Histogram Matching

forces the intensity distribution of an image to match the intensity distribution of a target [12]. It is a generalisation of histogram equalisation. The latter transforms an input image into an output image with equally many pixels at every grey level (i.e. a flat histogram) and is solved using the following point operation [12],

$${g}^{\ast} = Q\mathcal{P}[ {g} ],$$ (5.3.1)

where ${g}$ is the input image, ${g}^{\ast}$ is the image with a flat histogram, and $\mathcal{P}$ is the cumulative distribution function. Let the histogram of an image ${g}$ be denoted by $\mathbf{H}_{{g}}(q)$ as a function of grey level $q \in \mathbf{Q}$. The cumulative distribution function $\mathcal{P}$ of the image ${g}$ is as follows,

$$\mathcal{P}[ {g} ] = \frac{1}{\vert{g}\vert}\int_{0}^{Q}\mathbf{H}_{{g}}(q)dq$$ (5.3.2)

Given Eq. (5.3.1), the problem of matching the histogram of an image ${g}$ with the desired histogram of the image ${g}^{\circ}$ is solved as follows [12],

$${g}^{\ast}=Q \mathcal{P}[ {g} ] = Q\mathcal{P}[ g^{\circ} ] \Rightarrow$$

$$\mathcal{P}[{g}]=\mathcal{P}[ g^{\circ}] \Rightarrow$$

$$g^{\circ}=\mathcal{P}^{-1}\left[ \mathcal{P}[{g} ]\right]$$ (5.3.3)

In Eq. (5.3.3), the histogram matching involves two concatenated point operations, where $\mathcal{P}^{-1}$ is the inverse function of $\mathcal{P}$. Practically the cumulative distribution function and its inverse function are discrete, which could be implemented using lookup tables [46].