Found the solution myself. It appears you need to inject the editorService service.

The media picker can be triggered with the following function:

$scope.showMediaPicker = function () {
    var mediaPickerOptions = {
        multiPicker: false,
        show: false,
        submit: function (model) {
            var data = model.selection[0];
            setImage(data)
            editorService.close();
        },
        close: function () {
            editorService.close();
        }
    }

    editorService.mediaPicker(mediaPickerOptions);
}

As well as handling the media picker submit to set the image:

function setImage(image) {
    if (!image) {
        $scope.control.value.imageId = 0;
        $scope.image = null;
        $scope.imageUrl = null;
        return;
    }
    $scope.control.value.imageId = image.id;
    $scope.image = image;
    $scope.imageUrl = (image.image ? image.image : mediaHelper.resolveFileFromEntity(image)) + '?width=' + 500 + '&mode=crop';
}