Simple enough to add a function to set a scope variable in your controller that you can call from ng-click. Then set your filter to equal the value in the scope variable. Like this:
Add to controller
$scope.myFilter = 'no' // or whatever you want to default to;
$scope.setMyFilter = function(){
$scope.myFilter = $scope.myFilter === 'no' ? 'yes' : 'no';
};
HTML
<div ng-app="MyApp">
<div class="container" ng-controller="sentCtrl">
<div class="row">
<div><strong>Only show:</strong>
<div class="btn-group btn-group-xs">
<button class="btn btn-primary" ng-click="setMyFilter()">Images</button>
</div>
</div>
</div>
<div class="row content-sc">
<div class="item-sc" data-ng-repeat="d in things | filter: {largeImage.hasImage: myFilter}">
<h4>{{d.NAME}}<br/><small>ID: {{d.ID}}</small></h4>
</div>
</div>
</div>