updates

_pages/animations.md

@@ -2,10 +2,10 @@
 layout: page
 title: animations
 permalink: /animations/
-description: acoustics animations
+description:  
 nav: true
 nav_order: 3
-display_categories: [animations]
+display_categories: [acoustics animations, signal processing animations]
 horizontal: false
 ---
 

_projects/5_geometric_acoustics_calibration copy.md

@@ -10,8 +10,7 @@ category: work
 
 Geometric acoustics simulations are used across areas of architectural acoustics for improved sound design. Nonetheless, an important aspect of creating a numerical model is selecting material properties so that measured data agrees with the simulation input. One area of my research is to generate numerical methods to calibrate geometric acoustics models. 
 
-
-
+Geometric acoustics (GA) models use principles of specular and diffuse reflections to predict the temporal energetic response of a room. Methods can include ray tracing, beam or cone tracing, and image-source methods to determine ray paths between a source and receiver. In modeling sound propagation, random incidence absorption coefficients determine the amount of energy lost at each reflection while random incidence scattering coefficients influence the trajectory at each reflection. The cumulation of many rays and leads to the echogram, an energetic description of the energy decay in the room. 
 
 
 <div class="row">
@@ -24,6 +23,25 @@ Geometric acoustics simulations are used across areas of architectural acoustics
     </div>
 </div>
 <div class="caption">
-    Simple geometrical acoustics model. 
+    Simple geometrical acoustics model. The red line indicates a single ray path from the source to the receiver. 
 </div>
 
+
+Many room acoustic metrics, including reverberation time and clarity, may be calculated from the echogram, which may be written as follows:
+
+\begin{equation}
+E(t) = \sum_{m = 1}^M W_m\left(\prod_{n = 1}^N (1 - \alpha_n)^{p_{nm}}\right)\delta(t - \tau_m)
+\end{equation}
+with $$M$$ being the number of rays, $$W_m$$ being the amplitude of the $$m$$th ray, $$N$$ being the number of surfaces $$\alpha_n$$ being the absorption coefficient for the $$n$$th surface, and $$\tau_m$$ being the impact time of the $$m$$th ray. The integer $$p_{nm}$$ represents the number of times the $$m$$th ray hit the $$n$$th surface. 
+
+Because the echogram is a function of the absorption coefficients, one can calculate its gradient as
+\begin{equation}
+\frac{\partial E}{\partial \alpha_n}  = -\sum_{m = 1}^M \frac{p_{nm}}{(1 - \alpha_n)} W_m\left(\prod_{n = 1}^N (1 - \alpha_n)^{p_{nm}}\right)\delta(t - \tau_m).
+\end{equation}
+Using this expression, one may use a gradient-descent algorithm to iteratively adjust the absorption coefficients as
+\begin{equation}
+\alpha_n^{(k+1)} = \alpha_n^{(k)} - \mu (\tilde{C}^{(k)} - C)\frac{\partial}{\partial \alpha_n}\tilde{C}^{(k)}.
+\end{equation}
+where $$C$$ is some meaasured room acoustics metric, $$\tilde{C}$$ is its simulated value, and $$k$$ is the iteration step. 
+
+

_projects/directivity_animations.md

@@ -4,7 +4,7 @@ title: source directivity
 description: 
 img: assets/img/animations/point_source_directivity.gif
 importance: 4
-category: animations
+category: acoustics animations
 ---
 
 

_projects/geometric_acoustics_animations.md

@@ -0,0 +1,20 @@
+---
+layout: page
+title: geometric acoustics
+description: 
+img: assets/img/animations/specular_reflection.gif
+importance: 6
+category: acoustics animations
+---
+
+<div class="row">
+    <div class= "col-sm">
+        {% include figure.html path="assets/img/animations/specular_reflection.gif" class="img-fluid rounded z-depth-1" %}
+    </div>
+    <div class = "col-sm">
+        {% include figure.html path="assets/img/animations/diffuse_reflection.gif" class="img-fluid rounded z-depth-1" %}  
+    </div>
+</div>
+<div class="caption">
+    Specular and diffuse reflections used in geometric acoustics
+</div>

_projects/image_method_animations.md

@@ -4,7 +4,7 @@ title: image source methods
 description: 
 img: assets/img/animations/image_source_intro.gif
 importance: 4
-category: animations
+category: acoustics animations
 ---
 
 

_projects/optimization_animations.md

@@ -0,0 +1,46 @@
+---
+layout: page
+title: optimization methods
+description: 
+img: assets/img/animations/GradientDescent.gif
+importance: 6
+category: signal processing animations
+---
+
+Optimization algorithms try to minimize a function.
+
+In gradient descent, the iterative update is related to the local gradient of the objective function
+\begin{equation}
+\mathbf{x}^{(k+1)} = \mathbf{x}^{(k)} - \mu \nabla J(\mathbf{x}^{(k)}).
+\end{equation}
+
+<div class="row">
+    <div class = "col-sm">
+    </div>
+    <div class="col-sm">
+        {% include figure.html path="assets/img/animations/GradientDescent.gif" title="example image" class="img-fluid rounded z-depth-1" %}
+    </div>
+        <div class = "col-sm">
+    </div>
+</div>
+<div class="caption">
+    Gradient descent algorithm 
+</div>
+
+Newton's method uses the Hessian matrix to compute the local curvature. This leads to a more direct descent path. However, computing the Hessian matrix may be difficult for some problems.  
+\begin{equation}
+\mathbf{x}^{(k+1)} = \mathbf{x}^{(k)} - \mathbf{H}^{-1}(\mathbf{x}^{(k)})\nabla J(\mathbf{x}^{(k)}).
+\end{equation}
+
+<div class="row">
+    <div class = "col-sm">
+    </div>
+    <div class="col-sm">
+        {% include figure.html path="assets/img/animations/NewtonsMethod.gif" title="example image" class="img-fluid rounded z-depth-1" %}
+    </div>
+        <div class = "col-sm">
+    </div>
+</div>
+<div class="caption">
+    Newton's method
+</div>

_projects/plate_animations.md

@@ -4,7 +4,7 @@ title: membrane  vibrations
 description: 
 img: assets/img/animations/circular_plate_22.gif
 importance: 4
-category: animations
+category: acoustics animations
 ---
 
 

_projects/polyhedra_animations.md

@@ -4,7 +4,7 @@ title: rpl directivity patterns
 description: 
 img: assets/img/animations/dodec_directivity.gif
 importance: 4
-category: animations
+category: acoustics animations
 ---
 
 Regular polyhedral loudspeakers are special loudspeakers with several drivers 

_projects/rod_animations.md

@@ -4,7 +4,7 @@ title: rod bending modes
 description: 
 img: assets/img/animations/rod_demo.gif
 importance: 4
-category: animations
+category: acoustics animations
 ---
 
 A "rod" is an elastic solid whose cross-sectional area is very small relative to its length. A vibrational model constrained to one dimension serves as a good approximation to its motion. 

_projects/sphere_animations.md

@@ -4,7 +4,7 @@ title: radiation from spheres
 description: 
 img: assets/img/animations/point_source.gif
 importance: 4
-category: animations
+category: acoustics animations
 ---