improc ass3: Added backprojection section to report.

improc/ass3/back_projection.py

@@ -59,17 +59,21 @@ def hbp(image, environment, bins, model, radius, mask=None):
 
     for c in domainIterator(R, 3):
         if I[c] != 0:
-            R[c] = float(M[c]) / float(I[c])
+            R[c] = float(M[c]) / I[c]
 
-    # Create back projection
-    print 'Creating back projection...'
-    b = zeros(environment.shape[:2])
-    use = environment.astype(float) * map(lambda x: x - 1, bins)
+    # Apply color model
+    use = environment.astype(float)
 
     if model == 'hsv':
         for p in domainIterator(image):
             use[p] = rgb_to_hsv(*use[p].tolist())
 
+    use *= map(lambda x: x - 1, bins)
+
+    # Create back projection
+    print 'Creating back projection...'
+    b = zeros(environment.shape[:2])
+
     for p in domainIterator(b):
         b[p] = min(R[col2bin(use[p])], 1.)
 
@@ -98,7 +102,7 @@ if __name__ == '__main__':
 
     # Execute the back projection algorithm
     #import pickle
-    b = hbp(waldo, env, [64] * 3, 'rgb', 15, mask)
+    b = hbp(waldo, env, [32] * 3, 'rgb', 10, mask)
     #pickle.dump(b, open('projection.dat', 'w'))
     #b = pickle.load(open('projection.dat', 'r'))
 
@@ -116,7 +120,7 @@ if __name__ == '__main__':
     # Experimental: use K-means to locate clusters, and draw a rectangle
     # with the size of the model image around each cluster
     print 'Locating peaks...'
-    peaks = find_peaks(b, .75, sqrt(w ** 2 + h ** 2))
+    peaks = find_peaks(b, .25, sqrt(w ** 2 + h ** 2))
     print 'Done'
 
     subplot(121)

improc/ass3/report/report.tex

@@ -40,4 +40,86 @@ model, since it not possible to generate ... We chose the HSV color model.
 
 % Does your model improve the results compared with the RGB model?
 
+\section{Histogram Backprojection}
+
+\subsection{Finding Waldo}
+
+The assignment is to find Waldo (\emph{waldo.png}) in a large image containing
+Waldo and many other characters (\emph{waldo\_env.png}) using Histogram Backprojection.
+
+The idea of Histogram Backprojection is explained in the paper by Swain and Ballard,
+so we will not explain it here. The algorithm as described in the paper is as follows:
+
+Given histograms $M$ (model, Waldo) and $I$ (environment), create the back projection
+$b$ in the following steps:
+
+\begin{enumerate}
+	\item for each histogram bin $j$ do $R_j := frac{M_j}{I_j}$
+	\item for each $x, y$ do $b_{x,y} := min(R_{h(c_{x,y})}, 1)$
+	\item $b := D^r * b$
+	\item $(x_t, y_t) := loc(max_{x,y}, b_{x,y})$
+\end{enumerate}
+
+However, the assignment tells us to only implement steps 1-3.
+
+\subsection{Mask}
+
+The algorithm is implemented in \emph{back\_projection.py}. First, a mask is created
+to ignore the white background in the \emph{waldo.png}. This is needed because the
+white color is not part of Waldo himself,. In fact, Waldo in the \emph{waldo\_env.png}
+has a yellowish background behind him. The usage of a mask is simple: if the mask value
+of a pixel is \texttt{False}, the pixel's color is discarded in the creation of the
+color histogram. The mask for Waldo is created by discarding all pixels with RGB
+color (255, 255, 255), which has the following result:
+
+\begin{figure}[h]
+	\label{fig:mask}
+	\includegraphics{mask.png}
+	\caption{The mask used to ignore the white background in \emph{waldo.png}.}
+\end{figure}
+
+\subsection{Basic algorithm}
+
+The algorithm is implemented in the function \texttt{hbp}:
+First, $M$ and $I$ are created using \texttt{colHist}. $R$ is then created by
+division (step 1). Afterwards, the back projection is created using the loop in step 2.
+Finally, a convolution function is used to soften the back projection. To create the
+convolution, we have used the function \texttt{scipy.ndimage.correlate} with a circular
+weight mask.
+
+The following result is generated with 64 bins in each color dimension and a convolution
+radius of 15 pixels:
+
+\begin{figure}[h]
+	\hspace{-4cm}
+	\includegraphics[width=20cm]{found_waldo.png}
+	\caption{\emph{found\_waldo.png}: Back projection of Waldo in the larger image, the
+	         red spot is Waldo's location.}
+\end{figure}
+
+This result is created in roughly 27 seconds on a laptop from 2009.
+
+\subsection{Experimental cluster detection using K-means}
+
+In addition to the assignment, we thought it would be interesting to be able to draw a
+rectangle around possible occurences of Waldo in the image. Therefore, we created the
+function \texttt{find\_peaks}, which returns a number of possible locations averages.
+To do this, we eliminate all pixels in the back projection which are below a certain
+threshold (which is given as e number from 0 to 1, and then multiplied by the maximum
+value in the projection). This leaves us with a number of pixels clusters, of which we
+can calculate the centers using K-means
+algorithm\footnote{\url{http://en.wikipedia.org/wiki/K-means\_clustering}} (function
+\texttt{scipy.cluster.vq.kmeans}). To find the initial estimators, we find all pixels
+that are further away from all other estimators than the diagonal of the model image
+(\emph{waldo.png}). Around each center pixel, a rectangle with the size of the model
+image is drawn over the larger image. With 32 bins in each color dimension, a threshold
+of 0.25 and a convolution radius of 10 pixels, the result is as follows:
+
+\begin{figure}[h]
+	\hspace{-4cm}
+	\includegraphics[width=20cm]{k-means.png}
+	\caption{\emph{k-means.png}: Multiple possible locations using a low threshold
+	         and convolution radius.}
+\end{figure}
+
 \end{document}
\ No newline at end of file