**Introduction**

Haar Wavelet Transforms (HWTs)
are used to detect significant changes in signal values. In two previous posts (here and here), I formalizied 1D Haar Wavelet up-down and down-up spikes that can be used to detect changes in signals. Specifically, four types of spikes were proposed: up-down
triangle, up-down trapezoid, down-up triangle, and down-up trapezoid. In this post, I will formally outline a method for using 1D Haar Wavelet Spikes to count changes in images.

**Spike Pixels**

Let us assume that we will be detecting changes in image rows. Image columns will work just as well. A spike pixel then is a pixel covered by an up-down or down-up spike detected in some row. Formally, when spikes are
computed for row
,
the column indices of the actual pixels covered by each spike at
scale

*j*are computed by the formula in**Eq**.**1**, where*s*and*e*are the positions of the starting and ending wavelet coefficients in the 1D HWT at scale*j*, respectively. For up-down spikes and , whereas, for down-up spikes, andEq. 1. Column indices of pixels covered by spikes. |

Let

*n*be the number of rows in an image and let be the set of up-down spikes in row Then the set of pixel columns in row covered by the up-down spikes in is given in**Eq**.**2**, where*j*is a given scale and and are the beginning and end positions of an up-down spikedetected in row respectively.Eq. 2. Set of pixel columns in row covered by up-down spikes. |

Letis
the set of down-up spikes in row
Then the set of pixel columns in
covered by the down-up spikes is

**Eq**.**3**, where*j*is a given scale and and are the beginning and end positions of a down-up spike detected in row respectively.Eq. 3. Set of pixel columns in row covered by down-up spikes. |

The number of unique column pixels covered by the up-down and down-up
spikes in row
is given in

**Eq**.**4**. The formula in**Eq**.**5**gives the actual number of pixels covered by the up-down and down-up spikes in an image with*n*rows.Eq. 4. Set of unique pixel columns in row covered by down-up spikes. |

Eq. 5. Number of unique pixel columns covered by up-down and down-up spikes in image
with
n rows. |