); K (find(J==i-1)) = new_x; // i-1 referring to pixel values (range:0-255) end

Below is the new image now approximately conforming to the linear distribution.

Figure 3. Histogram manipulated image via backprojection towards a linear CDF. |

We can perform the same method with non-linear distributions such as the parabolic function, implemented similar to the one below.

function inv_parabolicCDF = invParabolicCDF(y) y = y*255^2; // denormalization inv_parabolicCDF = round(sqrt(y)); // parabolic_cdf(x) = x^2 endfunction

We may also apply this with a logarithmic function:

function inv_logarithmicCDF = invLogarithmicCDF(y) y = y*log(255); // denormalization inv_logarithmicCDF = round(exp(y)); // logarithmic_cdf(x) = log(x) endfunction

The following are the image results of the previous two non-linear functions, including their manipulated histograms and cumulative frequencies. We can observe how the images are similar to an overexposed and underexposed image.

Figure 4. Image backprojected towards (1) a parabolic CDF; (2) a logarithmic CDF. |

Photographers usually use image processing softwares to easily perform histogram manipulation on photographs.** GIMP** or the GNU Image Manipulation Program (a FOSS graphics editor), offers this capability via graphical interface. To navigate to this function, go to *Colors* > *Curves*. Make sure the image is initial set to grayscale by choosing *Image* > *Mode* > *Grayscale*. Here is a snapshot of an actual curve manipulation on GIMP.

Now, being able to adjust images via histogram manipulation, we can achieve improved images with better brightness and color contrast.

For this segment, I would rate myself 11 for understanding the important ideas and, in addition, for exploring a software that applies the image enhancement method.

———————————————————————————————–—————–————————————

References:

[1] Esporlas, C., 2009. Histogram backprojection. Applied Physics 186.

[2] Photoxels, 2010. Digital photography tutorial: Histogram.

[3] Soriano, M., 2010. Image enhancement by histogram manipulation. Applied Physics 186.

[4] Wikipedia, 2010. Histogram equalization.

[5] Wikipedia, 2010. Image histogram.

Hi,

Do you have the source doe for this activity? If you do, can you send it to me?

Thank you. You have a very interesting blog.

Hi Arvin, I had the code sent to your email just now. Enjoy. 🙂

Hi

It seems to late but just I leave the comments

Would you please send me source code?